Numerische Mathematik und Wissenschaftliches Rechnen
Seminar Numerische Mathematik / Numerical mathematics seminars
aktuelles Programm / current program
Donnerstag, 06. 2. 2020, 14:00 Uhr (ESH)
Dr. Holger Stephan (WIAS Berlin)
A general concept of majorization and a corresponding Robin Hood method
The Robin Hood method is a numerical method for constructing a double-stochastic matrix that maps a given vector to another vector that is also given. This is possible if the preimage majorizes the image. The majorization of vectors is a special order sometimes called the Lorentz order. A double-stochastic matrix is a special case of a Markov (or stochastic) matrix. In the case of a general Markov matrix, it is not yet clear when such a construction is possible, what kind of condition similar to the majorization must be fulfilled by the given vectors, nor is a numerical method for determining the matrix known (regardless some heuristic iterative versions of the Robin-Hood method).
In the talk, we give a complete solution to all these problems, including a general direct Robin Hood method. It turns out that the construction of a Markov matrix from two given pairs of two vectors is possible if and only if the pairs satisfy a certain order condition. If one interprets the vectors as states of a physical system, then this order is exactly the order known as the natural time order or the second law of thermodynamics.
Donnerstag, 19. 12. 2019, 14:00 Uhr (ESH)
Lia Strenge (Technische Universität Berlin)
A multilayer, multi-timescale model approach for economic and frequency control in power grids using Julia
Power systems are subject to fundamental changes due to the increasing infeed of decentralized renewable energy sources and storage. The decentral nature of the new actors in the system requires new concepts for structuring the power grid, and achieving a wide range of control tasks ranging from seconds to days. Here we introduce a multilayer dynamical network model covering a wide range of control time scales. Crucially we combine a decentralized, self-organized low-level control and a smart grid layer of devices that can aggregate information from remote sources. The stability critical task of frequency control is performed by the former, the economic objective of demand matching dispatch by the latter. Having both aspects present in the same model allows us to study the interaction between the layers. Remarkably we find that adding communication in the form of aggregation does not improve the performance in the cases considered. Instead the self-organised state of the system already contains the information required to learn the demand structure in the entire power grid. The model introduced here is highly flexible, and can accommodate a wide range of scenarios relevant to future power grids. We expect that it will be especially useful in the context of low-energy microgrids with distributed generation. All simulations and numerical experiments for control design and analysis with sampling-based methods are performed in Julia 1.1.0. The overall model is implemented as stiff nonlinear ordinary differential equation (ODE) with periodic callbacks for the control actions. The ODE has dimension 4 N, where N is the number of edges of the graph representing the power grid (i.e., N feed-in/load connections). It is planned to use automatic differentiation to learn more about the overall nonlinear model.
Donnerstag, 12. 12. 2019, 14:00 Uhr (ESH)
Dr. Nancy Hitschfeld Kahler (Universdad de Chile, Santiago)
GPU computing and meshing
GPUs are an efficient and low cost alternative to CPU clusters for solving problems that are data-parallel or close to data parallel, but there are some restrictions in the current GPU hardware that must be taken into account in order to get efficient solutions. In this talk, fundamental concepts of GPU computing will be first introduced along with relevant techniques to make optimal use of GPU hardware, such as thread branching, coalesced memory, effective use of shared memory, thread mappings onto a triangular mesh, dynamic memory allocation, thread-safe exclusion mechanisms for neighboring triangles, and indeterminate decisions and traversals over the discrete graph structure of unstructured meshes. Then, an algorithm to transform any triangulation into a Delaunay triangulation and an implementation of a particle tracking algorithm that uses previous algorithm will be discussed. Finally, other relevant aspects to take into account such as gpu-mapping techniques and the ongoing work will be mentioned.
Donnerstag, 07. 11. 2019, 14:00 Uhr (405/406)
Benoit Gaudeul (Universite de Lille , Frankreich)
Some numerical schemes for a reduced case of a Nernst--Planck--Poisson model
We compare several different numerical strategies to simulate a Nernst--Planck--Poisson model introduced in . Different equivalent formulations are exploited based either on concentrations or activities . As a first step, we focus on a simplified model for ionic liquid, for which four different schemes are discussed. A particular attention is given to the preservation at the discrete level of key features of the continuous problem, that are the positivity of the concentrations and the dissipation of the energy along time. The existence and convergence of a discrete solutions to the nonlinear system corresponding to each scheme is then established. Finally, numerical comparisons b etween the schemes are provided.
 Wolfgang Dreyer, Clemens Guhlke, and Rüdiger Müller. Overcoming the shortcomings of the Nernst--Planck model. Physical Chemistry Chemical Physics, 15, 2013.
