# Beam combining scheme for high-power broad-area semiconductor lasers with Lyot-filtered reinjection: Modeling, simulations, and experiments

*Authors*

- Brée, Carsten
- Raab, Volker
- Montiel-Ponsoda, Joan
- Garre-Werner, Guillermo
- Staliunas, Kestutis
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Radziunas, Mindaugas

*2010 Mathematics Subject Classification*

- 65Z05 78A60 78A05 78A55 78-04

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.15.Eq 42.79.Ci 42.79.Fm 42.25.Lc 02.60.Cb

*Keywords*

- Broad area lasers, high power modeling, traveling wave, optical feedback, external cavity, Lyot filter, polarization, beam combining, coupling efficiency

*DOI*

*Abstract*

A brightness- and power-scalable polarization beam combining scheme for high-power, broadarea semiconductor laser diodes is investigated numerically and experimentally. To achieve the beam combining, we employ Lyot-filtered optical reinjection from an external cavity, which forces lasing of the individual diodes on interleaved frequency combs with overlapping envelopes and enables a high optical coupling efficiency. Unlike conventional spectral beam combining schemes with diffraction gratings, the optical coupling efficiency is insensitive to thermal drifts of laser wavelengths. This scheme can be used for efficient coupling of a large number of laser diodes and paves the way towards using broad-area laser diode arrays for cost-efficient material processing, which requires high-brilliance emission and optical powers in the kW-regime.

*Appeared in*

- J. Opt. Soc. Amer. B Opt. Phys., 36 (2019), pp. 1721--1730, DOI 10.1364/JOSAB.36.001721 .

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# A continuum model for yttria-stabilised zirconia incorporating triple phase boundary, lattice structure and immobile oxide ions

*Authors*

- Vágner, Petr
- Guhlke, Clemens
- Miloš, Vojtěch
- Müller, Rüdiger

ORCID: 0000-0003-2643-722X - Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434

*2010 Mathematics Subject Classification*

- 65N08 78A57 80A17

*Keywords*

- solid oxide, double layer, interface, thermodynamics, finite volume method

*DOI*

*Abstract*

A continuum model for yttria-stabilised zirconia (YSZ) in the framework of non-equilibrium thermodynamics is developed. Particular attention is given to i) modeling of the YSZ-metal-gas triple phase boundary, ii) incorporation of the lattice structure and immobile oxide ions within the free energy model and iii) surface reactions. A finite volume discretization method based on modified Scharfetter-Gummel fluxes is derived in order to perform numerical simulations.

The model is used to study the impact of yttria and immobile oxide ions on the structure of the charged boundary layer and the double layer capacitance. Cyclic voltammograms of an air-half cell are simulated to study the effect of parameter variations on surface reactions, adsorption and anion diffusion.

*Appeared in*

- J. Solid State Electrochem., 23 (2019), pp. 2907--2926.

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# Optimal control of a phase field system modelling tumor growth with chemotaxis and singular potentials

*Authors*

- Colli, Pierluigi
- Signori, Andrea
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K55 35Q92 49J20 92C50

*Keywords*

- Distributed optimal control, tumor growth, cancer treatment, phase field system, evolution equations, chemotaxis, adjoint system, necessary optimality conditions

*DOI*

*Abstract*

A distributed optimal control problem for an extended model of phase field type for tumor growth is addressed. In this model, the chemotaxis effects are also taken into account. The control is realized by two control variables that design the dispensation of some drugs to the patient. The cost functional is of tracking type, whereas the potential setting has been kept quite general in order to allow regular and singular potentials to be considered. In this direction, some relaxation terms have been introduced in the system. We show the well-posedness of the state system, the Fréchet differentiability of the control-to-state operator in a suitable functional analytic framework, and, lastly, we characterize the first-order necessary conditions of optimality in terms of a variational inequality involving the adjoint variables.

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# A large-deviations approach to gelation

*Authors*

- Andreis, Luisa
- König, Wolfgang

ORCID: 0000-0002-4212-0065 - Patterson, Robert I. A.

ORCID: 0000-0002-3583-2857

*2010 Mathematics Subject Classification*

- 05C80 60F10 60K35 82B26

*Keywords*

- Coagulation process, multiplicative coalescent, gelation, phase transition, large deviations, Erdős-Rényi random graph

*DOI*

*Abstract*

A large-deviations principle (LDP) is derived for the state, at fixed time, of the multiplicative coalescent in the large particle number limit. The rate function is explicit and describes each of the three parts of the state: microscopic, mesoscopic and macroscopic. In particular, it clearly captures the well known gelation phase transition given by the formation of a particle containing a positive fraction of the system mass at time t=1. Via a standard map of the multiplicative coalescent onto a time-dependent version of the Erdős-Rényi random graph, our results can also be rephrased as an LDP for the component sizes in that graph. Our proofs rely on estimates and asymptotics for the probability that smaller Erdős-Rényi graphs are connected.

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# Existence of weak solutions to a dynamic model for smectic-A liquid crystals under undulations

*Authors*

- Emmrich, Etienne
- Lasarzik, Robert

*2010 Mathematics Subject Classification*

- 35Q35 35K52 76A15

*Keywords*

- Liquid crystal, smectic-A, existence, weak solution, Galerkin approximation

*DOI*

*Abstract*

A nonlinear model due to Soddemann et al. [37] and Stewart [38] describing incompressible smectic-A liquid crystals under flow is studied. In comparison to previously considered models, this particular model takes into account possible undulations of the layers away from equilibrium, which has been observed in experiments. The emerging decoupling of the director and the layer normal is incorporated by an additional evolution equation for the director. Global existence of weak solutions to this model is proved via a Galerkin approximation with eigenfunctions of the associated linear differential operators in the three-dimensional case.

*Appeared in*

- IMA J. Appl. Math., (2019), published online on 18.12.2019, DOI 10.1093/imamat/hxz030 .

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# Low-rank tensor reconstruction of concentrated densities with application to Bayesian inversion

*Authors*

- Eigel, Martin
- Gruhlke, Robert
- Marschall, Manuel

ORCID: 0000-0003-0648-1936

*2010 Mathematics Subject Classification*

- 62F15 62G07 35R60 60H35 65C20 65N12 65N22 65J10

*Keywords*

- Tensor train, uncertainty quantification, VMC, low-rank, reduced order model, Bayesian inversion, partial differential equations with random coefficients

*DOI*

*Abstract*

A novel method for the accurate functional approximation of possibly highly concentrated probability densities is developed. It is based on the combination of several modern techniques such as transport maps and nonintrusive reconstructions of low-rank tensor representations. The central idea is to carry out computations for statistical quantities of interest such as moments with a convenient reference measure which is approximated by an numerical transport, leading to a perturbed prior. Subsequently, a coordinate transformation leads to a beneficial setting for the further function approximation. An efficient layer based transport construction is realized by using the Variational Monte Carlo (VMC) method. The convergence analysis covers all terms introduced by the different (deterministic and statistical) approximations in the Hellinger distance and the Kullback-Leibler divergence. Important applications are presented and in particular the context of Bayesian inverse problems is illuminated which is a central motivation for the developed approach. Several numerical examples illustrate the efficacy with densities of different complexity.

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# A gradient-robust well-balanced scheme for the compressible isothermal Stokes problem

*Authors*

- Akbas, Mine
- Gallouët, Thierry
- Gaßmann, Almut
- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Merdon, Christian

*2010 Mathematics Subject Classification*

- 76D07 65N30 65N12

*2008 Physics and Astronomy Classification Scheme*

- 47.10.ad 47.11.Fg

*Keywords*

- compressible Stokes equations, finite element method, finite volume method, well-balanced scheme, upwind, convergence

*DOI*

*Abstract*

A novel notion for constructing a well-balanced scheme --- a gradient-robust scheme --- is introduced and a showcase application for a steady compressible, isothermal Stokes equations is presented. Gradient-robustness means that arbitrary gradient fields in the momentum balance are well-balanced by the discrete pressure gradient --- if there is enough mass in the system to compensate the force. The scheme is asymptotic-preserving in the sense that it degenerates for low Mach numbers to a recent inf-sup stable and pressure-robust discretization for the incompressible Stokes equations. The convergence of the coupled FEM-FVM scheme for the nonlinear, isothermal Stokes equations is proved by compactness arguments. Numerical examples illustrate the numerical analysis, and show that the novel approach can lead to a dramatically increased accuracy in nearly-hydrostatic low Mach number flows. Numerical examples also suggest that a straight-forward extension to barotropic situations with nonlinear equations of state is feasible.

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# Finite element pressure stabilizations for incompressible flow problems

*Authors*

- John, Volker

ORCID: 0000-0002-2711-4409 - Knobloch, Petr
- Wilbrandt, Ulrich

*2010 Mathematics Subject Classification*

- 65N30 65N12 65N15

*Keywords*

- Incompressible flows, Stokes equations, finite element methods, pressure stabilization, error analysis

*DOI*

*Abstract*

Discretizations of incompressible flow problems with pairs of finite element spaces that do not satisfy a discrete inf-sup condition require a so-called pressure stabilization. This paper gives an overview and systematic assessment of stabilized methods, including the respective error analysis.

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# Non-isothermal Scharfetter--Gummel scheme for electro-thermal transport simulation in degenerate semiconductors

*Authors*

- Kantner, Markus
- Koprucki, Thomas

ORCID: 0000-0001-6235-9412

*2010 Mathematics Subject Classification*

- 35K05 35K08 35Q79, 65N08, 80M12, 82B35, 82D37

*Keywords*

- Scharfetter--Gummel scheme, finite volume method, Fermi--Dirac statistics, non-isothermal drift-diffusion system, electro-thermal transport, Seebeck effect, self-heating

*DOI*

*Abstract*

Electro-thermal transport phenomena in semiconductors are described by the non-isothermal drift-diffusion system. The equations take a remarkably simple form when assuming the Kelvin formula for the thermopower. We present a novel, non-isothermal generalization of the Scharfetter? Gummel finite volume discretization for degenerate semiconductors obeying Fermi?Dirac statistics, which preserves numerous structural properties of the continuous model on the discrete level. The approach is demonstrated by 2D simulations of a heterojunction bipolar transistor.

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# Stability of the solution set of quasi-variational inequalities and optimal control

*Authors*

- Alphonse, Amal
- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Rautenberg, Carlos N.

ORCID: 0000-0001-9497-9296

*Keywords*

- Quasi-variational inequality, optimal control, multivalued solution maps

*DOI*

*Abstract*

For a class of quasivariational inequalities (QVIs) of obstacle-type the stability of its solution set and associated optimal control problems are considered. These optimal control problems are non-standard in the sense that they involve an objective with set-valued arguments. The approach to study the solution stability is based on perturbations of minimal and maximal elements to the solution set of the QVI with respect to monotonic perturbations of the forcing term. It is shown that different assumptions are required for studying decreasing and increasing perturbations and that the optimization problem of interest is well-posed.

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# On the Dynamics of Single Mode Lasers with Passive Dispersive Reflector

*Authors*

- Henneberger, Fritz
- Schneider, Klaus R.
- Sieber, Jan
- Tronciu, Vasile Z.
- Wünsche, Hans-Jürgen

*2010 Mathematics Subject Classification*

- 78A60 78-05 78A55

*Keywords*

- Semiconductor laser, DFB lasers, Self-Pulsations

*DOI*

*Abstract*

For passive dispersive reflector (PDR) lasers we investigate a single mode model containing two functions characterizing the influence of the PDR. We study numerically the effect of the shape of these functions on the existence and robustness of self-pulsations. The possibility of tuning the frequency and modulation depth of the self-pulsations has been demonstrated.

*Appeared in*

- Opt. Commun., 182 (2000), pp. 221-228.

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# On a half-space radiation condition

*Authors*

- Rathsfeld, Andreas

*2010 Mathematics Subject Classification*

- 74J20 76B15 78A45

*Keywords*

- Scattering problem, Dirichlet problem, Helmholtz equation, half space and rough surface, radiation condition

*DOI*

*Abstract*

For the Dirichlet problem of the Helmholtz equation over the half space or rough surfaces, a radiation condition is needed to guarantee a unique solution, which is physically meaningful. If the Dirichlet data is a general bounded continuous function, then the well-established Sommerfeld radiation condition, the angular spectrum representation, and the upward propagating radiation condition do not apply or require restrictions on the data, in order to define the involved integrals. In this paper a new condition based on a representation of the second derivative of the solution is proposed. The twice differentiable half-space Green's function is integrable and the corresponding radiation condition applies to general bounded functions. The condition is checked for special functions like plane waves and point source solution. Moreover, the Dirichlet problem for the half plane is discussed. Note that such a ``continuous'' radiation condition is helpful e.g. if finite sections of the rough-surface problem are analyzed.

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# Pressure reconstruction for weak solutions of the two-phase incompressible Navier--Stokes equations with surface tension

*Authors*

- Abels, Helmut
- Daube, Johannes
- Kraus, Christiane

*2010 Mathematics Subject Classification*

- 76T99 35Q30 35Q35, 35R35, 76D05, 76D45

*Keywords*

- Fluid mechanics, Navier--Stokes equations, free boundary problems, surface tension

*DOI*

*Abstract*

For the two-phase incompressible Navier--Stokes equations with surface tension, we derive an appropriate weak formulation incorporating a variational formulation using divergence-free test functions. We prove a consistency result to justify our definition and, under reasonable regularity assumptions, we reconstruct the pressure function from the weak formulation.

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# Modeling, simulation, and optimization of geothermal energy production from hot sedimentary aquifers

*Authors*

- Blank, Laura
- Meneses Rioseco, Ernesto
- Wilbrandt, Ulrich
- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645

*2010 Mathematics Subject Classification*

- 65M60 76S05 86A20

*Keywords*

- porous and fractured geothermal reservoir modeling, geothermal multi-well configurations, finite element method, thermo-hydraulic coupling, optimization, open-source software

*DOI*

*Abstract*

Geothermal district heating development has been gaining momentum in Europe with numerous deep geothermal installations and projects currently under development. With the increasing density of geothermal wells, questions related to the optimal and sustainable reservoir exploitation become more and more important. A quantitative understanding of the complex thermo-hydraulic interaction between tightly deployed geothermal wells in heterogeneous temperature and permeability fields is key for a maximum sustainable use of geothermal resources. Motivated by the geological settings of the Upper Jurassic aquifer in the Greater Munich region, we develop a computational model based on finite element analysis and gradient-free optimization to simulate groundwater flow and heat transport in hot sedimentary aquifers, and investigate numerically the optimal positioning and spacing of multi-well systems. Based on our numerical simulations, net energy production from deep geothermal reservoirs in sedimentary basins by smart geothermal multi-well arrangements provides significant amounts of energy to meet heat demand in highly urbanized regions. Our results show that taking into account heterogeneous permeability structures and variable reservoir temperature may drastically affect the results in the optimal configuration. We demonstrate that the proposed numerical framework is able to efficiently handle generic geometrical and geologocal configurations, and can be thus flexibly used in the context of multi-variable optimization problems. Hence, this numerical framework can be used to assess the extractable geothermal energy from heterogeneous deep geothermal reservoirs by the optimized deployment of smart multi-well systems.

