Research Seminar on Mathematical Optimization / Non-smooth Variational Problems and Operator Equations

Thursday, Nov 14, 2024, 10:00 a.m.   WIAS HVP5-7 R411 and online

Dr. Amal Alphonse (WIAS)
Free boundary problems as limits of a bulk-surface model for receptor-ligand interactions on evolving domains

We derive various free boundary problems as reaction or singular limits of a coupled bulk-surface reaction-diffusion system on an evolving domain. These limiting free boundary problems may be formulated as Stefan-type problems on an evolving hypersurface. These results, which are new even in the setting where there is no domain evolution, are of particular relevance to an application in cell biology. In this talk, I will discuss the modelling, sketch the analysis, show some numerical simulations and finish with some open questions. Based on a joint work with Charlie Elliott, Chandrasekhar Venkataraman and Diogo Caetano.

Tuesday, Oct 22, 2024, 2:00 p.m.   WIAS HVP5-7 R411 and online

Dr. Ioannis Papadopoulos (WIAS)
A semismooth Newton method for obstacle-type quasivariational inequalities

Quasivariational inequalities (QVIs) are ubiquitous but, in particular, arise in PDE-constrained optimization in cases where the constraint set depends on the solution itself. In obstacle-type QVIs, this manifests as an obstacle that bends according to the state of the system. QVIs are notoriously hard to analyze, especially in the infinite-dimensional setting, and developing fast solvers posed in infinite dimensions has proven particularly challenging. As such most solvers in the literature rely on fixed point algorithms which can be slow to converge. In this talk, we introduce the first semismooth Newton method, posed in a Banach space setting, for such problems. We will see that the solver enjoys favourable properties such as local superlinear convergence and mesh independence.

Tuesday, Sept 10, 2024, 10:30 a.m.   WIAS HVP11 R 3.13 and online

Aaron Sander (Technische Universität München)
Verifying the equivalence or non-equivalence of quantum circuits with tensor networks

The development of quantum computers and algorithms is currently rapidly accelerating and will likely continue to do so in the next few years and even decades. As these systems continue to increase in size and complexity, there is an increasing need for methods to aid in their design. In particular, there is a need for debugging tools at each level of the quantum computing stack, specifically in the compilation and optimization of quantum circuits. In this work, we bridge the gap between quantum many-body physics and computer science by using tensor network techniques to verify the equivalence or non-equivalence of quantum circuits in order to detect errors that may occur during the many steps of this process.

Wednesday, June 26, 2024, 2:00 p.m.   WIAS HVP5-7 R411 and online

Antonia Topalovic (Humboldt-Universität zu Berlin)
Model predictive control for generalized Nash Equilibrium problems

We study model predictive control (MPC) schemes for non-cooperative dynamic games. The dynamic games are modelled as jointly convex generalized Nash equilibrium problems (GNEP) governed by a jointly controlled linear time-discrete dynamics. We are particularly interested in the asymptotic stability of the resulting closed-loop dynamics. To this end, we introduce a family of auxiliary problems, α-Quasi-GNEPs, which approximate the original GNEPs. For MPC schemes based on α-Quasi-GNEPs, stability guarantees can be derived if stabilizing end-constraints are enforced. This analysis is based on showing that the underlying optimal-value function is a Lyapunov function for the closed-loop. Passing to a limit, we identify a suitable Lyapunov function for MPC schemes based on the original GNEPs.

Wednesday, May 29, 2024, 10:00 a.m.   WIAS HVP5-7 R411 and online

Max Fröhlich (WIAS)
Quantum noise characterization with a tensor network quantum jump method

In this talk, we will discuss a novel approach to characterizing the noise in noisy quantum circuits through the Tensorized Quantum Jump Method (TJM). The well-known Quantum Jump Method (Monte Carlo wave function), which is used to approximate Lindbladian dynamics, can be transferred to a tensor network algorithm via Strang splitting of the Lindbladian and the help of a dynamical low-rank approximation through the Time-Dependent Variational Principle. Choosing the sparse Pauli strings of the Sparse Pauli-Lindblad Model (SPLM) as Lindblad operators makes this method a new approach to characterizing quantum noise in large systems by learning the corresponding noise parameters.

Monday, May 27, 2024, 2:00 p.m.   WIAS ESH and online

Qi Wang (WIAS)
Robust Multilevel Training of Artificial Neural Networks

In this talk, we will introduce a multilevel optimizier for training of an artificial neural network. We are particularly interested in nerual networks to learn the hidden physical law or nonlinear mapping from the given data using algebraic multigrid strategies. And we would like to give some further insight into the potential of multilevel optimization methods in the end.

Tuesday, May 7, 2024, 2:30 p.m.   WIAS HVP5-7 R411 and online

Prof. Martin Brokate (Technische Universität München)
Rate independent evolutions: some basics, some progress

We discuss some elementary rate independent evolutions, in particular the stop and the play, and offer remarks on the historical development. We also elaborate on issues concerning related optimal control problems.

Wednesday, February 21, 2024, 2:00 p.m.   WIAS HVP5-7 R411 and online

Dr. Ioannis Papadopoulos (WIAS)
Computing multiple solutions of topology optimization problems

Topology optimization finds the optimal material distribution of a fluid or solid in a domain, subject to PDE and volume constraints. The models often result in a PDE, volume and inequality constrained, nonconvex, infinite-dimensional optimization problem that may support many local minima. In practice, heuristics are used to obtain the global minimum, but these can fail even in the simplest of cases. In this talk, we will introduce the deflated barrier method, a second-order algorithm that solves such problems, has local superlinear convergence, and can systematically discover many of these local minima. We will present examples which include finding 42 solutions of the topology optimization of a fluid satisfying the Navier-Stokes equations and more recent work involving the three-dimensional topology optimization of a fluid in Stokes flow.

Underpinning the algorithm is the deflation mechanism. Deflation prevents a Newton-like solver from converging to a solution that has already been discovered. Deflation is computationally cheap, it does not affect the conditioning of the discretized systems, it may be coupled with a finite difference, finite volume or finite element discretization, and it is easy to implement.

2023

Tuesday, December 19, 2023, 2:00 p.m.   WIAS ESH and online

Dr. Joachim Rehberg (WIAS)
Maximal parabolic regularity for the treatment of real world problems

This is a series of three lectures on non-smooth problems, the first dedicated to elliptic ones and the third to parabolic ones. We start by explainig what means 'nonsmooth' and describe effects which cancel classical regularity results. Then we introduce a general geometric setting which allows in consequence to prove elliptic regularity results sufficient for attacking two- and three dimensional real world problems.
Having this at hand, elliptic operators on the scale of (negatively indexed) Sobolev spaces and L^p spaces are introduced and their regularity properties are investigated, among them L^∞ estimates, Hölder estimates and W ^1,q estimates for the solution. After these preparations we pass to parabolic equations in the third and last lecture. Here the notion of 'maximal parabolic regularity' for an elliptic operator A is the central one - being now for some decades the ultimative instrument also for the investigation in particular of nonlinar problems. Since into this notion the domain of A enters explicitly, it becomes clear that an exact knowledge of dom(A), prepared in the first two lectures, is highly desirable. So, after having introduced maximal parabolic regularity and explained some of its properties, we show that second order divergence operators satisfy this property even if the domain is highly non-smooth, the coefficient function is only bounded, measurable and elliptic and the boundary conditions are mixed

Monday, December 18, 2023, 2:00 p.m.  Online talk and WIAS ESH

Dr. Clarice Poon (Mathematics Institute, University of Warwick)
Super-resolved Lasso

Super-resolution of pointwise sources is of utmost importance in various areas of imaging sciences. Specific instances of this problem arise in single molecule fluorescence, spike sorting in neuroscience, astrophysical imaging, radar imaging, and nuclear resonance imaging. In all these applications, the Lasso method (also known as Basis Pursuit or l1-regularization) is the de facto baseline method for recovering sparse vectors from low-resolution measurements. This approach requires discretization of the domain, which leads to quantization artifacts and consequently, an overestimation of the number of sources. While grid-less methods, such as Prony-type methods or non-convex optimization over the source position, can mitigate this, the Lasso remains a strong baseline due to its versatility and simplicity. In this work, we introduce a simple extension of the Lasso, termed ``super-resolved Lasso" (SR-Lasso). Inspired by the Continuous Basis Pursuit (C-BP) method, our approach introduces an extra parameter to account for the shift of the sources between grid locations. Our method is more comprehensive than C-BP, accommodating both arbitrary real-valued or complex-valued sources. Furthermore, it can be solved similarly to the Lasso as it boils down to solving a group-Lasso problem. A notable advantage of SR-Lasso is its theoretical properties, akin to grid-less methods. Given a separation condition on the sources and a restriction on the shift magnitude outside the grid, SR-Lasso precisely estimates the correct number of sources.