 Jürgen Fuhrmann. Comparison and numerical treatment of generalised Nernst--Planck models. Computer Physics Communications, 6, 2015.
Montag, 04. 11. 2019, 14:00 Uhr (ESH)
Dr. Christopher Rackauckas (Massachusetts Institute of Technology/University of Maryland, USA)
Neural differential equations as a basis for scientific machine learning
Scientific Machine Learning (SciML) is an emerging discipline which merges the mechanistic models of science and engineering with non-mechanistic machine learning models to solve problems which were previously intractable. Recent results have showcased how methods like Physics Informed Neural Networks (PINNs) can be utilized as a data-efficient learning method, embedding the structure of physical laws as a prior into a learnable structures so that small data and neural networks can sufficiently predict phenomena. Additionally, deep learning embedded within backwards stochastic differential equations has been shown to be an effective tool for solving high-dimensional partial differential equations, like the Hamilton-Jacobian-Bellman equation with 1000 dimensions. In this talk we will introduce the audience to these methods and show how these diverse methods are all instantiations of a neural differential equation, a differential equation where all or part of the equation is described by a latent neural network. Once this is realized, we will show how a computational tool, DiffEqFlux.jl, is being optimized to allow for efficient training of a wide variety of neural differential equations, explaining how the performance properties of these equation differ from more traditional uses of differential equations and some of the early results of optimizing for this domain. The audience will leave knowing how neural differential equations and DiffEqFlux.jl may be a vital part of next-generation scientific tooling.
Dienstag, 03. 09. 2019, 13:30 Uhr (ESH)
Dr. Yori Fournier (Leibniz-Institut für Astrophysik Potsdam (AIP))
Non-local effects in the solar dynamo
Model of the solar dynamo hardly reproduce the observations of the solar magnetic field. In the presence of strong magnetic field in a turbulent fully ionized medium, such as the solar interior, temporal and spacial re-correlations can occur. In the frame of mean-field MHD, these can be treated as spacial and temporal non-localities (memory effect). We found that the magnetic buoyancy occurring in the convective solar interior implies a non-linear temporal non-locality. This additional non-linearity leads to solutions that recover the morphological evolution of the solar magnetic field.
Donnerstag, 11. 07. 2019, 14:00 Uhr (ESH)
Arijit Hazra (Tata Institute of Fundamental Research, Indien)
Globally constraint-preserving FR/DG scheme for Maxwell's equations up to fifth order of accuracy
Computational electrodynamics (CED), the numerical solution of Maxwell?s equations, plays an incredibly important role in several problems in science and engineering. High accuracy solutions are desired, and the discontinuous Galerkin (DG)-based methods is one of the most sophisticated ways of delivering high accuracy in numerical CED. Maxwell?s equations also have a pair of involution constraints and mimetic schemes that globally satisfy the constraints at a discrete level are also highly desirable. Keeping the above desirable properties in mind, in this work, we propose a globally divergence-conforming DG-like scheme for CED upto fifth orders of accuracy. We achieve global constraint-preservation of involution constraints of CED by collocating the electric displacement, magnetic induction and their higher order modes in the faces of the mesh. In our numerical scheme, one has to evolve some zone-centered modes in addition to the face-centered modes at fourth and higher orders of accuracy. The novel features of our schemes are: retention of higher order accuracy without any limiter even when permittivity and permeability vary by almost an order of magnitude, superior control of numerical diffusion and preservation of electromagnetic energy with strongly varying material properties in absence of conductivity. Please do not hesitate to contact me if you have any further queries. I would like to thank you for you understanding and cooperation.
Dienstag, 02. 07. 2019, 13:30 Uhr (ESH)
Dr. Ankik Kumar Giri (Indian Institute of Technology Roorkee)
Recent developments in the theory of coagulation-fragmentation models
The coagulation-fragmentation equations describe the kinetics of particle growth in which particles can coagulate via binary interaction to form larger particles or fragment to form smaller ones. These models arise in many fields of science and engineering: kinetic of phase transformations in binary alloys such as segregation of binary alloys, aggregation of red blood cells in biology, fluidized bed granulation processes, raindrop breakup in clouds, aerosol physics, i.e. the evolution of a system of solid or liquid particles suspended in a gas, formation of planets in astrophysics, polymer science and many more.
The coagulation is in general a nonlinear process where the fragmentation is classified into two major categories, one of them is the linear fragmentation and another one is the nonlinear fragmentation. The linear fragmentation may occur due to the external forces or spontaneously (that depends on the nature of particles). However, the nonlinear fragmentation takes place due to the collision between a pair of particles. Therefore, it is also known as collision-induced fragmentation or collisional breakage.