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# On the $L^p$-theory for second-order elliptic operators in divergence form with complex coefficients

*Authors*

- ter Elst, A. F. M.
- Haller-Dintelmann, Robert
- Rehberg, Joachim
- Tolksdorf, Patrick

*2010 Mathematics Subject Classification*

- 35J15 47D06 47B44

*Keywords*

- Divergence form operators on open sets,
*p*-ellipticity, sectorial, operators, analytic semigroups, maximal regularity, reverse Hölder inequalities, Gaussian estimates, De Giorgi estimates

*DOI*

*Abstract*

Given a complex, elliptic coefficient function we investigate for which values of *p* the corresponding second-order divergence form operator, complemented with Dirichlet, Neumann or mixed boundary conditions, generates a strongly continuous semigroup on L^{p}(Ω). Additional properties like analyticity of the semigroup, H^{∞}-calculus and maximal regularity arealso discussed. Finally we prove a perturbation result for real coefficients that gives the whole range of ^{p}'s for small imaginary parts of the coefficients. Our results are based on the recent notion of ^{p}-ellipticity, reverse Hölder inequalities and Gaussian estimates for the real coefficients.

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# Topology optimization subject to additive manufacturing constraints

*Authors*

- Ebeling-Rump, Moritz
- Hömberg, Dietmar
- Lasarzik, Robert
- Petzold, Thomas

*2010 Mathematics Subject Classification*

- 49Q10 74P05 49Q20 65M60 74P10

*Keywords*

- Additive manufacturing, topology optimization, linear elasticity, phase field method, optimality conditions, numerical simulations

*DOI*

*Abstract*

In Topology Optimization the goal is to find the ideal material distribution in a domain subject to external forces. The structure is optimal if it has the highest possible stiffness. A volume constraint ensures filigree structures, which are regulated via a Ginzburg-Landau term. During 3D Printing overhangs lead to instabilities, which have only been tackled unsatisfactorily. The novel idea is to incorporate an Additive Manufacturing Constraint into the phase field method. A rigorous analysis proves the existence of a solution and leads to first order necessary optimality conditions. With an Allen-Cahn interface propagation the optimization problem is solved iteratively. At a low computational cost the Additive Manufacturing Constraint brings about support structures, which can be fine tuned according to engineering demands. Stability during 3D Printing is assured, which solves a common Additive Manufacturing problem.

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# Time-warping invariants of multidimensional time series

*Authors*

- Diehl, Joscha
- Kurusch, Ebrahimi-Fard
- Tapia, Nikolas

*2010 Mathematics Subject Classification*

- 16T05 68T10 62H30

*2008 Physics and Astronomy Classification Scheme*

- 68Q10

*Keywords*

- Time series analysis, time-warping invariants, signature, quasisymmetric functions, quasi-shuffle product, Hoffman's exponential, area-operation, Hopf algebra

*DOI*

*Abstract*

In data science, one is often confronted with a time series representing measurements of some quantity of interest. Usually, in a first step, features of the time series need to be extracted. These are numerical quantities that aim to succinctly describe the data and to dampen the influence of noise. In some applications, these features are also required to satisfy some invariance properties. In this paper, we concentrate on time-warping invariants.We show that these correspond to a certain family of iterated sums of the increments of the time series, known as quasisymmetric functions in the mathematics literature. We present these invariant features in an algebraic framework, and we develop some of their basic properties.

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# Optimal Neumann boundary control of a vibrating string with uncertain initial data and probabilistic terminal constraints

*Authors*

- Farshbaf Shaker, Mohammad Hassan
- Gugat, Martin
- Heitsch, Holger
- Henrion, René

*2010 Mathematics Subject Classification*

- 90C15 49J29 9J55

*Keywords*

- PDE constrained optimization, probabilistic constraints, uncertain initial data

*DOI*

*Abstract*

In optimal control problems, often initial data are required that are not known exactly in practice. In order to take into account this uncertainty, we consider optimal control problems for a system with an uncertain initial state. A finite terminal time is given. On account of the uncertainty of the initial state, it is not possible to prescribe an exact terminal state. Instead, we are looking for controls that steer the system into a given neighborhood of the desired terminal state with sufficiently high probability. This neighborhood is described in terms of an inequality for the terminal energy. The probabilistic constraint in the considered optimal control problem leads to optimal controls that are robust against the inevitable uncertainties of the initial state. We show the existence of such optimal controls. Numerical examples with optimal Neumann control of the wave equation are presented.

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# Numerical simulation of high-frequency induction welding in longitudinal welded tubes

*Authors*

- Asperheim, John Inge
- Das, Prerana

ORCID: 0000-0001-6040-0311 - Grande, Bjørnar
- Hömberg, Dietmar
- Petzold, Thomas

*2010 Mathematics Subject Classification*

- 35K05 35Q61 65N30

*Keywords*

- Simulation, multi-field problem, Joule heating, welding

*DOI*

*Abstract*

In the present paper the high-frequency induction (HFI) welding process is studied numerically. The mathematical model comprises a harmonic vector potential formulation of the Maxwell equations and a quasi-static, convection dominated heat equation coupled through the joule heat term and nonlinear constitutive relations. Its main novelties are twofold: A new analytic approach permits to compute a spatially varying feed velocity depending on the angle of the Vee-opening and additional spring-back effects. Moreover, a numerical stabilization approach for the finite element discretization allows to consider realistic weld-line speeds and thus a fairly comprehensive three-dimensional simulation of the tube welding process.

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# Tunable Kerr frequency combs and temporal localized states in time-delayed Gires--Tournois interferometers

*Authors*

- Schelte, Christian
- Pimenov, Alexander
- Vladimirov, Andrei G.
- Javaloyes, Julien
- Gurevich, Svetlana V.

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a, 02.30.Ks, 42.55.Px, 42.65.Pc, 42.65.Tg

*Keywords*

- Temporal localized states, time-delay model, large delay, Gires--Tournois interferometer, Kerr frequency combs

*DOI*

*Abstract*

In this Letter we study theoretically a new set-up allowing for the generation of temporal localized states and frequency combs. The setup is compact (a few cm) and can be implemented using established technologies, while offering tunable repetition rates and potentially high power operation. It consists of a vertically emitting micro-cavity, operated in the Gires?Tournois regime, containing a Kerr medium with strong time-delayed optical feedback as well as detuned optical injection. We disclose sets of multistable dark and bright temporal localized states coexisting on their respective bistable homogeneous backgrounds.

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# Semi-tractability of optimal stopping problems via a weighted stochastic mesh algorithm

*Authors*

- Belomestny, Denis
- Kaledin, Maxim
- Schoenmakers, John G. M.

ORCID: 0000-0002-4389-8266

*2010 Mathematics Subject Classification*

- 65C05 60H35 62P05

*Keywords*

- Optimal stopping, American options, Monte Carlo algorithms, complexity

*DOI*

*Abstract*

In this article we propose a Weighted Stochastic Mesh (WSM) algorithm for approximating the value of discrete and continuous time optimal stopping problems. It is shown that in the discrete time case the WSM algorithm leads to semi-tractability of the corresponding optimal stopping problem in the sense that its complexity is bounded in order by $varepsilon^-4log^d+2(1/varepsilon)$ with $d$ being the dimension of the underlying Markov chain. Furthermore we study the WSM approach in the context of continuous time optimal stopping problems and derive the corresponding complexity bounds. Although we can not prove semi-tractability in this case, our bounds turn out to be the tightest ones among the complexity bounds known in the literature. We illustrate our theoretical findings by a numerical example.

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# Longtime behavior for a generalized Cahn--Hilliard system with fractional operators

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K45 35K90 35R11 35B40

*Keywords*

- Fractional operators, Cahn--Hilliard systems, longtime behaviour

*DOI*

*Abstract*

In this contribution, we deal with the longtime behavior of the solutions to the fractional variant of the Cahn--Hilliard system, with possibly singular potentials, which we recently investigated in the paper "Well-posedness and regularity for a generalized fractional CahnHilliard system". More precisely, we give a complete characterization of the Omega-limit of the phase parameter. The characterization depends on the first eigenvalue of one of the involved operators: if this eigenvalue is positive, then the chemical potential vanishes at infinity, and every element of the Omega-limit is a stationary solution to the phase equation; if it is zero instead, then every element of the Omega-limit solves a problem containing a real function which is related to the chemical potential. Such a function is nonunique and time dependent, in general, as we show by means of an example; however, we give sufficient conditions for it to be uniquely determined and constant.

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# A mathematical model for Alzheimer's disease: An approach via stochastic homogenization of the Smoluchowski equation

*Authors*

- Franchi, Bruno
- Heida, Martin
- Lorenzani, Silvia

*2010 Mathematics Subject Classification*

- 74Q10 80M40 80A30 35Q79

*Keywords*

- Smoluchowski equation, stochastic homogenization, Alzheimer disease

*DOI*

*Abstract*

In this note, we apply the theory of stochastic homogenization to find the asymptotic behavior of the solution of a set of Smoluchowski's coagulation-diffusion equations with non-homogeneous Neumann boundary conditions. This system is meant to model the aggregation and diffusion of β-amyloid peptide (Aβ) in the cerebral tissue, a process associated with the development of Alzheimer's disease. In contrast to the approach used in our previous works, in the present paper we account for the non-periodicity of the cellular structure of the brain by assuming a stochastic model for the spatial distribution of neurons. Further, we consider non-periodic random diffusion coefficients for the amyloid aggregates and a random production of Aβ in the monomeric form at the level of neuronal membranes.

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# Structural multiscale topology optimization with stress constraint for additive manufacturing

*Authors*

- Auricchio, Ferdinando
- Bonetti, Elena
- Carraturo, Massimo
- Hömberg, Dietmar
- Reali, Alessandro
- Rocca, Elisabetta

*2010 Mathematics Subject Classification*

- 74P05 49M05 74B99

*Keywords*

- Additive manufacturing, first-order necessary optimality conditions, phase-field method, structural topology optimization, functionally graded material

*DOI*

*Abstract*

In this paper a phase-field approach for structural topology optimization for a 3D-printing process which includes stress constraint and potentially multiple materials or multiscales is analyzed. First order necessary optimality conditions are rigorously derived and a numerical algorithm which implements the method is presented. A sensitivity study with respect to some parameters is conducted for a two-dimensional cantilever beam problem. Finally, a possible workflow to obtain a 3D-printed object from the numerical solutions is described and the final structure is printed using a fused deposition modeling (FDM) 3D printer.

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# Weak solutions and weak-strong uniqueness for a thermodynamically consistent phase-field model

*Authors*

- Lasarzik, Robert
- Rocca, Elisabetta
- Schimperna, Giulio

*2010 Mathematics Subject Classification*

- 35D30 35D35 80A22

*Keywords*

- Existence of weak solutions, weak-strong uniqueness, phase transition, local solutions

*DOI*

*Abstract*

In this paper we prove the existence of weak solutions for a thermodynamically consistent phase-field model introduced in [26] in two and three dimensions of space. We use a notion of solution inspired by [18], where the pointwise internal energy balance is replaced by the total energy inequality complemented with a weak form of the entropy inequality. Moreover, we prove existence of local-in-time strong solutions and, finally, we show weak-strong uniqueness of solutions, meaning that every weak solution coincides with a local strong solution emanating from the same initial data, as long as the latter exists.

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# Asymptotics of the eigenvalues of the Anderson Hamiltonian with white noise potential in two dimensions

*Authors*

- Chouk, Khalil
- van Zuijlen, Willem

*2010 Mathematics Subject Classification*

- 60H25 60F15 35J10 35P15 60F10

*Keywords*

- Anderson Hamiltonian, white noise, paracontrolled distributions, operators with Dirichlet boundary, conditions

*DOI*

*Abstract*

In this paper we consider the Anderson Hamiltonian with white noise potential on the box [0,L]² with Dirichlet boundary conditions. We show that all the eigenvalues divided by log L converge as L → ∞ almost surely to the same deterministic constant, which is given by a variational formula.

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# Stochastic homogenization of Lambda-convex gradient flows

*Authors*

- Heida, Martin
- Neukamm, Stefan
- Varga, Mario

*2010 Mathematics Subject Classification*

- 49J40 74Q10 35K57

*Keywords*

- Stochastic homogenization, stochastic unfolding, two-scale convergence, gradient system

*DOI*

*Abstract*

In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a Λ-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen--Cahn type equations and evolutionary equations driven by the *p*-Laplace operator with *p* ∈ in (1, ∞). The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of (Λ-)convex functionals.

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# Adaptive gradient descent for convex and non-convex stochastic optimization

*Authors*

- Ogaltsov, Aleksandr
- Dvinskikh, Darina
- Dvurechensky, Pavel

ORCID: 0000-0003-1201-2343 - Gasnikov, Alexander
- Spokoiny, Vladimir

ORCID: 0000-0002-2040-3427

*2010 Mathematics Subject Classification*

- 90C25 90C30 90C06

*Keywords*

- Convex and non-convex optimization, stochastic optimization, first-order method, adaptive method, gradient descent, complexity bounds, mini-batch

*DOI*

*Abstract*

In this paper we propose several adaptive gradient methods for stochastic optimization. Our methods are based on Armijo-type line search and they simultaneously adapt to the unknown Lipschitz constant of the gradient and variance of the stochastic approximation for the gradient. We consider an accelerated gradient descent for convex problems and gradient descent for non-convex problems. In the experiments we demonstrate superiority of our methods to existing adaptive methods, e.g. AdaGrad and Adam.

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# Exponential moments for planar tessellations

*Authors*

- Tóbiás, András
- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065

*2010 Mathematics Subject Classification*

- 60K05 52A38 60G55

*Keywords*

- Poisson--Voronoi tessellation, Poisson--Delaunay tessellation, Poisson line tessellation, Johnson--Mehl tessellation, Manhattan grid, Cox--Voronoi tessellation, nested tessellation, iterated tessellation, exponential moments, total edge, length, number of cells, number of edges, Palm calculus

*DOI*

*Abstract*

In this paper we show existence of all exponential moments for the total edge length in a unit disc for a family of planar tessellations based on Poisson point processes. Apart from classical such tessellations like the Poisson--Voronoi, Poisson--Delaunay and Poisson line tessellation, we also treat the Johnson--Mehl tessellation, Manhattan grids, nested versions and Palm versions. As part of our proofs, for some planar tessellations, we also derive existence of exponential moments for the number of cells and the number of edges intersecting the unit disk.