Monday, November 20, 2023, 2:00 p.m.   WIAS R406 and online

Dr. Joachim Rehberg (WIAS)
Regularity for non-smooth elliptic problems II

This is a series of three lectures on non-smooth problems, the first dedicated to elliptic ones and the third to parabolic ones. We start by explainig what means 'nonsmooth' and describe effects which cancel classical regularity results. Then we introduce a general geometric setting which allows in consequence to prove elliptic regularity results sufficient for attacking two- and three dimensional real world problems.
Having this at hand, elliptic operators on the scale of (negatively indexed) Sobolev spaces and L^p spaces are introduced and their regularity properties are investigated, among them L^∞ estimates, Hölder estimates and W ^1,q estimates for the solution. After these preparations we pass to parabolic equations in the third and last lecture. Here the notion of 'maximal parabolic regularity' for an elliptic operator A is the central one - being now for some decades the ultimative instrument also for the investigation in particular of nonlinar problems. Since into this notion the domain of A enters explicitly, it becomes clear that an exact knowledge of dom(A), prepared in the first two lectures, is highly desirable. So, after having introduced maximal parabolic regularity and explained some of its properties, we show that second order divergence operators satisfy this property even if the domain is highly non-smooth, the coefficient function is only bounded, measurable and elliptic and the boundary conditions are mixed

Tuesday, October 24, 2023, 10:00 a.m.   WIAS ESH and online

Dr. Amal Alphonse (WIAS)
Analysis of a variational contact problem arising in thermoelasticity

We study a model of a thermoforming process involving a membrane and a mould as implicit obstacle problems. Mathematically, the model consists of parabolic and elliptic PDEs coupled to a (quasi-)variational inequality. We study the related stationary (elliptic) model. We look at the existence of weak solutions, and by exploring the fine properties of the contact set under non-degenerate data, we give sufficient conditions for the existence of regular solutions. Under certain contraction conditions, we also show a uniqueness result. This is based on a joint paper with Jose-Francisco Rodrigues (Lisbon, Portugal) and Carlos N. Rautenberg (Virginia, USA).

Tuesday, October 17, 2023, 2:00 p.m.   WIAS ESH and online

Dr. Joachim Rehberg (WIAS)
Regularity for non-smooth elliptic problems I

This is a series of three lectures on non-smooth problems, the first dedicated to elliptic ones and the third to parabolic ones. We start by explainig what means 'nonsmooth' and describe effects which cancel classical regularity results. Then we introduce a general geometric setting which allows in consequence to prove elliptic regularity results sufficient for attacking two- and three dimensional real world problems.
Having this at hand, elliptic operators on the scale of (negatively indexed) Sobolev spaces and L^p spaces are introduced and their regularity properties are investigated, among them L^∞ estimates, Hölder estimates and W ^1,q estimates for the solution. After these preparations we pass to parabolic equations in the third and last lecture. Here the notion of 'maximal parabolic regularity' for an elliptic operator A is the central one - being now for some decades the ultimative instrument also for the investigation in particular of nonlinar problems. Since into this notion the domain of A enters explicitly, it becomes clear that an exact knowledge of dom(A), prepared in the first two lectures, is highly desirable. So, after having introduced maximal parabolic regularity and explained some of its properties, we show that second order divergence operators satisfy this property even if the domain is highly non-smooth, the coefficient function is only bounded, measurable and elliptic and the boundary conditions are mixed

Thursday, October 12, 2023, 10:15 a.m.   WIAS R406 and online

Dr. Mazen Ali (Fraunhofer-Institut ITWM, Kaiserslautern)
Quantum Computing for Differential Equations and Surrogate Modeling

Quantum computing has transitioned from theoretical promise to practical reality, with multiple devices now accessible to the public. This technological evolution has catalyzed a multidisciplinary race to achieve the first 'quantum advantage,' drawing experts from fields as diverse as physics, computer science, finance, mathematics, and chemistry. However, despite the immense potential, the practical utility of current quantum computing implementations remains modest. Much of the research is concentrated on similar, easily-attainable goals, often accompanied by overstated claims and unwarranted optimism. Consequently, pivotal questions about the true nature of 'quantum advantage,' the roadmap to achieving it, and its fundamental relevance remain

Our focus is on harnessing the capabilities of quantum computing for material simulations at the macroscopic scale. In this presentation, I will offer an overview of the current state of quantum computing, discuss methodologies for solving differential equations directly on quantum platforms, and explore the use of quantum machine learning to create surrogate models for complex systems.

Tuesday, August 29, 2023, 2:00 p.m.   WIAS ESH and online

Robert Baraldi (Sandia National Laboratories, USA)
A proximal trust-region method for nonsmooth optimization with inexact function and gradient evaluations

We develop a novel trust-region method to minimize the sum of a smooth nonconvex function and a nonsmooth convex function. This class of problems that is ubiquitous in data science, learning, optimal control, and inverse problems. Our method is unique in that it permits and systematically controls the use of inexact objective function and derivative evaluations inherent in large-scale system solves and compression techniques, e.g. randomized sketching. When using a quadratic Taylor model for the trust-region subproblem, our algorithm is an inexact, matrix-free proximal Newton-type method that permits indefinite Hessians. We prove global convergence of our method in Hilbert space and elaborate on potential nonsmooth subproblem solvers based on ideas taken from their smooth counter-parts. Under additional assumptions, we can also prove superlinear, or even quadratic convergence to local minima. We demonstrate its efficacy on examples from data science and PDE-constrained optimization.

Thursday, July 20, 2023, 2:00 p.m.   WIAS R406 and online

Olaf Klein (WIAS)
Uncertainty quantification for models involving hysteresis operators

Parameters within models involving hysteresis operators that are supposed to describe with real world objects like, e.g. magneto mechanical devices, have to be identified from measurements. Hence, they are subject to corresponding errors. The methods of Uncertainty Quantification (UQ) are applied to investigate the influence of these errors. As an example, results of forward UQ for a play operator with uncertain yield limit will be presented. Afterwards, the model for a magneto mechanical devices involving a generalized Prandtl-Ishlinskiĭ operator considered in Sec. 5 in Davino-Krejčí--Visone-2013, Fully coupled modeling of magneto-mechanical hysteresis through `thermodynamic' compatibility. Smart Mater. Struct. https://doi.org/10.1088/0964-1726/22/9/095009 will be considered. Starting from data used to generated a First-Order-Reversal-Curves (FORC)-diagram inverse UQ is performed by formulating appropriate Bayesian Inverse Problems (BIPs) and applying Bayes' Theorem. The density of the resulting posterior density is represented by samples resulting from MCMC-computations using UQLab, the “The Framework for Uncertainty Quantification”, see https://www.uqlab.com/. Afterwards, forward UQ is performed and the results are compared to measurements. These are results of a joined work with Carmine Stefano Clemente and Daniele Davino of the Università degli Studi del Sannio, Benevento, Italy and Ciro Visone of Università di Napoli Federico II, Napoli, Italy, see also: K.-Davino-Visone-2020, On forward and inverse uncertainty quantification for models involving hysteresis operators, Math. Model. Nat. Phenom 15, https://doi.org/10.1051/mmnp/2020009 and Clemente-Davino-K.-Visone-2023, Forward and Inverse Uncertainty Quantification for a model for a magneto mechanical device involving a hysteresis operator, WIAS Preprint 3009

Monday, July 10, 2023, 2:00 p.m.   Online

Patrick Jaap (WIAS)
A semismooth Newton solver with automatic differentiation written in C++

In this talk we consider problems of the form F(x)=0 where F is a nonlinear Newton differentiable mapping between Solbolev spaces. It is well-known that a semismooth Newton method ensures local superlinear convergence towards a solution. The function spaces are discretized by suitable finite elements over a given grid. A major difficulty is the practical implementation of generalized Jacobians. To this end, we present automatic differentiation techniques to obtain discrete subgradients of F. The resulting sparse linear problems are solved by efficient linear solvers. The framework is easy to use and to implement: The user only needs to implement a local evaluation of F in the weak form for a given set of test functions. An example implementation is given for a thermoforming model from a recent paper. To verify the solver, the results of this model are reproduced.

Tuesday, June 20, 2023, 2:00 p.m.   WIAS ESH and online

Dr. Evelyn Herberg (Heidelberg University, Germany)
Deep Learning with variable time stepping

Feature propagation in Deep Neural Networks (DNNs) can be associated to nonlinear discrete dynamical systems. The novelty, in this talk, lies in letting the discretization parameter (time step-size) vary from layer to layer, which needs to be learned, in an optimization framework. The proposed framework can be applied to any of the existing networks such as ResNet, DenseNet or Fractional-DNN. This framework is shown to help overcome the vanishing and exploding gradient issues. Stability of some of the existing continuous DNNs such as Fractional-DNN is also studied. The proposed approach is applied to an ill-posed 3D-Maxwell's equation.