In general, the non-conservative approximation of coagulation and linear fragmentation equations may lead to the occurrence of gelation phenomenon i.e. the breakdown of mass conservation property of the solution. In the first half of this talk, it is shown that the non-conservative approximation of coagulation and linear fragmentation equations can also provide the existence of mass conserving solutions for large classes of unbounded coagulation and fragmentation kernels.The fragmentation kernel may have a singularity near the origin. Later on, this result is further generalized by including the singular coagulation kernels in the existence theory of mass-conserving solution to the nonlinear coagulation equation using non-conservative approximations.
In the second half, we introduce an existence result on weak solutions to the continuous coagulation equation with collision-induced fragmentation for certain classes of unbounded collision and breakup distribution kernels. The breakup kernel may have a possibility to attain a singularity at the origin. The proof is based on the weak compactness methods applied to suitably chosen conservative approximating equations. The question of uniqueness is also considered under additional growth conditions on the kernels which mainly relies on the integrability of higher moments. Moreover, It is observed that the unique solution is mass-conserving.
Freitag, 17. 05. 2019, 14:00 Uhr (ESH)
Prof. Peter Berg (University of Alberta, Kanada)
Energy conversion in electrokinetic flow through charged and viscoelastic nanochannels
The flow of ions and water through viscoelastic, nanoscopic domains forms the basis for many processes in biological materials. Surprisingly, such systems have rarely been explored for nanofluidic transport in artificial channels of technological applications such as energy harvesting, water desalination or DNA purification.
This talk explores the nonlinear coupling between wall deformation and quasi 1-D electrokinetic transport in a nanochannel with charged walls. Within the framework of nonequilibrium thermodynamics, formulae are derived for the electrokinetic transport parameters in terms of Onsager phenomenological coefficients and, subsequently, for energy conversion efficiencies. Results confirm that Onsager's reciprocity principle holds for rigid channels but breaks down in the 1-D formulation when the channel is deformed due to the introduction of a ''fictitious'' diffusion term of counter-ions. Furthermore, the model predicts a reduced efficiency of electrokinetic energy harvesting for channels with soft, deformable walls.
This research is conducted in collaboration with Michael Eikerling and Mpumelelo Matse.
Donnerstag, 16. 05. 2019, 14:00 Uhr (ESH)
Prof. Robert Eisenberg (Rush University Chicago, USA)
Voltage sensors of biological channels are nanomachines that perfectly conserve current, as Maxwell defined it
Biological channels produce the signals of the nervous system, and coordinate the contraction of muscle, including the heart, by responding to voltage. Biological channels are proteins with a specific piece of machinery that responds to voltage, called the voltage sensor. The voltage sensor moves charges through an electric field creating a polarization (i.e., dielectric) current that can be measured in the far field some 1e23 atoms away from the channel because Maxwell's equations enforce the perfect conservation of current, as Maxwell defined it. Maxwell's current includes the polarization of the vacuum, independent of the properties of matter (including proteins!) no matter how complex. We have built a precise electromechanical model of the voltage sensor based on its atomic scale structure that fits a wide range of experimental data.
Donnerstag, 09. 05. 2019, 14:00 Uhr (ESH)
Dr. Christoph Freysoldt (Max-Planck-Institut für Eisenforschung GmbH)
Concepts and algorithms in SPHInX
SPHInX is a C++ class library for computer simulations in material science, with a focus on electronic-structure theory. Our aim is to provide a framework that is on one hand enables physicists to implement easily new algorithms, and on the other hand makes use of developments in computer hardware and software design principles to achieve efficiency. I will give an overview over the hierarchy of available classes, and exemplify how mathematical and physical concepts are mapped onto C++ classes. I will then demonstrate the mutual benefit of concept and algorithm evolution. Last, some seemingly minor features of the SPHInX package will be highlighted, such as debugging checks, clocks, file parsing, or command line options, that proved to be immensely productive in daily work.