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# A numerical analysis focused comparison of several finite volume schemes for an unipolar degenerated drift-diffusion model

*Authors*

- Cancès, Clément
- Chainais-Hillairet, Claire
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Gaudeul, Benoît

*2010 Mathematics Subject Classification*

- 35Q35

*Keywords*

- Finite volume methods, drift-diffusion equations

*DOI*

*Abstract*

In this paper, we consider an unipolar degenerated drift-diffusion system where the relation between the concentration of the charged species *c* and the chemical potential *h* is *h(c) = log* ^{c}/_{1-c}. We design four different finite volume schemes based on four different formulations of the fluxes. We provide a stability analysis and existence results for the four schemes. The convergence proof with respect to the discretization parameters is established for two of them. Numerical experiments illustrate the behaviour of the different schemes.

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# Weak entropy solutions to a model in induction hardening, existence and weak-strong uniqueness

*Authors*

- Hömberg, Dietmar
- Lasarzik, Robert

*2010 Mathematics Subject Classification*

- 35Q61 35Q79 80A17

*Keywords*

- Induction hardening, existence, weak-strong uniqueness, weak-entropy solution

*DOI*

*Abstract*

In this paper, we investigate a model describing induction hardening of steel. The related system consists of an energy balance, an ODE for the different phases of steel, and Maxwell's equations in a potential formulation. The existence of weak entropy solutions is shown by a suitable regularization and discretization technique. Moreover, we prove the weak-strong uniqueness of these solutions, i.e., that a weak entropy solutions coincides with a classical solution emanating form the same initial data as long as the classical one exists. The weak entropy solution concept has advantages in comparison to the previously introduced weak solutions, e.g., it allows to include free energy functions with low regularity properties corresponding to phase transitions.

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# Asymptotic analysis of a tumor growth model with fractional operators

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35B40 35K55 35K90 35Q92 92C17

*Keywords*

- Fractional operators, Cahn--Hilliard systems, well-posedness, regularity of solutions, tumor growth models, asymptotic analysis

*DOI*

*Abstract*

In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalized and relaxed version of a phase field system of Cahn--Hilliard type modelling tumor growth that has originally been proposed in Hawkins-Daarud et al. (*Int. J. Numer. Math. Biomed. Eng. ***28** *(2012), 3--24*). The original phase field system and certain relaxed versions thereof have been studied in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn--Hilliard equation for the tumor cell fraction φ, coupled to a reaction-diffusion equation for a function S representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. Motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type, the present authors studied in a recent note a generalization of the systems investigated in the abovementioned works. Under rather general assumptions, well-posedness and regularity results have been shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth contributions of logarithmic or of double obstacle type to the energy density could be admitted. In this note, we perform an asymptotic analysis of the governing system as two (small) relaxation parameters approach zero separately and simultaneously. Corresponding well-posedness and regularity results are established for the respective cases; in particular, we give a detailed discussion which assumptions on the admissible nonlinearities have to be postulated in each of the occurring cases.

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# Well-posedness and regularity for a fractional tumor growth model

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35Q92 92C17 35K35 35K90

*Keywords*

- Fractional operators, Cahn--Hilliard systems, well-posedness, regularity of solutions, tumor growth models

*DOI*

*Abstract*

In this paper, we study a system of three evolutionary operator equations involving fractional powers of selfadjoint, monotone, unbounded, linear operators having compact resolvents. This system constitutes a generalization of a phase field system of Cahn--Hilliard type modelling tumor growth that has been proposed in Hawkins-Daarud et al. (*Int. J. Numer. Math. Biomed. Eng.* **28** *(2012), 3--24*) and investigated in recent papers co-authored by the present authors and E. Rocca. The model consists of a Cahn--Hilliard equation for the tumor cell fraction φ, coupled to a reaction-diffusion equation for a function *S* representing the nutrient-rich extracellular water volume fraction. Effects due to fluid motion are neglected. The generalization investigated in this paper is motivated by the possibility that the diffusional regimes governing the evolution of the different constituents of the model may be of different (e.g., fractional) type. Under rather general assumptions, well-posedness and regularity results are shown. In particular, by writing the equation governing the evolution of the chemical potential in the form of a general variational inequality, also singular or nonsmooth constributions of logarithmic or of double obstacle type to the energy density can be admitted.

*Appeared in*

- Adv. Math. Sci. Appl., 28 (2019), pp. 343--375.

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# A class of second-order geometric quasilinear hyperbolic PDEs and their application in imaging science

*Authors*

- Dong, Guozhi

ORCID: 0000-0002-9674-6143 - Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Zhang, Ye

*2010 Mathematics Subject Classification*

- 35L10 35L70 35L72

*Keywords*

- Quasilinear hyperbolic equation, geometric PDEs, total variation flow, mean curvature flow, level set, second-order dynamics, non-smooth and non-convex variational methods, image denoising, displacement error correction

*DOI*

*Abstract*

In this paper, we study damped second-order dynamics, which are quasilinear hyperbolic partial differential equations (PDEs). This is inspired by the recent development of second-order damping systems for accelerating energy decay of gradient flows. We concentrate on two equations: one is a damped second-order total variation flow, which is primarily motivated by the application of image denoising; the other is a damped second-order mean curvature flow for level sets of scalar functions, which is related to a non-convex variational model capable of correcting displacement errors in image data (e.g. dejittering). For the former equation, we prove the existence and uniqueness of the solution. For the latter, we draw a connection between the equation and some second-order geometric PDEs evolving the hypersurfaces which are described by level sets of scalar functions, and show the existence and uniqueness of the solution for a regularized version of the equation. The latter is used in our algorithmic development. A general algorithm for numerical discretization of the two nonlinear PDEs is proposed and analyzed. Its efficiency is demonstrated by various numerical examples, where simulations on the behavior of solutions of the new equations and comparisons with first-order flows are also documented.

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# A distributed control problem for a fractional tumor growth model

*Authors*

- Colli, Pierluigi
- Gilardi, Gianni
- Sprekels, Jürgen

*2010 Mathematics Subject Classification*

- 35K55 35Q92 49J20 92C50

*Keywords*

- Fractional operators, Cahn--Hilliard systems, well-posedness, regularity, optimal control, necessary optimality conditions

*DOI*

*Abstract*

In this paper, we study the distributed optimal control of a system of three evolutionary equations involving fractional powers of three selfadjoint, monotone, unbounded linear operators having compact resolvents. The system is a generalization of a Cahn--Hilliard type phase field system modeling tumor growth that goes back to Hawkins-Daarud et al. (*Int. J. Numer. Math. Biomed. Eng. ***28** *(2012), 3--24.*) The aim of the control process, which could be realized by either administering a drug or monitoring the nutrition, is to keep the tumor cell fraction under control while avoiding possible harm for the patient. In contrast to previous studies, in which the occurring unbounded operators governing the diffusional regimes were all given by the Laplacian with zero Neumann boundary conditions, the operators may in our case be different; more generally, we consider systems with fractional powers of the type that were studied in the recent work *Adv. Math. Sci. Appl.* **28** *(2019), 343--375 * by the present authors. In our analysis, we show the Fréchet differentiability of the associated control-to-state operator, establish the existence of solutions to the associated adjoint system, and derive the first-order necessary conditions of optimality for a cost functional of tracking type.

*Appeared in*

- Mathematics, 7 (2019), pp. 792/1--792/32, DOI 10.3390/math7090792 .

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# Self-consistent field theory for a polymer brush. Part I: Asymptotic analysis in the strong-stretching limit

*Authors*

- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 74H10 82B21

*2008 Physics and Astronomy Classification Scheme*

- 02.30.Mv, 61.25.H-

*Keywords*

- Self-consistent field theory, asymptotic analysis, Cole-Hopf transformation, polymer brush

*DOI*

*Abstract*

In this study we consider the self-consistent field theory for a dry, in- compressible polymer brush, densely grafted on a substrate, describing the average segment density of a polymer in terms of an effective chemical potential for the interaction between the segments of the polymer chain. We present a systematic singular perturbation analysis of the self-consistent field theory in the strong-stretching limit, when the length scale of the ratio of the radius of gyration of the polymer chain to the extension of the brush from the substrate vanishes. Our analysis yields, for the first time, an approximation for the average segment density that is correct to leading order in the outer scaling and resolves the boundary layer singularity at the end of the polymer brush in the strong-stretching limit. We also show that in this limit our analytical results agree increasingly well with our numerical solutions to the full model equations comprising the self-consistent field theory.

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# Regularization for optimal control problems associated to nonlinear evolution equations

*Authors*

- Meinlschmidt, Hannes
- Meyer, Christian
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 49K20 49J20 47J20 47J35 46E40

*Keywords*

- Optimal control, regularization, nonlinear evolution equations, compactness, function spaces

*DOI*

*Abstract*

It is well-known that in the case of a sufficiently nonlinear general optimal control problem there is very frequently the necessity for a compactness argument in order to pass to the limit in the state equation in the standard ``calculus of variations'' proof for the existence of optimal controls. For time-dependent state equations, i.e., evolution equations, this is in particular unfortunate due to the difficult structure of compact sets in Bochner-type spaces. In this paper, we propose an abstract function space and a suitable regularization- or Tychonov term for the objective functional which allows for the usual standard reasoning in the proof of existence of optimal controls and which admits a reasonably favorable structure in the characterization of optimal solutions via first order necessary conditions in, generally, the form of a variational inequality of obstacle-type in time. We establish the necessary properties of the function space and the Tychonov term and derive the aforementioned variational inequality. The variational inequality can then be reformulated as a projection identity for the optimal control under additional assumptions. We give sufficient conditions on when these are satisfied. The considerations are complemented with a series of practical examples of possible constellations and choices in dependence on the varying control spaces required for the evolution equations at hand.

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# Essential boundedness for solutions of the Neumann problem on general domains

*Authors*

- ter Elst, A. F. M.
- Meinlschmidt, Hannes
- Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 35D10 46E35

*Keywords*

- Regularity, Neumann problem, elliptic operators

*DOI*

*Abstract*

Let the domain under consideration be bounded. Under the suppositions of very weak Sobolev embeddings we prove that the solutions of the Neumann problem for an elliptic, second order divergence operator are essentially bounded, if the right hand sides are taken from the dual of a Sobolev space which is adapted to the above embedding.

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# Temporal dissipative solitons in time-delay feedback systems

*Authors*

- Yanchuk, Serhiy
- Ruschel, Stefan
- Sieber, Jan
- Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 34K13 34K25

*Keywords*

- Dissipative solitons, time-delay systems

*DOI*

*Abstract*

Localized states are a universal phenomenon observed in spatially distributed dissipative nonlinear systems. Known as dissipative solitons, auto-solitons, spot or pulse solitons, these states play an important role in data transmission using optical pulses, neural signal propagation, and other processes. While this phenomenon was thoroughly studied in spatially extended systems, temporally localized states are gaining attention only recently, driven primarily by applications from fiber or semiconductor lasers. Here we present a theory for temporal dissipative solitons (TDS) in systems with time-delayed feedback. In particular, we derive a system with an advanced argument, which determines the profile of the TDS. We also provide a complete classification of the spectrum of TDS into interface and pseudo-continuous spectrum. We illustrate our theory with two examples: a generic delayed phase oscillator, which is a reduced model for an injected laser with feedback, and the FitzHugh--Nagumo neuron with delayed feedback. Finally, we discuss possible destabilization mechanisms of TDS and show an example where the TDS delocalizes and its pseudo-continuous spectrum develops a modulational instability.

*Appeared in*

- Phys. Rev. Lett., 123 (2019), pp. 053901/1--053901/6, DOI 10.1103/PhysRevLett.123.053901 .

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# Generalized Scharfetter--Gummel schemes for electro-thermal transport in degenerate semiconductors using the Kelvin formula for the Seebeck coefficient

*Authors*

- Kantner, Markus

ORCID: 0000-0003-4576-3135

*2010 Mathematics Subject Classification*

- 35K57 35Q79 65N08 80A20

*Keywords*

- Finite volume Scharfetter--Gummel method, semiconductor device simulation, electro-thermal transport, non-isothermal drift-diffusion system, degenerate semiconductors, Fermi--Dirac statistics, Seebeck coefficient

*DOI*

*Abstract*

Many challenges faced in today's semiconductor devices are related to self-heating phenomena. The optimization of device designs can be assisted by numerical simulations using the non-isothermal drift-diffusion system, where the magnitude of the thermoelectric cross effects is controlled by the Seebeck coefficient. We show that the model equations take a remarkably simple form when assuming the so-called Kelvin formula for the Seebeck coefficient. The corresponding heat generation rate involves exactly the three classically known self-heating effects, namely Joule, recombination and Thomson--Peltier heating, without any further (transient) contributions. Moreover, the thermal driving force in the electrical current density expressions can be entirely absorbed in the (nonlinear) diffusion coefficient via a generalized Einstein relation. The efficient numerical simulation relies on an accurate and robust discretization technique for the fluxes (finite volume Scharfetter--Gummel method), which allows to cope with the typically stiff solutions of the semiconductor device equations. We derive two non-isothermal generalizations of the Scharfetter--Gummel scheme for degenerate semiconductors (Fermi--Dirac statistics) obeying the Kelvin formula. The approaches differ in the treatment of degeneration effects: The first is based on an approximation of the discrete generalized Einstein relation implying a specifically modified thermal voltage, whereas the second scheme follows the conventionally used approach employing a modified electric field. We present a detailed analysis and comparison of both schemes, indicating a superior performance of the modified thermal voltage scheme.

*Appeared in*

- J. Comput. Phys., (2019), published online on 07.11.2019, DOI 10.1016/j.jcp.2019.109091 .

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# Temporal cavity solitons in a delayed model of a dispersive cavity ring laser

*Authors*

- Pimenov, Alexander
- Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Vladimirov, Andrei G.

*2008 Physics and Astronomy Classification Scheme*

- 05.45.-a 02.30.Ks 42.55.Px 42.65.Pc 42.65.Tg

*Keywords*

- Temporal cavity solitons, semiconductor laser, chromatic dispersion, time-delay model, large delay

*DOI*

*Abstract*

Nonlinear localised structures appear as solitary states in systems with multistability and hysteresis. In particular, localised structures of light known as temporal cavity solitons were observed recently experimentally in driven Kerr-cavities operating in the anomalous dispersion regime when one of the two bistable spatially homogeneous steady states exhibits a modulational instability. We use a distributed delay system to study theoretically the formation of temporal cavity solitons in an optically injected ring semiconductor-based fiber laser, and propose an approach to derive reduced delay-differential equation models taking into account the dispersion of the intracavity fiber delay line. Using these equations we perform the stability and bifurcation analysis of injection-locked CW states and temporal cavity solitons.