Tuesday, June 6, 2023, 2:00 p.m.   WIAS HVP5-7 and online

Dr. Denis Korolev (WIAS)
Physics-informed neural control of partial differential equations with applications to numerical homogenisation

In this talk we discuss a model for numerical homogenisation based on the combination of physics-informed neural networks and standard numerical approximation techniques. From a continuous viewpoint, the formulation corresponds to a non-standard PDE-constrained optimisation problem with a neural network objective. From a discrete viewpoint, the formulation represents a hybrid neural network numerical solver. We discuss physics-informed neural networks, the numerical homogenisation modelling framework and its aforementioned optimisation interpretation, as well as related discrete concepts and routes towards applications in materials science and fluid flows in porous media.

Thursday, May 25, 2023, 2:00 p.m.   WIAS ESH and online

Dr. Brendan Keith (Brown University, USA)
Proximal Galerkin: Structure-preserving finite element analysis for free boundary problems, maximum principles, and optimal design

One of the longest-standing challenges in finite element analysis is to develop a stable, scalable, high-order Galerkin method that strictly enforces pointwise bound constraints. The latent variable proximal Galerkin finite element method is a nonlinear, structure-preserving method with these properties. This talk will introduce proximal Galerkin and describe its capability for treating free-boundary problems, enforcing discrete maximum principles, and designing scalable, mesh-independent algorithms for optimal design. The talk begins with a derivation of the latent variable proximal point (LVPP) method: an unconditionally stable alternative to the interior point method. LVPP is an infinite-dimensional optimization algorithm that may be viewed as having an adaptive (Bayesian) barrier function that is updated with a new informative prior at each (outer loop) optimization iteration. One of the main benefits of this algorithm is witnessed when analyzing the classical obstacle problem. Thereupon, we find that the original variational inequality can be replaced by a sequence of semilinear partial differential equations (PDEs) that are readily discretized and solved with, e.g., high-order finite elements. Throughout this talk, we arrive at several unexpected contributions that may be of independent interest. These include (1) a semilinear PDE we refer to as the entropic Poisson equation; (2) an algebraic/geometric connection between positivity-preserving discretizations and infinite-dimensional Lie groups; and (3) a gradient-based, structure-preserving algorithm for two-field density-based topology optimization. The overall latent variable proximal Galerkin combines ideas from nonlinear programming, functional analysis, tropical algebra, and differential geometry and can potentially lead to new synergies among these areas as well as within variational and numerical analysis.

Wednesday, May 24, 2023, 3:15 p.m.   WIAS ESH (joint with Langenbach-Seminar) and online

Prof. Dr. Pavel Krejči (Technical University, Prague, Czechoslovakia)
Degenerate hysteresis in partially saturated porous media

We propose a model for fluid diffusion in partially saturated porous media taking into account hysteresis effects in the pressure-saturation relation. The resulting mathematical problem leads to a diffusion equation with Robin boundary condition for the pressure in an N-dimensional domain with a Preisach hysteresis operator under the time derivative. The problem is doubly degenerate in the sense that the saturation range is bounded, and no a priori control of the time derivative of the pressure is available. A bootstrapping argument based on particular geometric properties of the hysteresis operator makes it possible to prove the existence and uniqueness of a strong solution to the problem for arbitrarily large data. This is a joint work with Chiara Gavioli from TU Wien.

Thursday, May 04, 2023, 2:00 p.m.   WIAS ESH and online

Dr. Eskil Hansen (Lund University, Sweden)
Convergence analysis of the nonoverlapping Robin-Robin method for nonlinear elliptic equations

The nonoverlapping Robin-Robin method is commonly encountered when discretizing elliptic equations, as it enables the usage of parallel and distributed hardware. Convergence has been derived in various linear contexts, but little has been proven for nonlinear equations. In this talk we present a convergence analysis for the Robin-Robin method applied to nonlinear elliptic equations with a p-structure, including degenerate diffusion equations governed by the p-Laplacian. The analysis relies on a new theory for nonlinear Steklov-Poincare operators based on the p-structure and the Lp-generalization of the Lions-Magenes spaces. This framework allows the reformulation of the Robin-Robin method into a Peaceman-Rachford splitting on the interfaces of the subdomains, and the convergence analysis then follows by employing elements of the abstract theory for monotone operators. This is joint work with Emil Engström (Lund University)

Tuesday, April 25, 2023, 2:00 p.m.   WIAS HVP5-7 R411 and online

Dr. Caroline Geiersbach (WIAS)
Optimality conditions for problems with probabilistic state constraints

In this talk, we discuss optimization problems subject to probabilistic constraints. Our focus is on the setting in which the control variable belongs to a reflexive and separable Banach space, which is of interest, for instance, in physics-based models where the control acts on a system described by a partial differential equation (PDE). Incorporating uncertainty into such models has been of increasing interest, since in practice, one might only have access to empirical measurements or ranges of values for model parameters and inputs. We present different possibilities for incorporating uncertainty in state constraints and derive their optimality conditions. The conditions are applied to a simple example from PDE-constrained optimization under uncertainty. Perspectives for the numerical solution of these problems are discussed, as well as planned research directions.

Wednesday, March 22, 2023, 10:00 a.m.   WIAS R406 and online

Dr. Constantin Christof (Technical University of Munich, Germany)
On the identification and optimization of nonsmooth superposition operators in semilinear elliptic PDEs

We study an infinite-dimensional optimization problem that aims to identify the Nemytskii operator in the nonlinear part of a prototypical semilinear elliptic partial differential equation which minimizes the distance between the PDE-solution and a given desired state. In contrast to previous works, we consider this identification problem in a low-regularity regime in which the function inducing the Nemytskii operator is a-priori only known to be an element of H1loc. This makes the studied problem class a suitable point of departure for the rigorous analysis of training problems for learning-informed PDEs in which an unknown superposition operator is approximated by means of a neural network with nonsmooth activation functions (ReLU, leaky-ReLU, etc.). We establish that, despite the low regularity of the controls, it is possible to derive a classical stationarity system for local minimizers and to solve the considered problem by means of a gradient projection method. It is also shown that the established first-order necessary optimality conditions imply that locally optimal superposition operators share various characteristic properties with commonly used activation unctions: They are always sigmoidal, continuously differentiable away from the origin, and typically possess a distinct kink at zero.

Thursday, Feb 16, 2023, 14:00 p.m.   Online talk and WIAS ESH

Dr. Christian Parkinson (University of Arizona, USA)
The Hamilton-Jacobi Formulation of Optimal Path Planning for Autonomous Vehicles

We present a partial-differential-equation-based optimal path planning framework for simple self-driving cars. This formulation relies on optimal control theory, dynamic programming, and a Hamilton-Jacobi-Bellman equation, and thus provides an interpretable alternative to black-box machine learning algorithms. We design grid-based numerical methods used to resolve the solution to the Hamilton-Jacobi-Bellman equation and generate optimal trajectories. We then describe how efficient and scalable algorithms for solutions of high dimensional Hamilton-Jacobi equations can be used to solve similar problems in higher dimensions and in nearly real-time. We demonstrate all of our methods with several examples.

Thursday, Jan 26, 2023, 10:00 a.m.   WIAS HVP5-7 R411 and online

Clemens Sirotenko (WIAS)
Machine Learning for Quantitative MRI

The field of quantitative Magnetic Resonance Imaging aims at extracting physical tissue parameters from a sequence of highly under sampled MR images. One recently proposed approach attempts to solve this problem by estimating a set of unknown parameters in a system of ordinary differential equations. While classical approaches such as the Levenberg Marquardt algorithm or Landweber iteration yield good results under small noise levels, numerical experiments with low sampling rates and noise of large magnitude lead to unsatisfactory outcomes and unstable convergence behavior. Therefore, a spatial regularization approach based on coupled dictionary learning is proposed. From a mathematical viewpoint this ends up in a variety of non convex and non smooth optimization problems. Iterative schemes to solve these problems are discussed and convergence to equilibrium points is studied. Moreover numerical results and open questions are presented.

2022

Monday, Dec 19, 2022, 3:00 p.m.   WIAS R406 and online

Mike Theiss (WIAS)
Deriving a constrained Mean-Field Game

Mean-Field Games (MFGs) have a wide area of applications, i.e. crowd motion, flocking models, or behavior of investors. In most of these applications, it makes sense to assume constraints to the control or the state. We will start with some basic properties of a specific linear quadratic N-player game with mean field interaction. Afterward, we let the number of players N go to infinity, for deriving a "constrained MFG". Therefore we have to analyze the mean-field interaction, which describes the behavior of the whole group modeled as a flow of probability measures. Interesting is also the connection of the MFG to the original N-player problem. In the end, we will discuss some ideas on how to solve such constrained MFGs.