Dienstag, 16. 04. 2019, 13:30 Uhr (ESH)
Dr. Li Jie (University of Cambridge, GB)
Macroscopic model for head-on binary droplet collisions in a gaseous medium
In this work, coalescence-bouncing transitions of head-on binary droplet collisions are predicted by a novel macroscopic model based entirely on fundamental laws of physics. By making use of an existing lubrication theory, we have modified the Navier-Stokes equations to accurately account for the rarefied nature of the interdroplet gas film. Through the disjoint pressure model, we have incorporated the intermolecular Van der Waals forces. Our model does not use any adjustable (empirical) parameters. It therefore encompasses an extreme range of length scales (more than 5 orders of magnitude): from those of the external flow in excess of the droplet size (a few hundred micros) to the effective range of the Van der Waals force around 10 nm. A state of the art moving adaptive mesh method, capable of resolving all the relevant length scales, has been employed. Our numerical simulations are able to capture the coalescence-bouncing and bouncing-coalescence transitions that are observed as the collision intensity increases. The predicted transition Weber numbers for tetradecane and water droplet collisions at different pressures show remarkably good agreement with published experimental values. Our study also sheds new light on the roles of gas density, droplet size and mean free path in the rupture of the gas film.
Donnerstag, 14. 02. 2019, 14:30 Uhr (ESH)
Philipp Schroeder (Georg-August-Universität Göttingen)
Building bridges: Pressure-robust FEM, Beltrami flows and structure preservation in incompressible CFD
In this talk, an attempt is made to explain why pressure-robust FEM are (by construction) superior when it comes to simulating incompressible flows with a large amount of large-scale/coherent structures. We show that flows with large-scale structures are frequently dominated by large gradient (curl-free) forces. Pressure-robust methods are designed in exactly such a way that they can treat those forces more accurately than non-pressure-robust methods. Furthermore, the class of generalised Beltrami flows is introduced and placed in context with structure preservation and pressure-robustness. Several numerical examples, both in 2D and in 3D, of laminar and turbulent flows, are shown which underline our statements. All pressure-robust computations make us of (high-order) exactly divergence-free H(div)-(H)DG methods.
Mittwoch, 23. 01. 2019, 15:00 Uhr (ESH)
Prof. Vladimir A. Garanzha (Russian Academy of Sciences)
Moving adaptive meshes based on the hyperelastic stress deformation
We suggest an algorithm for the time-dependent mesh deformation based on the minimization of hyperelastic quasi-isometric functional without introducing time derivatives. The source of deformation is a time-dependent metric tensor in Eulerian coordinates. Due to the nonlinearity of the Euler-Lagrange equations we can not assume that the norm of its residual is reduced to zero at each time step. Thus, time continuity of the moving mesh is not guaranteed since iterative minimization may result in considerable displacements even for infinitely small time steps. To solve this problem, we introduce a special variant of factorized representation of Lagrangian metric tensor and nonlinear interpolation procedure for factors of this metric tensor. Continuation problem with respect to interpolation parameters is similar to the hypoelastic deformation with a controlled stress relaxation. At each time step we start with a Lagrangian metric tensor which completely eliminates internal elastic stresses and makes current mesh an exact solution of the Euler-Lagrange equations. The continuation procedure gradually introduces internal stresses back while forcing the deformation to follow the prescribed Eulerian metric tensor. At each step of the continuation procedure the functional is approximately minimized using a few steps of the parallel preconditioned gradient search algorithm. We derive an auxiliary discrete evolution equation for target shape matrices (factors of Lagrangian metric tensor) which resembles stress relaxation equations in hypoelasticity. We present 2d and 3d examples of moving deforming meshes which serve to represent moving bodies for parallel immersed boundary flow solver.
Donnerstag, 10. 01. 2019, 14:00 Uhr (ESH)
Dr. Naveed Ahmed (Lahore University of Management Sciences, Pakistan)
Numerical comparisons of finite element stabilized methods for a 2D vortex dynamics simulation at high Reynolds number
In this talk, I will present an up-to-date and classical Finite Element (FE) stabiliz ed methods for time-dependent incompressible flows. All studied methods belong to the Variational MultiScale (VMS) framework. So, different realizations of stabilized FE-VMS methods are compared in a high Reynolds number vortex dynamics simulation. In particular, a fully Residual-Based (RB)-VMS method is compared with the classical Streamline-Upwind Petrov--Galerkin (SUPG) method together with grad-div stabilization, a standard one-level Local Projection Stabilization (LPS) method, and a recently proposed LPS method by interpolation. These procedures do not make use of the statistical theory of equilibrium turbulence, and no ad-hoc eddy viscosity modeling is required for all methods. Applications to the simulation of a high Reynolds numbers flow with vortical structures on relatively coarse grids are showcased, by focusing on a two-dimensional plane mixing-layer flow. Both Inf-Sup Stable (ISS) and Equal Order (EO) (H^1)-conforming FE pairs are explored, using a second-order semi-implicit Backward Differentiation Formula (BDF2) in time. Based on the numerical studies, it is concluded that the SUPG method using EO FE pairs performs best among all methods. Furthermore, there seems to be no reason to extend the SUPG method by the higher order terms of the RB-VMS method.