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# Optimal control of geometric partial differential equations

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Keil, Tobias

*2010 Mathematics Subject Classification*

- 49J20 49K20 35J87 35Q93 35Q35 35R35 90C33 90C46 76D45 76T10 65K10 65K15

*Keywords*

- Adaptive discretization, constraint degeneracy, diffuse interface model, electro-wetting on dielectric, geometric evolution, mathematical program with equilibrium constraints, multiphase fluids, numerical method, optimal control, sharp interface model, C stationarity conditions

*DOI*

*Abstract*

Optimal control problems for geometric (evolutionary) partial differential inclusions are considered. The focus is on problems which, in addition to the nonlinearity due to geometric evolution, contain optimization theoretic challenges because of non-smoothness. The latter might stem from energies containing non-smooth constituents such as obstacle-type potentials or terms modeling, e.g., pinning phenomena in microfluidics. Several techniques to remedy the resulting constraint degeneracy when deriving stationarity conditions are presented. A particular focus is on Yosida-type mollifications approximating the original degenerate problem by a sequence of nondegenerate nonconvex optimal control problems. This technique is also the starting point for the development of numerical solution schemes. In this context, also dual-weighted residual based error estimates are addressed to facilitate an adaptive mesh refinement. Concerning the underlying state model, sharp and diffuse interface formulations are discussed. While the former always allows for accurately tracing interfacial motion, the latter model may be dictated by the underlying physical phenomenon, where near the interface mixed phases may exist, but it may also be used as an approximate model for (sharp) interface motion. In view of the latter, (sharp interface) limits of diffuse interface models are addressed. For the sake of presentation, this exposition confines itself to phase field type diffuse interface models and, moreover, develops the optimal control of either of the two interface models along model applications. More precisely, electro-wetting on dielectric is used in the sharp interface context, and the control of multiphase fluids involving spinodal decomposition highlights the phase field technique. Mathematically, the former leads to a Hele-Shaw flow with geometric boundary conditions involving a complementarity system due to contact line pinning, and the latter gives rise to a Cahn-Hilliard Navier-Stokes model including a non-smooth obstacle type potential leading to a variational inequality constraint.

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# Uniformly positive correlations in the dimer model and phase transition in lattice permutations in $mathbbZ^d, d > 2$, via reflection positivity

*Authors*

- Taggi, Lorenzo

*2010 Mathematics Subject Classification*

- 82B20 82B26 82B41 05A05

*Keywords*

- Dimer model, spatial permutations, quantum Bose gas, self-avoiding walks, loop O(N) model, reflection positivity, infrared bound, phase transitions

*DOI*

*Abstract*

Our first main result is that correlations between monomers in the dimer model in ℤ^{d} do not decay to zero when d > 2. This is the first rigorous result about correlations in the dimer model in dimensions greater than two and shows that the model behaves drastically differently than in two dimensions, in which case it is integrable and correlations are known to decay to zero polynomially. Such a result is implied by our more general, second main result, which states the occurrence of a phase transition in the model of lattice permutations, which is related to the quantum Bose gas. More precisely, we consider a self-avoiding walk interacting with lattice permutations and we prove that, in the regime of fully-packed loops, such a walk is `long' and the distance between its end-points grows linearly with the diameter of the box. These results follow from the derivation of a version of the infrared bound from a new general probabilistic settings, with coloured loops and walks interacting at sites and walks entering into the system from some `virtual' vertices.

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# Generalized Nash equilibrium problems with partial differential operators: Theory, algorithms, and risk aversion

*Authors*

- Gahururu, Deborah
- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Stengl, Steven-Marian
- Surowiec, Thomas M.

ORCID: 0000-0003-2473-4984

*2010 Mathematics Subject Classification*

- 49J20 49J55 49K20 49K45 49M99 65K10 65K15 90C15 91A10

*Keywords*

- Generalized Nash equilibrium problems, PDE-constrained optimization, L-convexity, set-valued analysis, fixed-point theory, risk averse optimization, coherent risk measures, stochastic optimization, method of multipliers

*DOI*

*Abstract*

PDE-constrained (generalized) Nash equilibrium problems (GNEPs) are considered in a deterministic setting as well as under uncertainty. This includes a study of deterministic GNEPs with nonlinear and/or multivalued operator equations as forward problems and PDE-constrained GNEPs with uncertain data. The deterministic nonlinear problems are analyzed using the theory of generalized convexity for set-valued operators, and a variational approximation approach is proposed. The stochastic setting includes a detailed overview of the recently developed theory and algorithms for risk-averse PDE-constrained optimization problems. These new results open the way to a rigorous study of stochastic PDE-constrained GNEPs.

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# Generalized gradients for probabilistic/robust (probust) constraints

*Authors*

- van Ackooij, Wim
- Henrion, René
- Pérez-Aros, Pedro

*2010 Mathematics Subject Classification*

- 90C15

*Keywords*

- Stochastic optimization, probabilistic constraints, chance constraints, gradients of probability functions, probust constraints

*DOI*

*Abstract*

Probability functions are a powerful modelling tool when seeking to account for uncertainty in optimization problems. In practice, such uncertainty may result from different sources for which unequal information is available. A convenient combination with ideas from robust optimization then leads to probust functions, i.e., probability functions acting on generalized semi-infinite inequality systems. In this paper we employ the powerful variational tools developed by Boris Mordukhovich to study generalized differentiation of such probust functions. We also provide explicit outer estimates of the generalized subdifferentials in terms of nominal data.

*Appeared in*

- Optimization, published online on 14.02.2019, DOI 10.1080/02331934.2019.1576670 .

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# Multi-dimensional modeling and simulation of semiconductor nanophotonic devices

*Authors*

- Kantner, Markus

ORCID: 0000-0003-4576-3135 - Höhne, Theresa
- Koprucki, Thomas

ORCID: 0000-0001-6235-9412 - Burger, Sven
- Wünsche, Hans-Jürgen
- Schmidt, Frank
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Bandelow, Uwe

ORCID: 0000-0003-3677-2347

*2008 Physics and Astronomy Classification Scheme*

- 02.70.Dh, 03.50.De, 42.50.-p, 42.55.Px, 47.11.Df, 81.07.Ta, 85.60.-q

*Keywords*

- Nanophotonic devices, device simulation, multi-physics models, VCSELs, single-photon sources, waveguides, quantum dots, van Roosbroeck system, drift-diffusion equations, Maxwell equations, Lindblad master equation, GENERIC, optical resonance modes, degenerate semiconductors, finite volume method, finite element method

*DOI*

*Abstract*

Self-consistent modeling and multi-dimensional simulation of semiconductor nanophotonic devices is an important tool in the development of future integrated light sources and quantum devices. Simulations can guide important technological decisions by revealing performance bottlenecks in new device concepts, contribute to their understanding and help to theoretically explore their optimization potential. The efficient implementation of multi-dimensional numerical simulations for computer-aided design tasks requires sophisticated numerical methods and modeling techniques. We review recent advances in device-scale modeling of quantum dot based single-photon sources and laser diodes by self-consistently coupling the optical Maxwell equations with semiclassical carrier transport models using semi-classical and fully quantum mechanical descriptions of the optically active region, respectively. For the simulation of realistic devices with complex, multi-dimensional geometries, we have developed a novel hp-adaptive finite element approach for the optical Maxwell equations, using mixed meshes adapted to the multi-scale properties of the photonic structures. For electrically driven devices, we introduced novel discretization and parameter-embedding techniques to solve the drift-diffusion system for strongly degenerate semiconductors at cryogenic temperature. Our methodical advances are demonstrated on various applications, including vertical-cavity surface-emitting lasers, grating couplers and single-photon sources.

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# SINR percolation for Cox point processes with random powers

*Authors*

- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065 - Tóbiás, András

*2010 Mathematics Subject Classification*

- 82B43 60G55 60K35

*Keywords*

- Signal-to-interference ratio, Cox point process, Poisson point process, continuum percolation, SINR percolation, Gilbert graph, Boolean model, stabilization, random power, degree bound

*DOI*

*Abstract*

Signal-to-interference plus noise ratio (SINR) percolation is an infinite-range dependent variant of continuum percolation modeling connections in a telecommunication network. Unlike in earlier works, in the present paper the transmitted signal powers of the devices of the network are assumed random, i.i.d. and possibly unbounded. Additionally, we assume that the devices form a stationary Cox point process, i.e., a Poisson point process with stationary random intensity measure, in two or higher dimensions. We present the following main results. First, under suitable moment conditions on the signal powers and the intensity measure, there is percolation in the SINR graph given that the device density is high and interferences are sufficiently reduced, but not vanishing. Second, if the interference cancellation factor γ and the SINR threshold τ satisfy γ ≥ 1/(2τ), then there is no percolation for any intensity parameter. Third, in the case of a Poisson point process with constant powers, for any intensity parameter that is supercritical for the underlying Gilbert graph, the SINR graph also percolates with some small but positive interference cancellation factor.

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# Millions of Perrin pseudoprimes including a few giants

*Authors*

- Stephan, Holger

*2010 Mathematics Subject Classification*

- 11B37 11B39 11B50

*Keywords*

- pseudoprimes, recurrence sequences, fast algorithm, large numbers

*DOI*

*Abstract*

The calculation of many and large Perrin pseudoprimes is a challenge. This is mainly due to their rarity. Perrin pseudoprimes are one of the rarest known pseudoprimes. In order to calculate many such large numbers, one needs not only a fast algorithm but also an idea how most of them are structured to minimize the amount of numbers one have to test. We present a quick algorithm for testing Perrin pseudoprimes and develop some ideas on how Perrin pseudoprimes might be structured. This leads to some conjectures that still need to be proved.

We think that we have found well over 90% of all 20-digit Perrin pseudoprimes. Overall, we have been able to calculate over 9 million Perrin pseudoprimes with our method, including some very large ones. The largest number found has 1436 digits. This seems to be a breakthrough, compared to the previously known just over 100,000 Perrin pseudoprimes, of which the largest have 20 digits.

In addition, we propose two sequences that do not provide any pseudoprimes up to 1,000,000,000 at all.

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# Analysis of cross-diffusion systems for fluid mixtures driven by a pressure gradient

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500 - Jüngel, Ansgar

*2010 Mathematics Subject Classification*

- 35K45 35L65 35Q79 35M31 35Q92 92C17

*Keywords*

- Parabolic-hyperbolic system, cross diffusion, fluid mixture, existence of solutions, transport equation

*DOI*

*Abstract*

The convective transport in a multicomponent isothermal compressible fluid subject to the mass continuity equations is considered. The velocity is proportional to the negative pressure gradient, according to Darcy?s law, and the pressure is defined by a state equation imposed by the volume extension of the mixture. These model assumptions lead to a parabolic-hyperbolic system for the mass densities. The global-in-time existence of classical and weak solutions is proved in a bounded domain with no-penetration boundary conditions. The idea is to decompose the system into a porous-medium-type equation for the volume extension and transport equations for the modified number fractions. The existence proof is based on parabolic regularity theory, the theory of renormalized solutions, and an approximation of the velocity field.

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# Inverse elastic scattering from rigid scatterers with a single incoming wave

*Authors*

- Elschner, Johannes
- Hu, Guanghui

*2010 Mathematics Subject Classification*

- 35R30 65M30 74J25

*Keywords*

- Uniqueness, inverse elastic scattering, rigid polygonal obstacle, single plane wave, reflection principle

*DOI*

*Abstract*

The first part of this paper is concerned with uniqueness to inverse time-harmonic elastic scattering from bounded rigid obstacles in two dimensions. It is proved that a connected polygonal obstacle can be uniquely identified by the far-field pattern corresponding to a single elastic plane wave. Our approach is based on a new reflection principle for the first boundary value problem of the Navier equation. In the second part, we propose a revisited factorization method to recover a rigid elastic body with a single far-field pattern.

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# Thermoviscoelasticity in Kelvin--Voigt rheology at large strains

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Roubíček, Tomáš

*2010 Mathematics Subject Classification*

- 35K55 35Q74 74A15 74A30 80A17

*Keywords*

- Finite-strain elasticity, static frame indifference, time-dependent frame indifference, balance of total energy, energy estimates, hyperstress regularization, uniform positive of determinant, viscous heating, staggered time-incremental minimization scheme

*DOI*

*Abstract*

The frame-indifferent thermodynamically-consistent model of thermoviscoelasticity at large strain is formulated in the reference configuration with using the concept of the second-grade nonsimple materials. We focus on physically correct viscous stresses that are frame indifferent under time-dependent rotations. Also elastic stresses are frame indifferent under rotations and respect positivity of the determinant of the deformation gradient. The heat transfer is governed by the Fourier law in the actual deformed configuration, which leads to a nontrivial description when pulled back into the reference configuration. Existence of weak solutions in the quasistatic setting, i.e. inertial forces are ignored, is shown by time discretization.

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# Rate-independent evolution of sets

*Authors*

- Rossi, Riccarda
- Stefanelli, Ulisse
- Thomas, Marita

*2010 Mathematics Subject Classification*

- 35A15 35R37 49Q10 74R10

*Keywords*

- Unidirectional evolution of sets by competition of perimeter and volume, minimizers of perimeter perturbed by a nonsmooth functional, minimizing movements, stability, energetic solutions

*DOI*

*Abstract*

The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions.

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# Modelling and simulation of flame cutting for steel plates with solid phases and melting

*Authors*

- Arenas Jaén, Manuel J.
- Hömberg, Dietmar
- Lasarzik, Robert
- Mikkonen, Pertti
- Petzold, Thomas

*2010 Mathematics Subject Classification*

- 35F60 65N30 80A22

*Keywords*

- Flame cutting, finite element method, heat equation, phase transitions, transport equation

*DOI*

*Abstract*

The goal of this work is to describe in detail a quasi-stationary state model which can be used to deeply understand the distribution of the heat in a steel plate and the changes in the solid phases of the steel and into liquid phase during the flame cutting process. We use a 3D-model similar to previous works from Thiebaud [1] and expand it to consider phases changes, in particular, austenite formation and melting of material. Experimental data is used to validate the model and study its capabilities. Parameters defining the shape of the volumetric heat source and the power density are calibrated to achieve good agreement with temperature measurements. Similarities and differences with other models from literature are discussed.

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# Combinatorial considerations on the invariant measure of a stochastic matrix

*Authors*

- Stephan, Artur

*2010 Mathematics Subject Classification*

- 60Jxx

*Keywords*

- Markov chain, Markov process, invariant measure, stationary measure, stationary distribution, Theorem of Frobenius-Perron, Kirchhoff tree theorem, Markov tree theorem, directed and undirected acyclic graphs, spanning trees, detailed balance

*DOI*

*Abstract*

The invariant measure is a fundamental object in the theory of Markov processes. In finite dimensions a Markov process is defined by transition rates of the corresponding stochastic matrix. The Markov tree theorem provides an explicit representation of the invariant measure of a stochastic matrix. In this note, we given a simple and purely combinatorial proof of the Markov tree theorem. In the symmetric case of detailed balance, the statement and the proof simplifies even more.