Thursday, Nov 10, 2022, 11:00 a.m.   WIAS HVP5-7 R411

Dr. Jonas Latz (Heriot-Watt University, Edinburgh, Scotland)
Analysis of stochastic gradient descent in continuous time

Optimisation problems with discrete and continuous data appear in statistical estimation, machine learning, functional data science, robust optimal control, and variational inference. The 'full' target function in such an optimisation problem is given by the integral over a family of parameterised target functions with respect to a discrete or continuous probability measure. Such problems can often be solved by stochastic optimisation methods: performing optimisation steps with respect to the parameterised target function with randomly switched parameter values. In this talk, we discuss a continuous-time variant of the stochastic gradient descent algorithm. This so-called stochastic gradient process couples a gradient flow minimising a parameterised target function and a continuous-time 'index' process which determines the parameter. We first briefly introduce the stochastic gradient processes for finite, discrete data which uses pure jump index processes. Then, we move on to continuous data. Here, we allow for very general index processes: reflected diffusions, pure jump processes, as well as other Lévy processes on compact spaces. Thus, we study multiple sampling patterns for the continuous data space. We show that the stochastic gradient process can approximate the gradient flow minimising the full target function at any accuracy. Moreover, we give convexity assumptions under which the stochastic gradient process with constant learning rate is geometrically ergodic. In the same setting, we also obtain ergodicity and convergence to the minimiser of the full target function when the learning rate decreases over time sufficiently slowly.

Monday, July 11, 2022, 3:00 p.m.   WIAS HVP5-7 R411

Amal Alphonse (WIAS)
Some aspects of elliptic quasi-variational inequalities

Quasi-variational inequalities (QVIs) can be thought of as generalisations of variational inequalities where the constraint set in which the solution is sought depends on the unknown solution itself. In this talk, I'll discuss various aspects of elliptic quasi-variational inequalities of obstacle type including existence results, sensitivity analysis of the source-to-solution map as well as optimal control problems with QVI constraints and associated stationarity systems.

Wednesday, June 22, 2022, 2:15 p.m.   WIAS HVP5-7 R411

Denis Korolev (WIAS)
Model order reduction techniques for electrical machines

In this talk, I will discuss model order reduction methods for parameterized elliptic and parabolic partial differential equations and their application to the modelling of magnetic fields in electrical machines. If time permits, modern deep learning methods of model order reduction will be discussed.

Wednesday, June 8, 2022, 2:00 p.m.   WIAS HVP5-7 R411

Clemens Sirotenko (WIAS)
Dictionary learning for quantitative MRI

A nonlinear inverse problem related to quantitative Magnetic Resonance Imaging (qMRI) is under consideration. In general, qMRI summarizes techniques that aim at extracting physical tissue parameters from a sequence of highly under sampled MR images. Recently, a mathematical setup was introduced that addresses this problem by estimating a set of unknown parameters in a system of ODEs called Bloch equations. While classical approaches such as the Levenberg Marquardt algorithm or Landweber iteration yield good results under small noise levels, numerical experiments show that low sampling rates and noise of large magnitude lead to unsatisfactory outcomes and unstable convergence behavior. Therefore, a spatial regularization approach based on (coupled) dictionary learning is proposed, which has already shown excellent results in the linear inverse problem of classical MRI. From a mathematical viewpoint this ends up in a variety of non convex and non smooth optimization problems. Iterative schemes to solve these problems are discussed and convergence to equilibrium points is studied. Moreover numerical results are presented and open questions such as regularization properties, parameter choice and acceleration strategies are discussed.

Tuesday, May 31, 2022, 2:00 p.m.   WIAS HVP5-7 R411

Sarah Essadi (WIAS)
From N-player games to mean-field games

We consider deterministic differential games with a large, but finite, population of symmetric interacting players. The interaction term is of mean-field type and exhibits heterogeneity both via the linear dynamics of the players and in their non-smooth cost functionals. We proceed on a first-step with only constraints on the control and with no additional state constraints. We characterise optimal solutions by deriving first-order optimality conditions. However, due to the non-smoothness of the objectives, set-valued mappings appear in the adjoint equation. To overcome this issue, we make use of a Huber-type regularisation. Furthermore, we aim at analysing the asymptotic behaviour of this system, for infinitely many players. This limiting analysis renders possible the construction of approximate Nash equilibria for the N-player games based on a solution of the corresponding mean-field game.

Monday, May 16, 2022, 2:00 p.m.   WIAS HVP5-7 R411

Marcelo Bongarti (WIAS)
Nonlinear Transport in Gas Networks

In this talk, we discuss the nonlinear transport of gas in a network of pipelines. The evolution of the gas distribution on a given pipe is modeled by a suitable isothermal Euler semilinear system in one space dimension. On the network, solutions satisfying the so-called Kirchoff flux continuity conditions at the nodes are shown to exist within the vicinity of an equilibrium state. We consider deterministic differential games with a large, but finite, population of symmetric interacting players. The interaction term is of mean-field type and exhibits heterogeneity both via the linear dynamics of the players and in their non-smooth cost functionals. We proceed on a first-step with only constraints on the control and with no additional state constraints. We characterise optimal solutions by deriving first-order optimality conditions. However, due to the non-smoothness of the objectives, set-valued mappings appear in the adjoint equation. To overcome this issue, we make use of a Huber-type regularisation. Furthermore, we aim at analysing the asymptotic behaviour of this system, for infinitely many players. This limiting analysis renders possible the construction of approximate Nash equilibria for the N-player games based on a solution of the corresponding mean-field game.

Thursday, April 15, 2020, 2 p.m.   Online

Steven-Marian Stengl (WIAS)
Combined Regularization and Discretization of Equilibrium Problems and Primal-Dual Gap Estimators

In this talk, we adress the treatment of finite element discretizations of a class of equilibrium problems involving moving constraints. Therefore, a Moreau-Yosida based regularization technique, controlled by a parameter, is discussed. A generalized Γ-convergence concept is utilized to obtain a priori results. The same technique is applied to the discretization and the combination of both. In addition, a primal-dual gap technique is used for the derivation of error estimators and a strategy for balancing between a refinement of the mesh and an update of the regularization parameter is established. The theoretical findings are illustrated for the obstacle problem as well as numerical experiments are performed for two quasi-variational inequalities with application to thermoforming and biomedicine, respectively.

Thursday, March 12, 2020, 1:00 p.m.   WIAS-R 406

Olivier Huber (HU Berlin, Germany)
Topics in gas transport: Nash equilibrium and constrained exact boundary controllability

We present two results related to the transport of gas: the existence of a solution to a Generalized Nash Equilibrium Problem (GNEP) arising from the modeling of the gas market as an oligopoly, that is only the producers are players, and the consumers just react to the quantity of gas available. In a second part, the constrained exact boundary controllability of a semilinear hyperbolic PDE is investigated. The existence of an absolutely continuous solution and boundary control will be shown, under appropriate assumptions.

Thursday, February 27, 2020, 11:00 a.m.  WIAS ESH

Axel Kröner (WIAS)
Optimal control of a semilinear heat equation subject to state and control constraints

In this talk we consider an optimal control problem governed by a semilinear heat equation with bilinear control-state terms and subject to control and state constraints. The state constraints are of integral type, the integral being with respect to the space variable. The control is multidimensional and the cost functional is of tracking type and contains a linear term in the control variable. We derive second-order necessary and sufficient conditions relying on the concept of alternative costates, quasi-radial critical directions, and the Goh transformation.

2019

Monday, December 16, 2019, 2:00 p.m.   WIAS ESH

André Uschmajew (MPI Leipzig, Germany)
Optimization on low-rank manifolds

Low-rank matrix and tensor models are important in many applications for representing and embedding high-dimensional data or functions. They typically lead to non-convex optimization problems on sets of matrices or tensors of given rank. In this talk, we give a basic introduction to the geometry of such sets and how it can be used to derive and study optimization algorithms. Compared to direct optimization of the factors in the model, the geometric approach is more intrinsic and can lead to improved methods. For a class of quadratic cost functions on matrices we also discuss how the geometric viewpoint is useful for studying the non-convex optimization landscape under low-rank constraints.

Thursday, November 21, 2019, 11:00 a.m.   WIAS ESH

Jo Andrea Brüggemann (WIAS)
On the existence of solutions and solution methods for elliptic obstacle-type quasi-variational inequalities with volume constraints

In this talk, an elliptic obstacle-type quasi-variational inequality (QVI) with volume constraints is studied. This type of QVI is motivated by the reformulation of a compliant obstacle problem, where two elastic membranes are subject to external forces while enclosing a constant volume. The existence of solutions to this QVI is established building on fixed-point arguments and partly on the concept of Mosco-convergence. Since Mosco-convergence of the considered feasible sets usually requires complete continuity or compactness properties of the obstacle map, a two-fold approach is explored towards generalising the available existence results for the considered QVI. Based on the analytical findings, the solution of the QVI is approached by solving a sequence of variational inequalities (VIs). Each of these VIs is tackled in function space via a path-following semismooth Newton method. An a posteriori error estimator is derived towards enhancement of the algorithm's numerical performance by using adaptive finite element methods.