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# Self-consistent field theory for a polymer brush. Part II: The effective chemical potential

*Authors*

- Münch, Andreas
- Wagner, Barbara

*2010 Mathematics Subject Classification*

- 74H10 82B21

*2008 Physics and Astronomy Classification Scheme*

- 02.30.Mv, 61.25.H-

*Keywords*

- Self-consistent field theory, asymptotic analysis, chemical potential

*DOI*

*Abstract*

The most successful mean-field model to describe the collective behaviour of the large class of macromolecular polymers is the self-consistent field theory (SCFT). Still, even for the simple system of a grafted dry polymer brush, the mean-field equations have to be solved numerically. As one of very few alternatives that offer some analytical tractability the strong-stretching theory (SST) has led to explicit expressions for the effective chemical potential and consequently the free energy to promote an understanding of the underlying physics. Yet, a direct derivation of these analytical results from the SCFT model is still outstanding. In this study we present a systematic asymptotic theory based on matched asymtptotic expansions to obtain the effective chemical potential from the SCFT model for a dry polymer brush for large but finite stretching.

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# Numerical study of coherence of optical feedback in semiconductor laser dynamics

*Authors*

- Radziunas, Mindaugas
- Little, Douglas J.
- Kane, Deborah M.

*2010 Mathematics Subject Classification*

- 78A60 37N20 35Q60 78-04 60G15

*2008 Physics and Astronomy Classification Scheme*

- 42.55.Px 42.60.Mi 42.65.Sf 42.25.Kb 42.60.Da

*Keywords*

- Semiconductor lasers, modeling, dynamics, optical feedback, coherence, external cavity, traveling wave

*DOI*

*Abstract*

The nonlinear dynamics of semiconductor laser with coherent, as compared to incoherent, delayed optical feedback systems have been discussed and contrasted in prior research literature. Here, we report simulations of how the dynamics change as the coherence of the optical feedback is systematically varied from being coherent to incoherent. An increasing rate of phase disturbance is used to vary the coherence. An edge emitting, 830nm, Fabry Perot semiconductor laser with a long external cavity is simulated. Following this study, consideration of prior and future experimental studies should include evaluation of where on the continuum of partial coherence the delayed optical feedback sits. Partial coherence is a parameter that will affect the dynamics.

*Appeared in*

- Opt. Lett., 44 (2019), pp. 4207--4210, DOI 10.1364/OL.44.004207 .

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# Effective diffusion in thin structures via generalized gradient systems and EDP-convergence

*Authors*

- Frenzel, Thomas
- Liero, Matthias

ORCID: 0000-0002-0963-2915

*2010 Mathematics Subject Classification*

- 35K20 35K10 35K57 49S99

*Keywords*

- Fokker--Planck equation, dimension reduction, sandwich structure, Γ-convergence, gradient system, EDP-convergence, Wasserstein gradient flow

*DOI*

*Abstract*

The notion of Energy-Dissipation-Principle convergence (EDP-convergence) is used to derive effective evolution equations for gradient systems describing diffusion in a structure consisting of several thin layers in the limit of vanishing layer thickness. The thicknesses of the sublayers tend to zero with different rates and the diffusion coefficients scale suitably. The Fokker--Planck equation can be formulated as gradient-flow equation with respect to the logarithmic relative entropy of the system and a quadratic Wasserstein-type gradient structure. The EDP-convergence of the gradient system is shown by proving suitable asymptotic lower limits of the entropy and the total dissipation functional. The crucial point is that the limiting evolution is again described by a gradient system, however, now the dissipation potential is not longer quadratic but is given in terms of the hyperbolic cosine. The latter describes jump processes across the thin layers and is related to the Marcelin--de Donder kinetics.

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# Lipschitz lower semicontinuity moduli for linear inequality systems

*Authors*

- Cánovas, Maria Josefa
- Gisbert, María Jesús
- Henrion, René
- Parra, Juan

*2010 Mathematics Subject Classification*

- 90C31 49J53 49K40 90C05

*Keywords*

- Variational analysis, Lipschitz lower semicontinuity, Lipschitz modulus, Aubin property, feasible set mapping, linear programming

*DOI*

*Abstract*

The paper is focussed on the Lipschitz lower semicontinuity of the feasible set mapping for linear (finite and infinite) inequality systems in three different perturbation frameworks: full, right-hand side and left-hand side perturbations. Inspired by [14], we introduce the Lipschitz lower semicontinuity-star as an intermediate notion between the Lipschitz lower semicontinuity and the well-known Aubin property. We provide explicit point-based formulae for the moduli (best constants) of all three Lipschitz properties in all three perturbation settings.

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# Generating structured non-smooth priors and associated primal-dual methods

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Papafitsoros, Kostas

ORCID: 0000-0001-9691-4576

*2010 Mathematics Subject Classification*

- 94A08 68U10 49K20 49M37 49M15 26A45

*Keywords*

- Non-smooth priors, image processing, total variation, total generalized variation, bilevel optimization, regularization parameter selection

*DOI*

*Abstract*

The purpose of the present chapter is to bind together and extend some recent developments regarding data-driven non-smooth regularization techniques in image processing through the means of a bilevel minimization scheme. The scheme, considered in function space, takes advantage of a dualization framework and it is designed to produce spatially varying regularization parameters adapted to the data for well-known regularizers, e.g. Total Variation and Total Generalized variation, leading to automated (monolithic), image reconstruction workflows. An inclusion of the theory of bilevel optimization and the theoretical background of the dualization framework, as well as a brief review of the aforementioned regularizers and their parameterization, makes this chapter a self-contained one. Aspects of the numerical implementation of the scheme are discussed and numerical examples are provided.

*Appeared in*

- M. Hintermüller, K. Papafitsoros, Chapter 11: Generating Structured Nonsmooth Priors and Associated Primal-dual Methods, in: Processing, Analyzing and Learning of Images, Shapes, and Forms: Part 2 , R. Kimmel, X.-Ch. Tai, eds., vol. 20 of Handbook of Numerical Analysis, Elsevier, 2019, pp. 437--502, DOI 10.1016/bs.hna.2019.08.001 .

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# Hierarchical adaptive sparse grids and quasi Monte Carlo for option pricing under the rough Bergomi model

*Authors*

- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Hammouda, Chiheb Ben
- Tempone, Raúl F.

*2010 Mathematics Subject Classification*

- 91G60 91G20 65C05

*Keywords*

- Rough volatility, Monte Carlo, adaptive sparse grids, quasi Monte Carlo, Brownian bridge construction, Richardson extrapolation

*DOI*

*Abstract*

The rough Bergomi (rBergomi) model, introduced recently in [4], is a promising rough volatility model in quantitative finance. It is a parsimonious model depending on only three parameters, and yet exhibits remarkable fit to empirical implied volatility surfaces. In the absence of analytical European option pricing methods for the model, and due to the non-Markovian nature of the fractional driver, the prevalent option is to use the Monte Carlo (MC) simulation for pricing. Despite recent advances in the MC method in this context, pricing under the rBergomi model is still a timeconsuming task. To overcome this issue, we design a novel, hierarchical approach, based on i) adaptive sparse grids quadrature (ASGQ), and ii) quasi Monte Carlo (QMC). Both techniques are coupled with Brownian bridge construction and Richardson extrapolation. By uncovering the available regularity, our hierarchical methods demonstrate substantial computational gains with respect to the standard MC method, when reaching a sufficiently small relative error tolerance in the price estimates across different parameter constellations, even for very small values of the Hurst parameter. Our work opens a new research direction in this field, i.e., to investigate the performance of methods other than Monte Carlo for pricing and calibrating under the rBergomi model.

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# Computational modelling and simulation of cancer growth and migration within a 3D heterogeneous tissue: The effects of fibre and vascular structure

*Authors*

- Macnamara, Cicely K.
- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Ramis-Conde, Ignacio
- Chaplain, Mark A. J.

*2010 Mathematics Subject Classification*

- 65P99 65P99 92B05

*Keywords*

- Cancer modelling, individual-based model, cell-matrix interaction, vasculature, finite element method

*DOI*

*Abstract*

The term cancer covers a multitude of bodily diseases, broadly categorised by having cells which do not behave normally. Since cancer cells can arise from any type of cell in the body, cancers can grow in or around any tissue or organ making the disease highly complex. Our research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modeling. We present a 3D individual-based model which allows one to simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent (a single cell, for example) is fully realised within the model and interactions are primarily governed by mechanical forces between elements. However, as well as the mechanical interactions we also consider chemical interactions, for example, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells. The current state of the art of the model allows us to simulate tumour growth around an arbitrary blood-vessel network or along the striations of fibrous tissue.

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# Existence and uniqueness of solution for multidimensional parabolic PDAEs arising in semiconductor modeling

*Authors*

- Alì, Giuseppe
- Rotundo, Nella

*2010 Mathematics Subject Classification*

- 35K40 35K55 35K51 82D37 94C05

*Keywords*

- Existence and uniqueness of solution, multidimensional parabolic PDAEs, network and semiconductor coupling model

*DOI*

*Abstract*

This paper concerns with a compact network model combined with distributed models for semiconductor devices. For linear RLC networks containing distributed semiconductor devices, we construct a mathematical model that joins the differential-algebraic initial value problem for the electric circuit with multi-dimensional parabolic-elliptic boundary value problems for the devices. We prove an existence and uniqueness result, and the asymptotic behavior of this mixed initial boundary value problem of partial differential-algebraic equations.

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# On decomposition of embedded prismatoids in $R^3$ without additional points

*Authors*

- Si, Hang

*2010 Mathematics Subject Classification*

- 65D18 68U05 65M50 65N50

*Keywords*

- Twisted prisms, twisted prismatoids, torus knots, indecomposable polyhedra, Steiner points, Schönhardt polyhedron, Rambau polyhedron

*DOI*

*Abstract*

This paper considers three-dimensional *prismatoids* which can be embedded in ℝ³ A subclass of this family are *twisted prisms*, which includes the family of non-triangulable Scönhardt polyhedra [12, 10]. We call a prismatoid *decomposable* if it can be cut into two smaller prismatoids (which have smaller volumes) without using additional points. Otherwise it is *indecomposable*. The indecomposable property implies the non-triangulable property of a prismatoid but not vice versa.

In this paper we prove two basic facts about the decomposability of embedded prismatoid in ℝ³ with convex bases. Let *P* be such a prismatoid, call an edge *interior edge* of *P* if its both endpoints are vertices of *P* and its interior lies inside *P*. Our first result is a condition to characterise indecomposable twisted prisms. It states that a twisted prism is indecomposable without additional points if and only if it allows no interior edge. Our second result shows that any embedded prismatoid in ℝ³ with convex base polygons can be decomposed into the union of two sets (one of them may be empty): a set of tetrahedra and a set of indecomposable twisted prisms, such that all elements in these two sets have disjoint interiors.

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# Exploring families of energy-dissipation landscapes via tilting -- Three types of EDP convergence

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Montefusco, Alberto
- Peletier, Mark A.

*2010 Mathematics Subject Classification*

- 49S05 47J30 35K55 35A15 49J45

*Keywords*

- Gradient systems, energy-dissipation principle, wiggly-energy model, wiggly-dissipation model, simple EDP convergence, EDP convergence with tilting, contact EDP convergence with tilting

*DOI*

*Abstract*

This paper revolves around a subtle distinction between two concepts: passing to the limit in a family of gradient systems, on one hand, and deriving effective kinetic relations on the other. The two concepts are strongly related, and in many examples they even appear to be the same. Our main contributions are to show that they are different, to show that well-known techniques developed for the former may give incorrect results for the latter, and to introduce new tools to remedy this. The approach is based on the Energy-Dissipation Principle that provides a variational formulation to gradient-flow equations that allows one to apply techniques from Γ-convergence of functional on states and functionals on trajectories.

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# Lower large deviations for geometric functionals

*Authors*

- Hirsch, Christian
- Jahnel, Benedikt

ORCID: 0000-0002-4212-0065 - Tóbiás, András

*2010 Mathematics Subject Classification*

- 60K35 60F10 82C22

*Keywords*

- large deviations, lower tails, stabilizing functionals, random geometric graph, κ-nearest neighbor graph, relative neighborhood graph, Voronoi tessellation, clique count

*DOI*

*Abstract*

This work develops a methodology for analyzing large-deviation lower tails associated with geometric functionals computed on a homogeneous Poisson point process. The technique applies to characteristics expressed in terms of stabilizing score functions exhibiting suitable monotonicity properties. We apply our results to clique counts in the random geometric graph, intrinsic volumes of Poisson--Voronoi cells, as well as power-weighted edge lengths in the random geometric, κ-nearest neighbor and relative neighborhood graph.

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# An existence result for a class of electrothermal drift-diffusion models with Gauss--Fermi statistics for organic semiconductors

*Authors*

- Glitzky, Annegret
- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Nika, Grigor

*2010 Mathematics Subject Classification*

- 35J57 35R05 35D05 78A35 80A20 82A57

*Keywords*

- Drift-diffusion system, heat flow, charge and heat transport, entropy solutions, existence of weak solutions, organic semiconductor, Gauss--Fermi statistics

*DOI*

*Abstract*

This work is concerned with the analysis of a drift-diffusion model for the electrothermal behavior of organic semiconductor devices. A "generalized Van Roosbroeck'' system coupled to the heat equation is employed, where the former consists of continuity equations for electrons and holes and a Poisson equation for the electrostatic potential, and the latter features source terms containing Joule heat contributions and recombination heat. Special features of organic semiconductors like Gauss--Fermi statistics and mobilities functions depending on the electric field strength are taken into account. We prove the existence of solutions for the stationary problem by an iteration scheme and Schauder's fixed point theorem. The underlying solution concept is related to weak solutions of the Van Roosbroeck system and entropy solutions of the heat equation. Additionally, for data compatible with thermodynamic equilibrium, the uniqueness of the solution is verified. It was recently shown that self-heating significantly influences the electronic properties of organic semiconductor devices. Therefore, modeling the coupled electric and thermal responses of organic semiconductors is essential for predicting the effects of temperature on the overall behavior of the device. This work puts the electrothermal drift-diffusion model for organic semiconductors on a sound analytical basis.

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# Design and testing of 3D-printed micro-architectured polymer materials exhibiting a negative Poisson's ratio

*Authors*

- Agnelli, Filippo
- Constantinescu, Andrei
- Nika, Grigor

*2010 Mathematics Subject Classification*

- 35M10 35M12 35M30 35Q93 74B05

*Keywords*

- Auxetic material, topology optimization, 3D printing, polymer

*DOI*

*Abstract*

This work proposes the complete design cycle for several auxetic materials where the cycle consists of three steps (i) the design of the micro-architecture, (ii) the manufacturing of the material and (iii) the testing of the material. In more precise terms, we aim to obtain domain micro-architectured materials with a prescribed elasticity tensor and Poisson's ratio. In order to reach this goal we use topology optimization via the level set method for the material design process. Specimens are manufactured using a commercial stereo-lithography Ember printer and mechanically tested. The observed displacement and strain fields during tensile testing obtained by digital image correlation match the predictions from the FE simulation.

*Appeared in*

- Contin. Mech. Thermodyn., (2019), pp. published online on 20.11.2019 (1--17), DOI 10.1007/s00161-019-00851-6 .

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# On the feasibility of using open source solvers for the simulation of a turbulent air flow in a dairy barn

*Authors*

- Janke, David
- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Ahmed, Naveed

ORCID: 0000-0002-9322-0373 - Alia, Najib
- Knoth, Oswald
- Moreau, Baptiste
- Wilbrandt, Ulrich
- Willink, Dilya
- Amon, Thomas
- John, Volker

ORCID: 0000-0002-2711-4409

*2010 Mathematics Subject Classification*

- 76F65

*Keywords*

- naturally ventilated barns, turbulent flows, wind tunnel model, experimental data, computational fluid dynamics, turbulence modeling, open source software

*DOI*

*Abstract*

Two transient open source solvers, OpenFOAM and ParMooN, are assessed with respect to the simulation of the turbulent air flow inside and around a dairy barn. For this purpose, data were obtained in an experimental campaign at a *1:100* scaled wind tunnel model. Both solvers used different meshes, discretization schemes, and turbulence models. The experimental data and numerical results agree well for time-averaged stream-wise and vertical-wise velocities. In particular, the air exchange was predicted with high accuracy by both solvers with relative errors less than 5 % compared to the experimental results. With respect to the turbulent quantities, good agreements at the second (downwind) half of the barn inside and especially outside the barn could be achieved, where both codes accurately predicted the flow separation and the root-mean-square velocities. Deviations between simulations and experimental results regarding turbulent quantities could be observed in the first part of the barn, due to different inlet conditions between the experimental setup and the numerical simulations. Both solvers proved to be promising tools for the accurate prediction of time-dependent phenomena in an agricultural context, e.g., like the transport of particulate matter or pathogen-laden aerosols in and around agricultural buildings.

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# Comparison of monomorphic and polymorphic approaches for uncertainty quantification with experimental investigations

*Authors*

- Drieschner, Martin
- Eigel, Martin
- Gruhlke, Robert
- Hömberg, Dietmar
- Petryna, Yuri

*2010 Mathematics Subject Classification*

- 35R60 47B80 60H35 65C20 65N12 65N22 65N55 65J10

*Keywords*

- Structural failure, experimental and numerical investigations, monomorphic uncertainty modeling, polymorphic uncertainty modeling, artificial neural networks, fuzzy, uncertainty quantification

*DOI*

*Abstract*

Unavoidable uncertainties due to natural variability, inaccuracies, imperfections or lack of knowledge are always present in real world problems. To take them into account within a numerical simulation, the probability, possibility or fuzzy set theory as well as a combination of these are potentially usable for the description and quantification of uncertainties. In this work, different monomorphic and polymorphic uncertainty models are applied on linear elastic structures with non-periodic perforations in order to analyze the individual usefulness and expressiveness. The first principal stress is used as an indicator for structural failure which is evaluated and classified. In addition to classical sampling methods, a surrogate model based on artificial neural networks is presented. With regard to accuracy, efficiency and resulting numerical predictions, all methods are compared and assessed with respect to the added value. Real experiments of perforated plates under uniaxial tension are validated with the help of the different uncertainty models.

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# Dynamical regimes in a class A model of a nonlinear mirror mode-locked laser

*Authors*

- Vladimirov, Andrei G.
- Kovalev, Anton V.
- Viktorov, Evgeny A.
- Rebrova, Natalia
- Huyet, Guillaume

*2010 Mathematics Subject Classification*

- 78A60 78M35

*2008 Physics and Astronomy Classification Scheme*

- 42.65.Sf 42.65.Re 42.60.Fc

*Keywords*

- Nonlinear mirror mode-locked lasers, delay differential equations, stability analysis, modulational instability, square waves, cavity solitons

*DOI*

*Abstract*

Using a simple delay differential equation model we study theoretically the dynamics of a unidirectional class-A ring laser with a nonlinear amplifying loop mirror. We perform analytical linear stability analysis of the CW regimes in the large delay limit and demonstrate that these regimes can be destabilized via modulational and Turing-type instabilities, as well as by a bifurcation leading to the appearance of square-waves. We investigate the formation of square-waves and mode-locked pulses in the system. We show that mode-locked pulses are very asymmetric with exponential decay of the trailing and superexponential growth of the leading edge. We discuss asymmetric interaction of these pulses leading to a formation of harmonic mode-locked regimes.

*Appeared in*

- Phys. Rev. E, 100 (2019), pp. 012216/1--012216/7, DOI 10.1103/PhysRevE.100.012216 .

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# Surface energy and boundary layers for a chain of atoms at low temperature

*Authors*

- Jansen, Sabine
- König, Wolfgang

ORCID: 0000-0002-4212-0065 - Schmidt, Bernd
- Theil, Florian

*2010 Mathematics Subject Classification*

- 82B21 74B20 74G65 60F10

*Keywords*

- Atomistic models of elasticity, surface energy and boundary layers, semi-classical limit of transfer operators, uniform decay of correlations, path large deviations for, stationary processes

*DOI*

*Abstract*

We analyze the surface energy and boundary layers for a chain of atoms at low temperature for an interaction potential of Lennard-Jones type. The pressure (stress) is assumed small but positive and bounded away from zero, while the temperature goes to zero. Our main results are: (1) As the temperature goes to zero and at fixed positive pressure, the Gibbs measures for infinite chains and semi-infinite chains satisfy path large deviations principles. The rate functions are bulk and surface energy functionals. The minimizer of the surface functional corresponds to zero temperature boundary layers. (2) The surface correction to the Gibbs free energy converges to the zero temperature surface energy, characterized with the help of the minimum of the surface energy functional. (3) The bulk Gibbs measure and Gibbs free energy can be approximated by their Gaussian counterparts. (4) Bounds on the decay of correlations are provided, some of them uniform in the inverse temperature.

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# Simulation and control of a nonsmooth Cahn--Hilliard Navier--Stokes system with variable fluid densities

*Authors*

- Gräßle, Carmen
- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Hinze, Michael
- Keil, Tobias

*2010 Mathematics Subject Classification*

- 35B65 35J87 35K55 65K15 49K20

*Keywords*

- Two phase flow, POD model order reduction, adaptivity, nonsmooth systems, mathematical programming with equilibrium constraints, optimal control

*DOI*

*Abstract*

We are concerned with the simulation and control of a two phase flow model governed by a coupled Cahn--Hilliard Navier--Stokes system involving a nonsmooth energy potential.We establish the existence of optimal solutions and present two distinct approaches to derive suitable stationarity conditions for the bilevel problem, namely C- and strong stationarity. Moreover, we demonstrate the numerical realization of these concepts at the hands of two adaptive solution algorithms relying on a specifically developed goal-oriented error estimator.In addition, we present a model order reduction approach using proper orthogonal decomposition (POD-MOR) in order to replace high-fidelity models by low order surrogates. In particular, we combine POD with space-adapted snapshots and address the challenges which are the consideration of snapshots with different spatial resolutions and the conservation of a solenoidal property.

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# Low rank surrogates for polymorphic fields with application to fuzzy-stochastic partial differential equations

*Authors*

- Eigel, Martin
- Grasedyck, Lars
- Gruhlke, Robert
- Moser, Dieter

*2010 Mathematics Subject Classification*

- 15A69 35R13 35R60 60H35 65C20 65N12 65N22 65J10 74B05 97N50

*Keywords*

- Fuzzy-stochastic partial differential equations, possibility, polymorphic uncertainty modeling, uncertainty quantification, low-rank hierachical tensor formats, parameteric partial differential equations, polymorphic domain

*DOI*

*Abstract*

We consider a general form of fuzzy-stochastic PDEs depending on the interaction of probabilistic and non-probabilistic ("possibilistic") influences. Such a combined modelling of aleatoric and epistemic uncertainties for instance can be applied beneficially in an engineering context for real-world applications, where probabilistic modelling and expert knowledge has to be accounted for. We examine existence and well-definedness of polymorphic PDEs in appropriate function spaces. The fuzzy-stochastic dependence is described in a high-dimensional parameter space, thus easily leading to an exponential complexity in practical computations. To aleviate this severe obstacle in practise, a compressed low-rank approximation of the problem formulation and the solution is derived. This is based on the Hierarchical Tucker format which is constructed with solution samples by a non-intrusive tensor reconstruction algorithm. The performance of the proposed model order reduction approach is demonstrated with two examples. One of these is the ubiquitous groundwater flow model with Karhunen-Loeve coefficient field which is generalized by a fuzzy correlation length.

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# Variable step mollifiers and applications

*Authors*

- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Papafitsoros, Kostas

ORCID: 0000-0001-9691-4576 - Rautenberg, Carlos N.

ORCID: 0000-0001-9497-9296

*2010 Mathematics Subject Classification*

- 46E35 42B20 49J40 26A45

*Keywords*

- Mollification, integral operators, boundary values, density of convex sets

*DOI*

*Abstract*

We consider a mollifying operator with variable step that, in contrast to the standard mollification, is able to preserve the boundary values of functions. We prove boundedness of the operator in all basic Lebesgue, Sobolev and BV spaces as well as corresponding approximation results. The results are then applied to extend recently developed theory concerning the density of convex intersections.

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# Rough nonlocal diffusions

*Authors*

- Coghi, Michele

ORCID: 0000-0002-4198-0856 - Nilssen, Torstein

*2010 Mathematics Subject Classification*

- 60H05 60H15 35K55

*Keywords*

- Rough paths, stochastic PDEs, McKean-Vlasov, non-local equations

*DOI*

*Abstract*

We consider a nonlinear Fokker-Planck equation driven by a deterministic rough path which describes the conditional probability of a McKean-Vlasov diffusion with "common" noise. To study the equation we build a self-contained framework of non-linear rough integration theory which we use to study McKean-Vlasov equations perturbed by rough paths. We construct an appropriate notion of solution of the corresponding Fokker-Planck equation and prove well-posedness.

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# Global-in-time existence for liquid mixtures subject to a generalised incompressibility constraint

*Authors*

- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35M33 35Q30 76N10 76D05 35D30 35B65

*Keywords*

- Multicomponent flow, complex fluid, fluid mixture, incompressible fluid, low Mach-number, mathematical analysis, global weak solutions, hard-sphere pressure law, defect measure

*DOI*

*Abstract*

We consider a system of partial differential equations describing diffusive and convective mass transport in a fluid mixture of N > 1 chemical species. A weighted sum of the partial mass densities of the chemical species is assumed to be constant, which expresses the incompressibility of the fluid, while accounting for different reference sizes of the involved molecules. This condition is different from the usual assumption of a constant total mass density, and it leads in particular to a non-solenoidal velocity field in the Navier-Stokes equations. In turn, the pressure gradient occurs in the diffusion fluxes, so that the PDE-system of mass transport equations and momentum balance is fully coupled. Another striking feature of such incompressible mixtures is the algebraic formula connecting the pressure and the densities, which can be exploited to prove a pressure bound in L^{1}. In this paper, we consider incompressible initial states with bounded energy and show the global existence of weak solutions with defect measure.

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# Mass transport in multicomponent compressible fluids: Local and global well-posedness in classes of strong solutions for general class-one models

*Authors*

- Bothe, Dieter
- Druet, Pierre-Étienne

ORCID: 0000-0001-5303-0500

*2010 Mathematics Subject Classification*

- 35M33 35Q30 76N10, 35D35, 35B65, 35B35, 35K57, 35Q35, 35Q79, 76R50, 80A17, 80A32, 92E20

*Keywords*

- Multicomponent flow, fluid mixture, compressible fluid, diffusion, reactive fluid, well-posedness analysis, strong solutions

*DOI*

*Abstract*

We consider a system of partial differential equations describing mass transport in a multicomponent isothermal compressible fluid. The diffusion fluxes obey the Fick-Onsager or Maxwell- Stefan closure approach. Mechanical forces result into one single convective mixture velocity, the barycentric one, which obeys the Navier-Stokes equations. The thermodynamic pressure is defined by the Gibbs-Duhem equation. Chemical potentials and pressure are derived from a thermodynamic potential, the Helmholtz free energy, with a bulk density allowed to be a general convex function of the mass densities of the constituents. The resulting PDEs are of mixed parabolic-hyperbolic type. We prove two theoretical results concerning the well-posedness of the model in classes of strong solutions: 1. The solution always exists and is unique for short-times and 2. If the initial data are sufficiently near to an equilibrium solution, the well-posedness is valid on arbitrary large, but finite time intervals. Both results rely on a contraction principle valid for systems of mixed type that behave like the compressible Navier- Stokes equations. The linearised parabolic part of the operator possesses the self map property with respect to some closed ball in the state space, while being contractive in a lower order norm only. In this paper, we implement these ideas by means of precise a priori estimates in spaces of exact regularity.

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# Lower and upper bounds for the number of limit cycles on a cylinder

*Authors*

- Schneider, Klaus R.
- Grin, Alexander

*2010 Mathematics Subject Classification*

- 34C05 34C07

*Keywords*

- Autonomous systems with cylindrical phase space, location and number of limit cycles of second kind, Dulac--Cherkas function, factorized Dulac function

*DOI*

*Abstract*

We consider autonomous systems with cylindrical phase space. Lower and upper bounds for the number of limit cycles surrounding the cylinder can be obtained by means of an appropriate Dulac-Cherkas function. We present different possibilities to improve these bounds including the case that the exact number of limit cycles can be determined. These approaches are based on the use of several Dulac-Cherkas functions or on applying some factorized Dulac function.

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# Bilinear coagulation equations

*Authors*

- Heydecker, Daniel
- Patterson, Robert I. A.

*2010 Mathematics Subject Classification*

- 60K35 82C2 82C26

*Keywords*

- Coagulation, bilinear kernel, gelation, phase transition, Smoluchowski equation,, Flory equation, random graph

*DOI*

*Abstract*

We consider coagulation equations of Smoluchowski or Flory type where the total merge rate has a bilinear form π(y) · Aπ (x) for a vector of conserved quantities π, generalising the multiplicative kernel. For these kernels, a gelation transition occurs at a finite time t_{g} ∈ (0,∞), which can be given exactly in terms of an eigenvalue problem in finite dimensions. We prove a hydrodynamic limit for a stochastic coagulant, including a corresponding phase transition for the largest particle, and exploit a coupling to random graphs to extend analysis of the limiting process beyond the gelation time.

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# Coarse-graining via EDP-convergence for linear fast-slow reaction systems

*Authors*

- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Stephan, Artur

*2010 Mathematics Subject Classification*

- 49S05 47D07 47J30 92E20 60J20

*Keywords*

- Markov process with detailed balance, linear reaction system, entropic gradient structure, energy-dissipation balance, EDP-convergence, microscopic equilibrium, coarse graining, reconstruction operators

*DOI*

*Abstract*

We consider linear reaction systems with slow and fast reactions, which can be interpreted as master equations or Kolmogorov forward equations for Markov processes on a finite state space. We investigate their limit behavior if the fast reaction rates tend to infinity, which leads to a coarse-grained model where the fast reactions create microscopically equilibrated clusters, while the exchange mass between the clusters occurs on the slow time scale. Assuming detailed balance the reaction system can be written as a gradient flow with respect to the relative entropy. Focusing on the physically relevant cosh-type gradient structure we show how an effective limit gradient structure can be rigorously derived and that the coarse-grained equation again has a cosh-type gradient structure. We obtain the strongest version of convergence in the sense of the Energy-Dissipation Principle (EDP), namely EDP-convergence with tilting.