Friday, November 1, 2019, 2:00 p.m.   WIAS ESH

Ana Djurdjevac (FU Berlin, Germany)
Random PDEs on moving hypersurfaces

It is well-known that in a variety of applications, especially in the biological modeling, PDEs that appear can be better formulated on evolving curved domains. Most of these equations contain various parameters and often there is a degree of uncertainty regarding the given data. We investigate the uncertainty which comes from unknown parameters or geometry. In the first part we study the advection-diffusion equation with random coefficients that is posed on an evolving hypersurface. We consider both cases, uniformly bounded and log-normal distributions of the coefficient. We introduce the solution space and prove the well-posedness using Banach - Nechas - Babushka theorem. Furthermore, we will introduce and analyse the evolving surface finite element discretization of the equation, introduced by Dziuk and Elliott. In the uniformly bounded case, we show unique solvability of the resulting semi-discrete problem and prove optimal error bounds for the semi-discrete solution and Monte Carlo of its expectation. In the second part we study PDEs that evolve with a given random velocity. Utilizing the domain mapping method, we transfer the problem into a PDE with random coefficients on a fixed domain and analyse this equation. Our theoretical convergence rates are confirmed by numerical experiments. This work is supported by DFG through project AA1-3 of MATH+.

Wednesday, October 23, 2019, 11:00 a.m.   WIAS HVP11A R 3.13

Martin Holler (University Graz, Austria)
A variational model for learning convolutional image atoms from incomplete data

Using lifting and relaxation strategies, we present a convex variational model for learning a convolutional sparse representation of image data via a few basic atoms. Such a representation provides a model for repeating patterns, but is also of interest for classification or as structural prior. We ensure well-posedness results for the proposed model in a general inverse problems setting and provide numerical experiments, where an atom-based representation is computed from incomplete, noisy and blurry data.

Thursday, September 26, 2019, 3:00 p.m.   WIAS R406

Martin Brokate (Technical University Munich, Germany)
Sensitivity in rate independent evolutions

As a topic in science, rate independent evolutions have appeared more than 100 years ago; their study as a mathematical subject in its own began in the 1960's. We will present some basic issues and then discuss in particular the question of differential sensitivity, that is, whether the associated solution operators possess weak derivatives.

Friday, August 2, 2019, 3:15 p.m. (WIAS ESH)

Ya-xiang Yuan (Academy of Mathematics and Systems Science, Chinese Academy of Sciences)
Efficient optimization algorithms for large scale data analysis

In this talk, two classes of problems in large scale data analysis and their optimization algorithms will be discussed. The first class focuses on composite convex program problems, where I introduce algorithms including a regularized semi-smooth Newton method, a stochastic semi-smooth Newton method and a parallel subspace correction method. The second class is on optimization with orthogonality constraints, particularly on parallelizable approaches for linear eigenvalue problems and nonlinear eigenvalue problems, and quasi-Newton type methods. Numerical results of applications, e.g., electronic structure calculations, $l_1$-regularized logistic regression problems, Lasso problems and Hartree-Fock total energy minimization problems, will be highlighted.

Thursday, July 11, 2019, 3:00 p.m. (WIAS-405-406)

Georg Stadler (Courant Institute of Mathematical Sciences, New York university)
Sparse optimal control of PDEs with uncertain coefficients

I will discuss sparse solutions of optimal control problems governed by elliptic PDEs with uncertain coefficients. Sparsity of controls is achieved by incorporating the L^1-norm of the mean of the pointwise squared controls in the objective. The main focus is on stochastic controls that share the same sparsity structure, i.e., controls that depend on the realization of the random parameters but have identical support. We propose an iterative norm reweighting formulation, which iterates over functions defined over the physical space only and thus avoids approximation of the random space. Combining a Newton method with low-rank operator approximations, this results in an efficient solution method that avoids approximation of the uncertain parameter random space. The qualitative structure of the optimal controls and the performance of the solution algorithm are studied numerically using control problems governed by the Laplace and Helmholtz equations. This is joint work with Chen Li (NYU).

Wednesday, July 10, 2019, 1:00 p.m. (WIAS ESH)

Georg Stadler (Courant Institute of Mathematical Sciences, New York university)
Estimation of extreme event probabilities by combining large deviation theory and PDE-constrained optimization, with application to tsunami waves

Tsunami waves are caused by a sudden change of ocean depth (bathymetry) after an earthquake below the ocean floor. Since large tsunami waves are extreme events, they correspond to the tail part of a probability distribution, whose exploration would require impractically many samples of a Monte Carlo method. We propose an alternative method to estimate extreme probabilities using large deviation theory (LDT), which relates the probabilities of extreme events to the solutions of a one-parameter family of optimization problems. We model tsunami waves with the shallow water equations and thus these equations appear as PDE-constraints in this optimization problem. The optimization objective includes a term that measures how extreme the event is, and a term corresponding to the likelihood of certain bathymetry changes, which are modeled as a Gaussian random field. Preliminary numerical results with the 1D inviscid shallow water equation are presented. This is joint work with Shanyin Tong and Eric Vanden-Eijnden (both NYU).

Monday, June 3, 2019, 11:30 a.m. (WIAS ESH)

Guozhi Dong (Humboldt Universität zu Berlin)
Quantitative magnetic resonance imaging: From fingerprinting to integrated physics-based models

In this talk, we introduce a novel method for quantitative MRI. The proposed approach simultaneously recovers the tissue parameters as the recently developed method magnetic resonance fingerprinting (MRF). However, in comparison with MRF and many of its variants, our new model is dictionary free. Thus, its accuracy is independent of the discretization size of the dictionary. The efficiency of our new method is proved theoretically and also verified by numerical examples.

Monday, June 3, 2019, 11 a.m. (WIAS ESH)

Selma Metzer (PTB - TU Berlin)
Approximate large-scale Bayesian inference with application to magnetic resonance fingerprinting

A class of nonlinear, large-scale regression problems is considered where the parameters model the spatial distribution of some property. Specific assumptions about the regression function are made and a homoscedastic Gaussian sampling distribution is assumed. The class of problems includes the task of Magnetic Resonance Fingerprinting (MRF), a new approach for quantitative Magnetic Resonance Imaging that allows for the estimation of absolute values of tissue related parameters like proton density and spin relaxation times. The properties of a Bayesian inference for the considered class of regression problems is investigated when different types of priors are employed, including Gaussian Markov random field priors expressing spatial smoothness. The properties of the resulting posteriors are explored and conditions for propriety of the posterior and existence of its moments established. An approximate calculation scheme is proposed and its practicability demonstrated up to dimensions of 105. Finally, the approach is applied to MRF. In using simulated data with known ground truth, it is shown that by using the prior knowledge of smoothness in the spatial distribution of the sought parameters, the results are significantly better than those achieved through maximum likelihood estimation.

Wednesday, January 23, 2019, 1 p.m. (WIAS ESH)

Maria Soledad Aronna (EMAp/FGV - TU Berlin)
On second order optimality conditions for control-affine problems: the finite and infinite dimensional case

In this talk I will present the main features of first and second order optimality conditions for optimal control problems of ordinary differential equations that are affine with respect to the control. Problems will be considered in the presence of control constraints. Both necessary and sufficient conditions for optimality will be presented. I will then show the extension of those results for a class of bilinear partial differential equations. Finally, if time allows it, I will briefly discuss the state-constrained case.

Wednesday, January 16, 2019, 1 p.m. (WIAS ESH)

Tobias Keil (WIAS Berlin)
Optimal control of a coupled Cahn-Hilliard-Navier-Stokes system with variable fluid densities

This talk is concerned with the optimal control of two immiscible fluids with non-matched densities. For the mathematical formulation of the fluid phases, we use a coupled Cahn- Hilliard/Navier-Stokes system by Abels, Garcke and Grün, which involves a variational ine- quality of fourth order. We verify the existence of solutions to a suitable time discretization of the system and formulate an associated optimal control problem. We further discuss the differentiability properties of the control-to-state operator and the corresponding stationari- ty concepts and present strong stationarity conditions for the problem. This enables us to provide a numerical solution algorithm which terminates at an at least C-stationary point which - in the best case - is even strongly stationary. The method is based on an adaptation of a bundle-free implicit programming approach for MPECs in function space presented by Hintermüller and Surowiec in 2016.

2018

Wednesday, December 19, 2018, 11:00 a.m. (WIAS ESH)

O. Huber (WIAS)
Using reformulations in nonsmooth mathematical programming: the examples of friction contact problems and optimal value functions

In this talk we illustrate how reformulations can be used to deal with nonsmoothness. In particular, we look at two specific structures arising from applications: the first one concerns finite dimensional variational inequalities with a nonsmooth functional. This is motivated by the study of contact problems with Coulomb friction. The second example involves optimization models with Optimal Value Functions (OVF) in the problem data. The OVF concept subsumes regularizers from fitting problems and coherent risk measures from Stochastic Optimization. Each reformulation scheme rely on tools from convex and variational analysis. In both instances, we obtain equivalent problems that are solvable by off-the-shelf solvers. Finally, numerical results are presented and discussed: in the OVF case, the example is a risk-averse equilibrium problem from the electricity market.