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# Orthogonality of fluxes in general nonlinear reaction networks

*Authors*

- Renger, D. R. Michiel

ORCID: 0000-0003-3557-3485 - Zimmer, Johannes

*2010 Mathematics Subject Classification*

- 60J27 80A30 82C22 82C35

*Keywords*

- Chemical reactions, reaction fluxes, large deviations, orthogonality

*DOI*

*Abstract*

We consider the chemical reaction networks and study currents in these systems. Reviewing recent decomposition of rate functionals from large deviation theory for Markov processes, we adapt these results for reaction networks. In particular, we state a suitable generalisation of orthogonality of forces in these systems, and derive an inequality that bounds the free energy loss and Fisher information by the rate functional.

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# The role of the self-steepening effect in soliton compression due to cross-phase modulation by dispersive waves

*Authors*

- Pickartz, Sabrina

*2010 Mathematics Subject Classification*

- 78A60 78M22

*2008 Physics and Astronomy Classification Scheme*

- 42.65.-k 42.81.Dp 42.65.Re 05.45.Yv

*Keywords*

- Ultrashort pulses, pulse compression, soliton, optical event horizons

*DOI*

*Abstract*

We consider the compression and amplification of an ultrashort soliton pulse through the interaction with a weaker velocity-matched dispersive wave, in the so-called optical event horizon regime. We demonstrate that in this interaction scheme the self-steepening effect plays the key role in producing a strong soliton compression. While the interaction between the two pulses is mediated through cross phase modulation, the self-steepening effect produces an energy exchange, which enhances soliton compression. We provide numerical results and an analytical expression for energy transfer and compression rate.

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# Reconstruction of quasi-local numerical effective models from low-resolution measurements

*Authors*

- Caiazzo, Alfonso

ORCID: 0000-0002-7125-8645 - Maier, Roland
- Peterseim, Daniel

*2010 Mathematics Subject Classification*

- 65N21 65N30 74Q05

*Keywords*

- Inverse problem, multiscale methods, numerical homogenization

*DOI*

*Abstract*

We consider the inverse problem of reconstructing an effective model for a prototypical diffusion process in strongly heterogeneous media based on low-resolution measurements. We rely on recent quasi-local numerical effective models that, in contrast to conventional homogenized models, are provably reliable beyond periodicity assumptions and scale separation. The goal of this work is to show that the identification of the matrix representation of these effective models is possible. Algorithmic aspects of the inversion procedure and its performance are illustrated in a series of numerical experiments.

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# The geometry of the space of branched rough paths

*Authors*

- Tapia, Nikolas
- Zambotti, Lorenzo

*2010 Mathematics Subject Classification*

- 60H10 16T05

*Keywords*

- Rough paths, Hopf algebras, renormalization

*DOI*

*Abstract*

We construct an explicit transitive free action of a Banach space of Hölder functions on the space of branched rough paths, which yields in particular a bijection between theses two spaces. This endows the space of branched rough paths with the structure of a principal homogeneous space over a Banach space and allows to characterize its automorphisms. The construction is based on the Baker-Campbell-Hausdorff formula, on a constructive version of the Lyons-Victoir extension theorem and on the Hairer-Kelly map, which allows to describe branched rough paths in terms of anisotropic geometric rough paths.

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# Multiscale modeling of magnetorheological suspensions

*Authors*

- Nika, Grigor
- Vernescu, Bogdan

*2010 Mathematics Subject Classification*

- 35M10 35M12 35M30 76D07 76T20

*Keywords*

- Magnetorheological fluids, homogenizatio, chain structures, poiseuille, couette

*DOI*

*Abstract*

We develop a multiscale approach to describe the behavior of a suspension of solid magnetizable particles in a viscous non-conducting fluid in the presence of an externally applied magnetic field. By upscaling the quasi-static Maxwell equations coupled with the Stokes' equations we are able to capture the magnetorheological effect. The model we obtain generalizes the one introduced by Neuringer & Rosensweig for quasistatic phenomena. We derive the macroscopic constitutive properties explicitly in terms of the solutions of local problems. The effective coefficients have a nonlinear dependence on the volume fraction when chain structures are present. The velocity profiles computed for some simple flows, exhibit an apparent yield stress and the flowprofile resembles a Bingham fluid flow.

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# Unipolar drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices

*Authors*

- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Doan, Duy Hai
- Glitzky, Annegret
- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Nika, Grigor

*2010 Mathematics Subject Classification*

- 65M08 35J92 80M12 80A20

*Keywords*

- Non-isothermal drift-diffusion, organic semiconductors, finite volumes, generalized Scharfetter--Gummel scheme, path following

*DOI*

*Abstract*

We discretize a unipolar electrothermal drift-diffusion model for organic semiconductor devices with Gauss--Fermi statistics and charge carrier mobilities having positive temperature feedback. We apply temperature dependent Ohmic contact boundary conditions for the electrostatic potential and use a finite volume based generalized Scharfetter-Gummel scheme. Applying path-following techniques we demonstrate that the model exhibits S-shaped current-voltage curves with regions of negative differential resistance, only recently observed experimentally.

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# Mathematical modeling of semiconductors: From quantum mechanics to devices

*Authors*

- Kantner, Markus
- Mielke, Alexander

ORCID: 0000-0002-4583-3888 - Mittnenzweig, Markus
- Rotundo, Nella

*2010 Mathematics Subject Classification*

- 35K57 80M30 81S22, 82D37

*Keywords*

- Semiconductor modeling, drift-diffusion system, open quantum system,, Lindblad operator, reaction-diffusion systems, detailed balance condition, gradient structure, thermodynamically consistent coupling

*DOI*

*Abstract*

We discuss recent progress in the mathematical modeling of semiconductor devices. The central result of this paper is a combined quantum-classical model that self-consistently couples van Roosbroeck's drift-diffusion system for classical charge transport with a Lindblad-type quantum master equation. The coupling is shown to obey fundamental principles of non-equilibrium thermodynamics. The appealing thermodynamic properties are shown to arise from the underlying mathematical structure of a damped Hamitlonian system, which is an isothermal version of so-called GENERIC systems. The evolution is governed by a Hamiltonian part and a gradient part involving a Poisson operator and an Onsager operator as geoemtric structures, respectively. Both parts are driven by the conjugate forces given in terms of the derivatives of a suitable free energy.

*Appeared in*

- Topics in Applied Analysis and Optimisation, M. Hintermüller, J.F. Rodrigues, eds., CIM Series in Mathematical Sciences, Springer Nature Switzerland AG, Cham, 2019, pp. 269--293, DOI 10.1007/978-3-030-33116-0 .

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# Maximal dissipative solutions for incompressible fluid dynamics

*Authors*

- Lasarzik, Robert

*2010 Mathematics Subject Classification*

- 35D99 35Q30 76D05 76N10

*Keywords*

- Existence, uniqueness, Navier--Stokes, Euler, incompressible, fluid dynamics

*DOI*

*Abstract*

We introduce the new concept of maximal dissipative solutions for the Navier--Stokes and Euler equations and show that these solutions exist and the solution set is closed and convex. The concept of maximal dissipative solutions coincides with the concept of weak solutions as long as the weak solutions inherits enough regularity to be unique. A maximal dissipative solution is defined as the minimizer of a convex functional and we argue that this definition bears several advantages.

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# The link between coherence echoes and mode locking

*Authors*

- Eydam, Sebastian
- Wolfrum, Matthias

*2010 Mathematics Subject Classification*

- 34C15 37N20

*2008 Physics and Astronomy Classification Scheme*

- 42.60.Fc 05.45.Xt 05.45.-a

*Keywords*

- phase oscillators, mode-locking

*DOI*

*Abstract*

We investigate the appearance of sharp pulses in the mean field of Kuramoto-type globally- coupled phase oscillator systems. In systems with exactly equidistant natural frequencies self- organized periodic pulsations of the mean field, called mode locking, have been described re- cently as a new collective dynamics below the synchronization threshold. We show here that mode locking can appear also for frequency combs with modes of finite width, where the natu- ral frequencies are randomly chosen from equidistant frequency intervals. In contrast to that, so called coherence echoes, which manifest themselves also as pulses in the mean field, have been found in systems with completely disordered natural frequencies as the result of two consecutive stimulations applied to the system. We show that such echo pulses can be explained by a stimula- tion induced mode locking of a subpopulation representing a frequency comb. Moreover, we find that the presence of a second harmonic in the interaction function, which can lead to the global stability of the mode-locking regime for equidistant natural frequencies, can enhance the echo phenomenon significantly. The non-monotonous behavior of echo amplitudes can be explained as a result of the linear dispersion within the self-organized mode-locked frequency comb. Fi- nally we investigate the effect of small periodic stimulations on oscillator systems with disordered natural frequencies and show how the global coupling can support the stimulated pulsation by increasing the width of locking plateaus.

*Appeared in*

- Chaos, 29 (2019), published online on 08.10.2019, DOI 10.1063/1.5114699 .

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# The sharp-interface limit for the Navier--Stokes--Korteweg equations

*Authors*

- Abels, Helmut
- Daube, Johannes
- Kraus, Christiane
- Kröner, Dietmar

*2010 Mathematics Subject Classification*

- 35B40 76T10 35Q35, 35R35

*Keywords*

- Sharp-interface limit, diffuse-interface model, liquid-vapour flow, Navier--Stokes--Korteweg system, free-boundary problem.

*DOI*

*Abstract*

We investigate the sharp-interface limit for the Navier--Stokes--Korteweg model, which is an extension of the compressible Navier--Stokes equations. By means of compactness arguments, we show that solutions of the Navier--Stokes--Korteweg equations converge to solutions of a physically meaningful free-boundary problem. Assuming that an associated energy functional converges in a suitable sense, we obtain the sharp-interface limit at the level of weak solutions.

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# Global bifurcation analysis of limit cycles for a generalized van der Pol system

*Authors*

- Schneider, Klaus R.
- Grin, Alexander

*2010 Mathematics Subject Classification*

- 34C05 34C23 34E15

*Keywords*

- Global bifurcation analysis, generalized van der Pol system, Dulac--Cherkas function, rotated vector field, singularly perturbed system

*DOI*

*Abstract*

We present a new approach for the global bifurcation analysis of limit cycles for a generalized van der Pol system. It is based on the existence of a Dulac-Cherkas function and on applying two topologically equivalent systems: one of them is a rotated vector field, the other one is a singularly perturbed system.

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# Pricing American options by exercise rate optimization

*Authors*

- Bayer, Christian

ORCID: 0000-0002-9116-0039 - Tempone , Raúl F.
- Wolfers, Sören

*2010 Mathematics Subject Classification*

- 91G60 91G20 49M20

*Keywords*

- Computational finance, American option pricing, stochastic optimization problem, Monte Carlo, multivariate approximation, rough volatility

*DOI*

*Abstract*

We present a novel method for the numerical pricing of American options based on Monte Carlo simulation and the optimization of exercise strategies. Previous solutions to this problem either explicitly or implicitly determine so-called optimal exercise regions, which consist of points in time and space at which a given option is exercised. In contrast, our method determines the exercise rates of randomized exercise strategies. We show that the supremum of the corresponding stochastic optimization problem provides the correct option price. By integrating analytically over the random exercise decision, we obtain an objective function that is differentiable with respect to perturbations of the exercise rate even for finitely many sample paths. The global optimum of this function can be approached gradually when starting from a constant exercise rate. Numerical experiments on vanilla put options in the multivariate Black-Scholes model and a preliminary theoretical analysis underline the efficiency of our method, both with respect to the number of time-discretization steps and the required number of degrees of freedom in the parametrization of the exercise rates. Finally, we demonstrate the flexibility of our method through numerical experiments on max call options in the classical Black-Scholes model, and vanilla put options in both the Heston model and the non-Markovian rough Bergomi model.

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# Drift-diffusion simulation of S-shaped current-voltage relations for organic semiconductor devices

*Authors*

- Doan, Duy Hai
- Fischer, Axel
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Glitzky, Annegret
- Liero, Matthias

ORCID: 0000-0002-0963-2915

*2010 Mathematics Subject Classification*

- 65M08 35J92 80M12 80A2

*Keywords*

- Non-isothermal drift-diffusion, organic semiconductors, finite volumes, generalized Scharfetter-Gummel scheme, path following

*DOI*

*Abstract*

We present an electrothermal drift-diffusion model for organic semiconductor devices with Gauss-Fermi statistics and positive temperature feedback for the charge carrier mobilities. We apply temperature dependent Ohmic contact boundary conditions for the electrostatic potential and discretize the system by a finite volume based generalized Scharfetter-Gummel scheme. Using path-following techniques we demonstrate that the model exhibits S-shaped current-voltage curves with regions of negative differential resistance, which were only recently observed experimentally.

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# Modeling polycrystalline electrode-electrolyte interfaces: The differential capacitance

*Authors*

- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Landstorfer, Manuel
- Müller, Rüdiger

ORCID: 0000-0003-2643-722X

*2010 Mathematics Subject Classification*

- 65N30 78A57 80A17

*Keywords*

- Electrochemistry, double layer, polycrystal, finite-elements, stochastic representation

*DOI*

*Abstract*

We present and analyze a model for polycrystalline electrode surfaces based on an improved continuum model that takes finite ion size and solvation into account. The numerical simulation of finite size facet patterns allows to study two limiting cases: While for facet size diameter $d^facet to 0$ we get the typical capacitance of a spatially homogeneous but possible amorphous or liquid surface, in the limit $L^Debye << d^facet$ , an ensemble of non-interacting single crystal surfaces is approached. Already for moderate size of the facet diameters, the capacitance is remarkably well approximated by the classical approach of adding the single crystal capacities of the contributing facets weighted by their respective surface fraction. As a consequence, the potential of zero charge is not necessarily attained at a local minimum of capacitance, but might be located at a local capacitance maximum instead. Moreover, the results show that surface roughness can be accurately taken into account by multiplication of the ideally flat polycrystalline surface capacitance with a single factor. In particular, we find that the influence of the actual geometry of the facet pattern in negligible and our theory opens the way to a stochastic description of complex real polycrystal surfaces.