Friday, November 30, 2018, 1:00 p.m. (WIAS ESH)

S.-M. Stengl (WIAS)
Uncertainty Quantification of the Ambrosio--Tortorelli approximation in image segmentation

In this talk we want to deal with quantification of uncertainties in image segmentation based on the Mumford--Shah model. The aim is to address the error propagation of noise and other error types in the original image to the segmentation and especially the edges. We analyze therefore the in the literature well-known Ambrosio--Tortorelli approximation and discuss the existence of measurable selections of its solutions as well as sampling-based methods. We end the talk with numerical examples.

Monday, September 10, 2018, 1:30 p.m. (WIAS ESH)

K. Ito (North Carolina State University)
Semismooth Newton method for variational inequalities and MPEC

 

Wednesday, June 20, 2018, 1:00 p.m. (WIAS-Mo 39, 4th floor)

L. Goubergrits (Charité - Universitätsmedizin Berlin)
Modelling and simulation for treatment planning: CFD methods for valve treatment

In aging population, the prevalence of heart valve diseases and heart failure with more than 8 per cent of the population in their 70s (valve disease) and 10 per cent in their 80s (heart failure) is one of the most relevant diseases for the healthcare system in industrial countries. Both diseases are chronic and can amplify each other. Wrongly treated valve diseases can lead to severe heart failure and vice versa. The prognosis of heart failure is still very poor (6-year mortality rate more than 67 per cent). CFD approach promises precise diagnosis without invasive procedures. Furthermore, CFD allows predictiv modelling allowing to support clinicians with treatment decision as well as treatment planing and optimization. Finally CFD approach promises risk and cost minimization. Current CFD abilities, challenges and requirements for CFD translation into the clinical practice are presented and discussed.

Monday, June 18, 2018, 2018, 3:00 p.m. (WIAS-Mo 39, 4th floor)

J. Pfefferer (TU Munich)
hp-finite elements for fractional diffusion

In this talk we introduce and analyze a numerical scheme based on hp-finite elements to solve boundary value problems involving the spectral fractional Laplacian. The approach is based on a reformulation of the problem posed on a semi-infinite cylinder in one more spatial dimension. After a suitable truncation of this cylinder, the resulting problem is discretized with linear finite elements in the original domain and with hp-finite elements in the extended direction. The proposed approach yields a reduction of the computational complexity in terms of degrees of freedom and even has slightly improved convergence properties compared to the state-of-the-art discretization using linear finite elements for both the original domain and the extended direction. The performance of the method is illustrated by numerical experiments.

Wednesday, June 13, 2018, 1:00 p.m. (WIAS ESH)

K. Welker (U Trier)
Constrained shape optimization problems in shape spaces

Shape optimization problems arise frequently in technological processes which are modelled in the form of partial differential equations (PDEs) or variational inequalities (VIs). In many practical circumstances, the shape under investigation is parametrized by finitely many parameters, which on the one hand allows the application of standard optimization approaches, but on the other hand limits the space of reachable shapes unnecessarily. In this talk, the theory of shape optimization is connected to the differential-geometric structure of shape spaces. In particular, efficient algorithms in terms of shape spaces and the resulting framework from infinite dimensional Riemannian geometry are presented. In this context, the space of H^1/2-shapes is defined. The H^1/2-shapes are a generalization of smooth shapes and arise naturally in shape optimization problems. Moreover, VI constrained shape optimization problems are treated from an analytical and numerical point of view in order to formulate approaches aiming at semi-smooth Newton methods on shape vector bundles. Shape optimization problems constrained by VIs are very challenging because of the necessity to operate in inherently non-linear and non-convex shape spaces. In classical VIs, there is no explicit dependence on the domain, which adds an unavoidable source of non-linearity and non-convexity due to the non-linear and non-convex nature of shape spaces.

Monday, May 28, 2018, 2:00 p.m. (WIAS ESH)

F. J. Romero Hinrichsen (ETH Zurich)
Dynamical super-resolution with applications to ultrafast ultrasound

Recently there has been a successful development in ultrasound imaging, increasing significantly the sampling rate and therefore enhancing this imaging's capacities. In particular, for vessel imaging, the use of microbubble tracking allows us to super-resolve blood vessels, and by estimating the particles' speeds inside them, it is possible to calculate the vessels' diameters. In this context, we model the microbubble tracking problem, formulating it in terms of a sparse spike recovery problem in the phase space (the position and velocity space), that allows us to obtain simultaneously the speed of the microbubbles and their location. This leads to an L1 minimization algorithm for point source tracking, that promises to be faster than current alternatives.

Wednesday, May 23, 2018, 1:00 p.m. (WIAS ESH)

A. Alphonse (WIAS)
Directional differentiability for elliptic QVIs of obstacle type

Quasi-variational inequalities (QVIs) are generalisations of variational inequalities where the associated constraint set is no longer explicitly given but it depends on the solution itself. In this talk, we present some work on the directional differentiability of the multi-valued mapping that takes the source term of a QVI onto the set of solutions. This result represents a first step in the study of differential sensitivity of QVIs in infinite dimensions. We also discuss an application to thermoforming and show some numerical experiments.

Wednesday, May 16, 2018, 1:00 p.m. (WIAS ESH)

L. Banz (U Salzburg)
A posteriori error estimates for $hp$-dual mixed finite elements by implicit reconstruction

A posteriori error estimates are derived for dual mixed methods of $hp$-adaptive finite elements for variational equations as well as for variational inequalities. The error control relies on the use of a special, but never computed, $H^1$-reconstruction of the non-smooth discrete potential. Thus, no post-processing reconstruction (and therewith no additional computation) is needed. Moreover, the use of the discrete potential instead of its reconstruction improves significantly the error estimation in terms of the numerical efficiency indices which are nearly constant and close to one in the numerical experiments. Numerical experiments demonstrate the convergence rates and the efficiency indices of these a posteriori error estimates in $h$- and $hp$-adaptivity.

Wednesday, April 18, 2018, 3:00 p.m. (WIAS-Mo 39, 4th floor)

P. Dvurechensky (WIAS)
First-order optimization methods with inexact information about the objective function value and its gradient

In this talk I will discuss first-order methods with inexact oracle for finite-dimensional optimization. Oracle model of optimization methods assumes that, given a point, the oracle returns some information on the objective function at this point. In the case of first-order optimization methods, this information is the function value and its gradient at this point. I will start with convex problems, inexact oracle defined in the work by O. Devolder, F. Glineur, Yu. Nesterov, Math. Prog., 2014, and convergence rates for gradient descent and accelerated gradient descent in this case. I will also describe an extension for non-convex problems. Then I will discuss some ideas on how these methods potentially can be extended and applied for infinite-dimensional problems. If time allows, I will cover other optimization problems and methods, which I work with. Among others, optimal transport problem and an accelerated gradient descent for its solution, randomized optimization methods, such as random coordinate descent and random derivative-free method, variational inequalities, saddle-point problems and first-order methods for their solution.

Friday, February 2, 2018, 2:30 p.m. (WIAS ESH)

Th. Ruf (U Augsburg)
On a variational approach to the nonlinear wave equation

In a 2012 paper, E. Serra and P. Tilli proved a conjecture by E. de Giorgi stating that global weak solutions of wave equations such as w'' - Delta w + w|w|^(p-2) = 0 on R^+ times R^d can be obtained as limits of minimizers of suitable variational functionals. We generalize this proof to equations of the form w'' - Delta w + f_w (t; x; w) = 0 with p-growth conditions on f, also replacing R^d by an arbitrary open set O subset R^d and suitable boundary conditions.

Friday, January 19, 2018, 2:00 p.m. (WIAS ESH)

S. Bartels (Albert-Ludwigs-Universität Freiburg)
Finite element methods for nonsmooth problems and application to a problem in optimal insulation

Nonsmooth problems arise in the mathematical modeling of contact and obstacle problems, the description of plastic material behavior, and mathematical image processing. The unknown functions are typically characterized as minimizers of nondifferentiable functionals. Numerical schemes approximately solve these problems either via duality methods or classically by making use of appropriate regularizations. In the talk we discuss the discretization and iterative solution of a model problem defined on functions of bounded variation. The numerical analysis of finite element discretizations leads to reduced convergence rates which can be improved using adaptive mesh refinement. Suitable iterative solution procedures are ADMM schemes, for which we propose an automatic step size adjustment strategy, and gradient flows, for which we demonstrate the unconditional stability of a semi-implicit time discretization. The methods are applicable in the numerical determination of optimal insulating films for heat conducting bodies. Below a critical value of available insulation mass an unexpected break of symmetry occurs.