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# Simulation and design of a compact GaAs based tunable dual-wavelength diode laser system

*Authors*

- Koester, Jan-Philipp
- Radziunas, Mindaugas
- Zeghuzi, Anissa
- Wenzel, Hans
- Knigge, Andrea

*2010 Mathematics Subject Classification*

- 78A60 35Q60 78-04 78A50

*Keywords*

- Tunable laser diodes, dual-wavelength laser, traveling-wave model

*DOI*

*Abstract*

We present our design of a compact, integrated and tunable dual-wavelength diode laser system emitting around 785 nm, which is of interest for several applications like Raman spectroscopy and the generation of THz radiation. To achieve a more compact device compared to previous GaAs based designs two etch depths are realized, leading to shallowly etched ridge waveguides in regions were optical gain is applied and deeply etched waveguides used to enable compact integrated waveguide components. The device parameters are optimized using a numerically efficient simulation tool for passive waveguides. Subsequently, the entire laser system is further analyzed applying a sophisticated traveling-wave equation based model for active devices giving access to internal intensity and carrier density distributions. It is shown that active laser simulations are crucial to deduce critical and performance limiting design aspects not accessible via an all-passive simulation.

*Appeared in*

- Opt. Quantum Electron., 51 (2019), pp. 334/1--334/12, DOI 10.1007/s11082-019-2050-2 .

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# Phase transitions for chase-escape models on Gilbert graphs

*Authors*

- Hinsen, Alexander

ORCID: 0000-0002-4212-0065 - Jahnel, Benedikt

ORCID: 0000-0002-4212-0065 - Cali, Eli
- Wary, Jean-Philippe

*2010 Mathematics Subject Classification*

- 60J25 60K35 60K37

*Keywords*

- Interacting particle systems, random graphs, survival, extinction, percolation, Boolean model

*DOI*

*Abstract*

We present results on phase transitions of local and global survival in a two-species model on Gilbert graphs. At initial time there is an infection at the origin that propagates on the Gilbert graph according to a continuous-time nearest-neighbor interacting particle system. The Gilbert graph consists of susceptible nodes and nodes of a second type, which we call white knights. The infection can spread on susceptible nodes without restriction. If the infection reaches a white knight, this white knight starts to spread on the set of infected nodes according to the same mechanism, with a potentially different rate, giving rise to a competition of chase and escape. We show well-definedness of the model, isolate regimes of global survival and extinction of the infection and present estimates on local survival. The proofs rest on comparisons to the process on trees, percolation arguments and finite-degree approximations of the underlying random graphs.

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# Time-dependent simulation of thermal lensing in high-power broad-area semiconductor lasers

*Authors*

- Zeghuzi, Anissa
- Wünsche, Hans-Jürgen
- Wenzel, Hans
- Radziunas, Mindaugas
- Fuhrmann, Jürgen

ORCID: 0000-0003-4432-2434 - Klehr, Andreas
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Knigge, Andrea

*2010 Mathematics Subject Classification*

- 78A60 35Q60 78-04 70K70 35Q79 81Q37

*Keywords*

- Broad-area laser, heat flow, pulsed laser operation, traveling-wave model, thermal lensing, temperature fluctuation

*DOI*

*Abstract*

We propose a physically realistic and yet numerically applicable thermal model to account for short and long term self-heating within broad-area lasers. Although the temperature increase is small under pulsed operation, a waveguide that is formed within a few-ns-long pulse can result in a transition from a gain-guided to an index-guided structure, leading to near and far field narrowing. Under continuous wave operation the longitudinally varying temperature profile is obtained self-consistently. The resulting unfavorable narrowing of the near field can be successfully counteracted by etching trenches.

*Appeared in*

- IEEE J. Select. Topics Quantum Electron., 25 (2019), 1502310, DOI 10.1109/JSTQE.2019.2925926 .

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# Analysis of a hybrid model for the electrothermal behavior of semiconductor heterostructures

*Authors*

- Glitzky, Annegret
- Liero, Matthias

ORCID: 0000-0002-0963-2915 - Nika, Grigor

*2010 Mathematics Subject Classification*

- 35J57 35K05 35R05 78A35

*Keywords*

- Drift-diffusion, charge & heat transport, electro-thermal interaction, semiconductor heterostructures, hybrid modeling, weak solutions

*DOI*

*Abstract*

We prove existence of a weak solution for a hybrid model for the electro-thermal behavior of semiconductor heterostructures. This hybrid model combines an electro-thermal model based on drift-diffusion with thermistor type models in different subregions of the semiconductor heterostructure. The proof uses a regularization method and Schauder's fixed point theorem. For boundary data compatible with thermodynamic equilibrium we verify, additionally, uniqueness. Moreover, we derive bounds and higher integrability properties for the electrostatic potential and the quasi Fermi potentials as well as the temperature.

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# Stabilization of optical pulse transmission by exploiting fiber nonlinearities

*Authors*

- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Amiranashvili, Shalva

ORCID: 0000-0002-8132-882X - Pickartz, Sabrina

*2010 Mathematics Subject Classification*

- 78A40 78A60 35Q60

*2008 Physics and Astronomy Classification Scheme*

- 42.25.Bs, 42.65.Sf, 42.65.Wi, 42.81.Dp

*Keywords*

- Nonlinear fibers, pulse propagation, pulse manipulation, controlling light by light, solitons, Raman scattering

*DOI*

*Abstract*

We prove theoretically, that the evolution of optical solitons can be dramatically influenced in the course of nonlinear interaction with much smaller group velocity matched pulses. Even weak pump pulses can be used to control the solitons, e.g., to compensate their degradation caused by Raman-scattering.

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# On a thermodynamic framework for developing boundary conditions for Korteweg fluids

*Authors*

- Souček, Ondřej
- Heida, Martin
- Málek, Josef

*2010 Mathematics Subject Classification*

- 00A71 76Txx 76T25

*Keywords*

- Continuum thermodynamics, Korteweg fluid, van der Waals fluid, boundary conditions, diffuse interface, contact angle hysteresis

*DOI*

*Abstract*

We provide a derivation of several classes of boundary conditions for fluids of Korteweg-type using a simple and transparent thermodynamic approach that automatically guarentees that the derived boundary conditions are compatible with the second law of thermodynamics. The starting assumption of our approach is to describe the boundary of the domain as the membrane separating two different continua, one inside the domain, and the other outside the domain. With this viewpoint one may employ the framework of continuum thermodynamics involving singular surfaces. This approach allows us to identify, for various classes of surface Helmholtz free energies, the corresponding surface entropy production mechanisms. By establishing the constitutive relations that guarantee that the surface entropy production is non-negative, we identify a new class of boundary conditions, which on one hand generalizes in a nontrivial manner the Navier's slip boundary conditions, and on the other hand describes dynamic and static contact angle conditions. We explore the general model in detail for a particular case of Korteweg fluid where the Helmholtz free energy in the bulk is that of a van der Waals fluid. We perform a series of numerical experiments to document the basic qualitative features of the novel boundary conditions and their practical applicability to model phenomena such as the contact angle hysteresis.

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# Nucleation chronology and electronic properties of In(As,Sb,P) graded composition quantum dots grown on InAs(100) substrate

*Authors*

- Gambaryan, Karen M.
- Boeck, Torsten
- Trampert, Achim
- Marquardt, Oliver

*2008 Physics and Astronomy Classification Scheme*

- 73.21.La 73.22.Dj 81.10.Dn

*Keywords*

- Quantum dots, electronic properties

*DOI*

*Abstract*

We provide a detailed study of nucleation process, characterization, electronic and optical properties of graded composition quantum dots (GCQDs) grown from In-As-Sb-P composition liquid phase on an InAs(100) substrate in the Stranski-Krastanov growth mode. Our GCQDs exhibit diameters from 10 to 120 nm and heights from 2 to 20 nm with segregation profiles having a maximum Sb content of approximately 20% at the top and a maximum P content of approximately 15% at the bottom of the GCQDs so that hole confinement is expected in the upper parts of the GCQDs. Using an eight-band **k** · **p** model taking strain and built-in electrostatic potentials into account, we have computed the hole ground state energies and charge densities for a wide range of InAs_{1-x-y}Sb_{x}P_{y} GCQDs as close as possible to the systems observed in experiment. Finally, we have obtained an absorption spectrum for an ensemble of GCQDs by combining data from both experiment and theory. Excellent agreement between measured and simulated absorption spectra indicates that such GCQDs can be grown following a theory-guided design for application in specific devices.

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# On the numerical range of sectorial forms

*Authors*

- ter Elst, A.F. M.
- Linke, Alexander

ORCID: 0000-0002-0165-2698 - Rehberg, Joachim

*2010 Mathematics Subject Classification*

- 47A12 47B44 47A60

*Keywords*

- Numerical range, sectorial form, H
^{∞}-angle

*DOI*

*Abstract*

We provide a sharp and optimal generic bound for the angle of the sectorial form associated to a non-symmetric second-order elliptic differential operator with various boundary conditions. Consequently this gives an, in general, sharper H^{∞}-angle for the H^{∞}-calculus on L_{p} for all p ∈ (1, ∞) if the coefficients are real valued.

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# Existence, iteration procedures and directional differentiability for parabolic QVIs

*Authors*

- Alphonse, Amal
- Hintermüller, Michael

ORCID: 0000-0001-9471-2479 - Rautenberg, Carlos N.

ORCID: 0000-0001-9497-9296

*2010 Mathematics Subject Classification*

- 47J20 49J40 49J52

*2008 Physics and Astronomy Classification Scheme*

- 49J50

*Keywords*

- Quasi-variational inequality, obstacle problem, conical derivative, directional differentiability, parabolic

*DOI*

*Abstract*

We study parabolic quasi-variational inequalities (QVIs) of obstacle type. Under appropriate assumptions on the obstacle mapping, we prove the existence of solutions of such QVIs by two methods: one by time discretisation through elliptic QVIs and the second by iteration through parabolic variational inequalities (VIs). Using these results, we show the directional differentiability (in a certain sense) of the solution map which takes the source term of a parabolic QVI into the set of solutions, and we relate this result to the contingent derivative of the aforementioned map. We finish with an example where the obstacle mapping is given by the inverse of a parabolic differential operator.

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# On the complexity of approximating Wasserstein barycenter

*Authors*

- Kroshnin, Alexey
- Dvinskikh, Darina
- Dvurechensky, Pavel

ORCID: 0000-0003-1201-2343 - Gasnikov, Alexander
- Tupitsa, Nazarii
- Uribe, César A.

*2010 Mathematics Subject Classification*

- 90C25 90C30 90C06

*2008 Physics and Astronomy Classification Scheme*

- 90C90

*Keywords*

- Optimal transport, Wasserstein barycenter, Sinkhorn's algorithma, accelerated gradient descent, distributed optimization

*DOI*

*Abstract*

We study the complexity of approximating Wassertein barycenter of discrete measures, or histograms by contrasting two alternative approaches, both using entropic regularization. We provide a novel analysis for our approach based on the Iterative Bregman Projections (IBP) algorithm to approximate the original non-regularized barycenter. We also get the complexity bound for alternative accelerated-gradient-descent-based approach and compare it with the bound obtained for IBP. As a byproduct, we show that the regularization parameter in both approaches has to be proportional to ", which causes instability of both algorithms when the desired accuracy is high. To overcome this issue, we propose a novel proximal-IBP algorithm, which can be seen as a proximal gradient method, which uses IBP on each iteration to make a proximal step. We also consider the question of scalability of these algorithms using approaches from distributed optimization and show that the first algorithm can be implemented in a centralized distributed setting (master/slave), while the second one is amenable to a more general decentralized distributed setting with an arbitrary network topology.

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# Pathwise McKean--Vlasov theory with additive noise

*Authors*

- Coghi, Michele

ORCID: 0000-0002-4198-0856 - Deuschel, Jean-Dominique
- Friz, Peter

ORCID: 0000-0003-2571-8388 - Maurelli, Mario

ORCID: 0000-0002-3028-1742

*2010 Mathematics Subject Classification*

- 60F99 60F10 60G09

*Keywords*

- McKean-Vlasov, interacting particles, lithium-ion batteries, large deviations

*DOI*

*Abstract*

We take a pathwise approach to classical McKean-Vlasov stochastic differential equations with additive noise, as e.g. exposed in Sznitmann [34]. Our study was prompted by some concrete problems in battery modelling [19], and also by recent progress on rough-pathwise McKean-Vlasov theory, notably Cass--Lyons [9], and then Bailleul, Catellier and Delarue [4]. Such a ``pathwise McKean-Vlasov theory'' can be traced back to Tanaka [36]. This paper can be seen as an attempt to advertize the ideas, power and simplicity of the pathwise appproach, not so easily extracted from [4, 9, 36]. As novel applications we discuss mean field convergence without a priori independence and exchangeability assumption; common noise and reflecting boundaries. Last not least, we generalize Dawson--Gärtner large deviations to a non-Brownian noise setting.

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# What company does my news article refer to? Tackling multiclass problems with topic modeling

*Authors*

- Lübbering, Max
- Kunkel, Julian
- Farrell, Patricio

ORCID: 0000-0001-9969-6615

*2010 Mathematics Subject Classification*

- 91G50 68Q32

*Keywords*

- Text classification, latent dirichlet allocation, Kullback--Leibler divergence, company prediction, news articles

*DOI*

*Abstract*

While it is technically trivial to search for the company name to predict the company a new article refers to, it often leads to incorrect results. In this article, we compare the two approaches *bag-of-words with k-nearest neighbors* and *Latent Dirichlet Allocation with k-nearest neighbor* by assessing their applicability for predicting the S&P 500 company which is mentioned in a business news article or press release. Both approaches are evaluated on a corpus of 13k documents containing 84% news articles and 16% press releases. While the *bag-of-words* approach yields accurate predictions, it is highly inefficient due to its gigantic feature space. The *Latent Dirichlet Allocation* approach, on the other hand, manages to achieve roughly the same prediction accuracy (0.58 instead of 0.62) but reduces the feature space by a factor of seven.

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# Traveling wave analysis of non-thermal far-field blooming in high-power broad-area lasers

*Authors*

- Zeghuzi, Anissa
- Radziunas, Mindaugas
- Wünsche, Hans-Jürgen
- Koester, Jan-Philipp
- Wenzel, Hans
- Bandelow, Uwe

ORCID: 0000-0003-3677-2347 - Knigge, Andrea

*2010 Mathematics Subject Classification*

- 78A60 35Q60 78-04 58C40 35P10 81Q37

*Keywords*

- Broad-area laser, current spreading, waveguide modes, far-field, beam quality, field dynamics

*DOI*

*Abstract*

With rising current the lateral far-field angle of high-power broad-area lasers widens (far-field blooming) which can be partly attributed to non-thermal effects due to carrier induced refractive index and gain changes that become the dominant mechanism under pulsed operation. To analyze the non-thermal contribution to far-field blooming we use a traveling wave based model that properly describes the injection of the current into and the diffusion of the carriers within the active region. Although no pre-assumptions regarding the modal composition of the field is made and filamentation is automatically accounted for, the highly dynamic time-dependent optical field distribution can be very well represented by only few modes of the corresponding stationary waveguide equation obtained by a temporal average of the carrier density and field intensity. The reduction of current spreading and spatial holeburning by selecting proper design parameters can substantially improve the beam quality of the laser.

*Appeared in*

- IEEE J. Quantum Electron., 55 (2019), pp. 2000207/1--2000207/7, DOI 10.1109/JQE.2019.2893352 .

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