Thursday, January 11, 2018, 10:00 a.m. (WIAS ESH)

T. Kluth (Universität Bremen)
Model-based magnetic particle imaging

Magnetic particle imaging (MPI) is a tracer-based imaging modality developed to detect the concentration of superparamagnetic iron oxide nanoparticles. It is highly sensitive to the nanoparticle's nonlinear response to a dynamic applied magnetic field. Model-based reconstruction techniques are still not able to reach the quality of data-based approaches in which the linear system function is determined by a time-consuming measurement process. Possible reasons include the relaxation behavior of nanoparticles in fast changing magnetic fields. However, the equilibrium model described by the Langevin function is still used to predict the system behavior. In this context we discuss the ill-posedness of the imaging problem. We further focus on the model-based MPI reconstruction problem incorporating deviations in the forward operator. This is illustrated by initial results from real data using a regularized total least squares approach.

2017

Thursday, December 21, 2017, 2:15 p.m. (WIAS, HVP11A, 4.13)

Guozhi Dong (HU Berlin)
Regularization methods and nonlinear PDEs for solving inverse and imaging problems

In this talk, I will present some recent developments on regularization methods in image sciences. I shall show a tiny background and also mention some of the state of the art in this area. The focus will be then to discuss some non-convex regularization models for specific problems which suffer from displacement errors. The non-convex energy functionals reveal to have tight connections to some nonlinear (geometric) partial differential equations, e.g. mean curvature flows. Finally, I will show some numerical results with some discussions on the algorithms.

Monday, November 27, 2017, 9:30 a.m. (WIAS, HVP11A, 4.13)

Andrea Ceretani (HU Berlin)
Anomalous diffusion with free boundaries

Recently, time fractional Stefan-like problems have been used to model anomalous diffusion with free boundaries and long memory retention. Nevertheless, the physical foundations of this usage is still unclear. We present a model for acid water neutralization with anomalous and fast diffusion. Though this problem presents short memory retention, it is a first step in deriving mathematical models for anomalous diffusion under memory effects based on commonly accepted physical laws. The problem consists in the neutralization of an acid solution in which the hydrogen ions are transported according to Cattaneo's diffusion, and we consider the specific case of a marble slab reacting with a sulphuric acid solution in a one-dimensional geometry. The mathematical problem is reduced to a hyperbolic free boundary problem where the consumption of the slab is described by a nonlinear differential equation. We prove global well-posedness and present some numerical simulations.

Thursday, October 12, 2017, 10:30 a.m. (HU Adlershof, room 2.417)

Stephan Schmidt (University of Wuerzburg)
SQP Methods for shape optimization based on weak shape Hessians

Many PDE constrained optimization problems fall into the category of shape optimization, meaning the geometry of the domain is the unknown to be found. Most natural applications are drag minimization in fluid dynamics, but many tomography and image reconstruction problems also fall into this category. The talk introduces shape optimization as a special sub-class of PDE constraint optimization problems. The main focus here will be on generating Newton-like methods for large scale applications. The key for this endeavor is the derivation of the shape Hessian, that is the second directional derivative of a cost functional with respect to geometry changes in a weak form based on material derivatives instead of classical local shape derivatives. To avoid human errors, a computer aided derivation system is also introduced. The methodologies are tested on problem from fluid dynamics and geometric inverse problems.

Wednesday, July 26, 2017, 1:15 p.m. (WIAS ESH)

Hasnaa Zidani (ENSTA ParisTech)
Multi-objective control problems under state constraints

In this talk, we shall present a new approach, based on Hamilton-Jacobi theory, for characterizing the Pareto front for multi-objective optimal control problems in presence of state constraints. We define an auxiliary control problem for an augmented dynamical system and show that the pareto front is a subset of the zero level set of the auxiliary value function. This characterization allows to deduce an efficient numerical procedure for computing the entire Pareto front and the corresponding optimal trajectories. Moreover, the approach allows to consider objective functions of different structures (minimum time cost, Bolza cost and/or infinite horizon objective). A numerical example will be considered to show the relevance of this approach.

Thursday, June 22, 2017, 9:30 a.m. (HU Berlin, Adlershof, RUD25, 2.417)

Caroline Löbhard (WIAS)
An adaptive space-time discretization for parabolic optimal control problem with state constraints

We present a space-time discretization which is based on a reformulation of the stationarity conditions of a Moreau-Yosida regularized parabolic optimal control problem, which involves only the state variable. The resulting nonlinear partial differential equation is of fourth order in space, and of second order in time. In order to cope with the disbalance of regularity of the respective solutions, we develop a taylored discontinuous Galerkin scheme and derive convergence rates in the mesh size as well as an integrated update strategy for the regularization parameter related to the state constraints. We also propose an adaptive mesh refinement strategy and illustrate the performance of our method in numerical test cases.

Wednesday, June 21, 2017, 3:30 p.m. (WIAS, HVP11A, 4.13)

Mattia Bongini (Barcelona Graduate School of Economics)
Optimal Control Problems in Transport Dynamics

In this talk we shall discuss several results concerning the "indirect control of populations", i.e., how to influence a group of individuals by means of external agents with a directly controlled dynamics. By using the general theory of functionals defined on spaces of measures, we give sufficient conditions for the existence of optimal control strategies and then we present a Pontryagin Maximum Principle for such controls in the form of an Hamiltonian flow in the Wasserstein space of probability measure. Finally, we present an application of the above framework to the evacuation problem of a crowd from an unknown environment with the help of undercover stewards.

Wednesday, Mai 31, 2017, 1:15 p.m. (WIAS, HVP11A, 4.13)

Jo Andrea Brüggemann (HU Berlin / WIAS)
Solution methods for a contact model motivated by the human heart's pericardium

Introducing a function space description of the contact within the human heart's pericardium, in this talk, we will study a relaxation of the latter. The quasi-variational inequality (QVI) in the focus of interest is attacked with a fixed-point approach and the hereby arising sequence of variational problems can be efficiently solved with a path-following semi-smooth Newton method.

Wednesday, Mai 24, 2017, 1:15 p.m. (WIAS ESH)

Hongpeng Sun (Renmin University of China)
Weak convergence of proximal ADMM and its relaxations in Hilbert spaces

ADMM (Alternating Direction Method of Multipliers) is a popular first order method for mathematical imaging and inverse problems. However, the weak convergence of ADMM in infinite dimensional spaces is not clear yet, which is different from the classical Augmented Lagrangian Method. We will discuss the weak convergence of ADMM and its proximal variants with relaxations in Hilbert spaces.

Tuesday, Mai 9, 2017, 3:30 p.m. (WIAS, HVP11A, 4.13)

Kazufumi Ito (North Carolina State University, USA)
Value function calculus and applications

In this talk the sensitivity analysis is discussed for the parameter-dependent optimization. The sensitivity of the optimal value function with respect to the change in parameters plays a significant role in the optimization theory, including economics, finance, the Hamilton-Jacobi theory, the inf-sup duality and the structural design and the bi-level optimization. We develop the calculus for the value function and present its applications in the variational calculus, the bi-level optimization and the optimal control and optimal design and inverse problems.

Friday, Mai 5, 2017, 2:15 p.m. (WIAS ESH)

Behzad Azmi (Univ. Graz, Austria)
On the stabilizability of infinite dimensional systems via receding horizon control

One efficient strategy for dealing with optimal control problems on an infinite time horizon is the receding horizon framework. In this approach, an infinite horizon optimal control problem is approximated by a sequence of finite horizon problems in a receding horizon fashion. Stability is not generally ensured due to the use of a finite prediction horizon. Thus, in order to ensure the asymptotic stability of the controlled system, additional terminal cost functions and/or terminal constraints are often needed to add to the finite horizon problems. In this presentation, we are concerned with the stabilization of several classes of infinite-dimensional controlled systems by means of a Receding Horizon Control (RHC) scheme. In this scheme, no terminal costs or terminal constraints are used to ensured the stability. The key assumption is the stabilizability of the underlying system. Based on this condition the suboptimality and stability of RHC are investigated. To justify the applicability of this framework, we consider controlled systems governed by different types of partial differential equations. Numerical examples are presented as well.

Thursday, Mai 4, 2017, 9:30 a.m. (HU Berlin, Adlershof, RUD25, 2.417)

Axel Kröner (HU Berlin / CMAP, Ecole Polytechnique, Paris-Saclay)
Optimal control of infinite dimensional systems

In this talk we analyze second order optimality conditions for a bilinear optimal control problem governed by a strongly continuous semigroup operator, the control entering linearly in the cost function. We derive first and second order optimality conditions, taking advantage of the Goh transform. The general framework allows the application to heat, wave, and Schr\"odinger equation. This is joint work with S. Aronna und F. Bonnans.

Wednesday, Mar 29, 2017, 1:15 p.m. (WIAS ESH)

Christian Clason (Universität Duisburg-Essen)
Convex relaxation of hybrid discrete-continuous control problems

We consider control problems for partial differential equations where the distributed control should take on values only from a given discrete and hence non-convex set. Such problems occur for example in parameter identification or topology optimization. Similar to their use in sparse optimization, L1-type norms can be used to formulate a convex relaxation which can be solved by semi-smooth Newton methods. We illustrate this approach using linear model problems and discuss the extension to vector-valued and nonlinear control problems.

Tuesday, Feb 7, 2017, 10:15 a.m.   WIAS ESH

Sara Merino-Aceituno (Imperial College London, UK)
Kinetic theory to study emergent phenomena in biology: an example on swarming

Classical methods in kinetic theory are challenged when studying emergent phenomena in the biological and social sciences. New methodologies are needed to study the problems at hand which typically involve many agents that interact locally. The aim of this talk is to introduce the general framework of kinetic theory and some of the new challenges of applying it to biological systems. We illustrate it in the case of the so-called collective motion of self-propelled particles, like swarming of birds. Particularly, based on the Vicsek model, we study systems of agents (birds) that move at a constant speed while trying to align their body orientation with those of their neighbors. Starting from a particle description, we find the macroscopic dynamics.

Tuesday, Jan 17, 2017, 10:15 a.m.   WIAS ESH

Tim Sullivan (FU Berlin)
Well-Posedness of Bayesian Inverse Problems - Stable Priors on Quasi-Banach Spaces

The Bayesian perspective on inverse problems has attracted much mathematical attention in recent years, and particular attention has been paid to Bayesian inverse problems (BIPs) in which the parameter to be inferred lies in an infinite-dimensional space, a typical example being a scalar or tensor field coupled to some observed data via an ordinary or partial differential equation. Numerical solution of such infinite-dimensional BIPs must necessarily be performed in an approximate manner on a finite-dimensional subspace, but it is profitable to delay discretisation to the last possible moment and consider the original infinite-dimensional problem as the primary object of study, since infinite-dimensional well-posedness results and algorithms descend to any finite-dimensional subspace in a discretisation-independent way, whereas careless early discretisation may lead to a sequence of well-posed finite-dimensional BIPs or algorithms whose stability properties degenerate as the discretisation dimension increases.

This presentation will give an introduction to the framework of well-posed BIPs in infinite-dimensional parameter spaces, as advocated by Stuart (Acta Numer. 19:451-559, 2010) and others. Recently, this framework has been extended to the case of a heavy-tailed prior measure in the family of stable distributions, such as an infinite-dimensional Cauchy distribution, for which polynomial moments are infinite or undefined. It is shown that analogues of the Karhunen-Loeve expansion for square-integrable random variables can be used to sample such measures on quasi-Banach spaces. Furthermore, under weaker regularity assumptions than those used to date, the Bayesian posterior measure is shown to depend Lipschitz continuously in the Hellinger and total variation metrics upon perturbations of the misfit function and observed data.

Tuesday, Jan 10, 2017, 10:15 a.m. (WIAS ESH)

Andrew Lam (Uni Regensburg)
Diffuse interface models of tumor growth and optimizing cancer treatment times

There has been a recent focus in modeling tumor growth with diffuse interface models, due to their ability to capture topological transitions and the nature of the equations allows for further mathematical treatment. In the first part of this talk I will introduce a class of Cahn-Hilliard systems that are used to capture the basic dynamics of tumor growth. Then, we will discuss an optimal control problem for chemotherapy, which is a cancer treatment using drugs to eliminate tumor cells. Treatments are usually conducted in cycles, and long treatment times may cause harm to the patient. Thus, it is important to optimize both the treatment time and drug dosage to minimize patient suffering. In the second part of this talk, we will analyze an optimal control problem with an objective functional depending on a free time variable, which represents the unknown treatment time to be optimized.

2016

Tuesday, Dec 13, 2016, 10:15 a.m. (WIAS ESH)

Amal Alphonse (WIAS)
Existence for a fractional porous medium equation on an evolving surface

In this talk, which is based on joint work with Prof. Charlie Elliott, I will present an existence theory for a porous medium equation with a fractional diffusion on an evolving surface. The nonlocal nature of the fractional diffusion (which in our case is the square root of the Laplacian) in combination with the nonlinearity and the moving domain makes the problem interesting. After defining the fractional Laplacian and giving a Dirichlet-to-Neumann characterisation of it in a general setting of closed Riemannian manifolds, I will define what we mean by a weak solution and then proceed with the proof of existence. This will involve harmonic extensions on semi-infinite and truncated cylinders, convergence/decay estimates and some technical results in order to deal with the time-evolving surface. I will finish by discussing some ideas for further work.

Tuesday, Dec 6, 2016, 10:15 a.m. (WIAS ESH)

Luca Calatroni (Ecole Polytechnique, Paris)
Infimal convolution of data discrepancies for mixed noise removal in images

In several real-world imaging applications such as microscopy, astronomy and medical imaging, transmission and/or acquisition faults result in a combination of multiple noise statistics in the observed image. Variational data discrepancy models designed to deal with such mixtures linearly combine standard data fidelities used for single-noise removal or make use of either approximated and cheap or exact but computationally expensive log-likelihood functionals. Via a joint MAP estimation, we derive a statistically consistent variational model combining data fidelities associated to single noise distributions in a handy infimal convolution fashion by which individual noise components corrupting the data are modeled appropriately and decomposed from each other. After showing the well-posedness of the model in suitable function spaces, we propose a semi-smooth Newton-type scheme to compute its numerical solution efficiently.

Tuesday, Nov 29, 2016, 10:15 a.m.   WIAS, HVP11A 4.13

Martin Kanitsar (University of Graz, Austria)
Numerical shape optimization for an industrial application

For industrial applications in fluid dynamics, finding an optimal shape with respect to some cost functional is important. Moreover, constraints due to inflow and outflow boundaries as well as a respecified construction space need to be taken care of. For the implementation of a gradient descent scheme, the shape sensitivity calculus is used to derive the shape gradient of the cost functional with respect to changes in the shape. The underlying physics are described by the stationary Navier-Stokes equation, the primal equation, and an adjoint equation is used for calculating the shape derivative. Highlighting some details on the numerical realization for a 3D application is the main part of this talk, but results on existence of an optimal shape will be pointed out too. In addition to illustrating interests of the industry, hints are given for appropriate usage of the software OpenFOAM and Star-CD CCM+.

Tuesday, Nov 22, 2016, 10:15 a.m.   WIAS ESH

Soheil Hajian (WIAS/HU Berlin)
Total variation diminishing RK methods for the optimal control of conservation laws

Optimal control problems subject to conservation laws are among challenging optimization problems due to difficulties that arise when shocks are present in the solution. Such optimization problems have application, for instance, in gas networks. Beside theoretical difficulties at the continuous level, discretization of such optimal control problems should be done with care in order to guarantee convergence. In this talk, we present stability results of the total variation diminishing (TVD) Runge-Kutta (RK) methods for such optimal control problems. In particular we show that enforcing strong stability preserving (SSP) for both forward and adjoint problem results to a first order time-discretization. However, requiring SSP only for forward problem is sufficient to obtain a stable discrete adjoint. We also present order-conditions for the TVD-RK methods in the optimal control context.

Tuesday, Nov 15, 2016, 10:15 a.m.   WIAS ESH

M. Hintermüller (WIAS/HU Berlin)
Bilevel optimization with applications to image processing

 

Friday, Oct 21, 2016, 3:00 p.m   WIAS ESH

M. Holler (Karl-Franzens-Universität Graz, Austria)
Higher order regularization and applications to medical image processing and data decompression

Variational methods are a powerful tool for tackling ill-posed problems in image processing. As such, they rely heavily on appropriate regularization terms which render a stable recovery possible and strongly influence qualitative solution properties. In this talk, we consider regularization concepts for both static and dynamic data that are based on higher order differentiation. Beginning with the static setting, we first discuss analytical properties of Total Generalized Variation (TGV) regularization which allow for well-posedness results for standard inverse problems. We then consider the application of TGV in the context of a variational model for image decompression, being in particular applicable to JPEG or JPEG 2000 compressed images. As second application, we introduce a nuclear-norm-based vectorial TGV functional for joint MR-PET reconstruction that exploits structural similarities between the two modalities. Moving to the dynamic setting, we motivate and introduce a suitable extension of derivative based regularization for spatio-temporal data. After establishing essential analytical properties, we deal with applications to the reconstruction of highly subsampled dynamic MR data and the decompression of MPEG compressed movies.

Wednesday, Oct 12, 2016, 2:00 p.m. (WIAS-HVP11A 4.01)

C. Geldhauser (WIAS)
Scaling limits of interacting diffusions

Abstract: In this talk we consider a system of N coupled stochastic differential equations, which we interpret as a system of N particles evolving according to the dynamics given by the SDEs. Due to the properties of the driving force and the noise, the limit as N goes to infinity does not lead in general to a well-posed equation. We develop conditions on the interaction strength between the particles to ensure existence of solutions to the limiting stochastic PDE. Moreover, we investigate the long-time behaviour of the solution. This is joint work with Anton Bovier.