From September 19-21, the Berlin SIAM student chapter hosted the Meeting of the European SIAM and GAMM student chapters (MESIGA18).

Organizers: Ines Ahrens, Daniel Bankmann, Felix Black, Stefan Ruschel, Lia Strenge, Benjamin Unger, Christoph Zimmer

##### List of Participants
Name, Forename Affiliation
Ahrens, Ines TU Berlin
Bankmann, Daniel TU Berlin
Bauer, Christian RWTH Aachen University
Black, Felix TU Berlin
Bünger, Alexandra TU Chemnitz
Cebulla, Dominik Heidelberg University
Fieberg, Debora Universität Heidelberg
Froidevaux, Marine TU Berlin
Geuter, Lukas TU Berlin
HosseiniMehr, Mousa TU Delft
Hüttenhain, Jesko Crowd Strike
Koellermeier, Julian Freie Universität Berlin
Morandin, Riccardo TU Berlin
Netušil, Marek Charles University
Penke, Carolin Max Planck Institute for Dynamics of Complex Technical Systems
Ruschel, Stefan TU Berlin
Scholz, Robert TU Berlin
Schulze, Philipp TU Berlin
Strenge, Lia TU Berlin
Unger, Benjamin TU Berlin
Varga, Dora Universität Augsburg
von Allwörden, Hannes Universität Hamburg
Werner, Steffen W. R. Max Planck Institute for Dynamics of Complex Technical Systems
Wiedemann, David Universität Augsburg
##### Keynote lecture: Of Puzzles and Riddles
###### Jesko Hüttenhain (CrowdStrike)

I will talk briefly about what I do as a Malware Reverse Engineer in Cyber Security, why I am so happy with this job, and why it does in no way contradict that I remain a passionate Mathematician.

##### A Structural Approach for Differential-Algebraic Equations with Delay
###### Ines Ahrens (TU Berlin)

The strangeness index for differential-algebraic equations (DAEs) is based on the derivative array, which consists of the system itself plus its time derivatives. Index reduction is performed by selecting certain important equations from the derivative array. In a large-scale setting with high index, this might become computationally infeasible. However, if it is known a priori which equations of the original systems need to be differentiated, then the computational cost can be reduced. One way to determine these equations is by means the Sigma method.

If the DAE features in addition a delay term then taking derivatives might not be sufficient and instead, the derivative array must additionally be shifted in time thus increasing the computational complexity even further. In this talk I will explain how one can modify the Sigma method such that it determines the equations which need to be shifted and/or differentiated. This is joint work with Benjamin Unger.

##### On the Imposition of Dirichlet Boundary Conditions at Moving Boundaries with Applications in Laser Processing
###### Christian Bauer (RWTH Aachen University)

One of the most demanding challenges in the simulation of e.g. Laser Welding (LW) and Laser Additive Manufacturing (LAM) is the description of a moving free boundary. This boundary separates the molten phase, the melt pool, from the solid phase of the processed material. The laser feed causes the material to melt on one front and solidify again on the opposite front. Therefore, the melt pool moves through the component. In the simulation of this process, two difficulties occur. Firstly, how can we describe the moving boundary efficiently? And secondly, how can we solve for the velocity field in the melt and the heat distribution in the workpiece without the need to remesh the domain in every time step?

In order to describe a moving boundary, the Level Set Method (LSM) is often applied. A common method to bypass remeshing the domain is to use a fixed background mesh, indicate which nodes are within the actual physical domain, and incorporate the boundary condition at the Fluid Structure Interface (FSI) into the underlying equations. In the talk, the focus will be on the imposition of boundary conditions at the FSI. We’ll provide an overview of the various methods to enforce a Dirichlet boundary condition at an embedded boundary that lies within the computational domain, namely, the Immersed Boundary Method, Penalty Methods, Nitsche’s Method, and the Lagrange Multiplier Techniques. Afterward, a sketch of a novel approach will be given where we illustrate challenges and present early results.

##### Low Rank Methods for Isogeometric Analysis in PDE-constrained Optimization
###### Alexandra Bünger (TU Chemnitz)

Isogeometric analysis has become a popular method for the discretization of partial differential equations motivated by the use of NURBS for geometric representations in industry and science. Opposed to the widely popular Finite Element method, the computational domain does not have to be simplified and we can work directly with an exact geometric representation, e.g.\, from a CAD model.

However, a crucial challenge lies in the assembly and solution of the discretized equations. To solve a PDE-constrained optimal control problem the discretization results in a system of large mass and stiffness matrices, which are typically very costly to assemble. To reduce the computing time and storage requirements low-rank tensor methods have become a promising tool. We present a framework for the assembly of these matrices in low-rank form as the sum of a small number of Kronecker products.

The resulting low rank Kronecker product structure of the mass and stiffness matrices can be used to solve a PDE-constrained optimization problem without assembly of the actual system matrices. We present a framework which preserves and exploits the attained Kronecker product format in MATLAB to efficiently solve the corresponding KKT system of the optimization problem. We illustrate the performance of the method on various examples.

##### Parameter Identification of a Ion-Exchange Chromatography Process
###### Dominik Cebulla (Heidelberg University)

Separation of proteins plays a significant role in the development and production of pharmaceuticals, where a high quality of the product has to be ensured. Over the last decades, chromatography has emerged as a powerful tool to fulfill this task. In this talk, I will briefly describe the concept of ion-exchange chromatography and present a mechanistic model for this type of chromatography, yielding a nonlinear partial differential equation. For the identification of the unknown model parameters, we discretize in space and time which yields a high-dimensional nonlinear least squares problem. To overcome this high dimensionality we present a strategy which reduces the number of degrees of freedom significantly (essentially to the number of parameters to be estimated). We present numerical results to highlight the benefits of our approach and to discuss further challenges.

##### Using Agent-Based Modeling to Explore the Emergence of Echo Chambers
###### Debora Fieberg (Universität Heidelberg)

We implemented an agent-based model (ABM) based on Schelling's Model of Segregation (1971) to explore the influence of confirmation bias and openness on propagation and polarization of opinions in a heterogeneous belief environment. The extension by a third dimension of belief allows us to analyze how these two parameters contribute to the emergence of echo chambers and how individual differences in openness lead to the occurrence of polarization. In a second step, we further equipped each agent with a simple working memory parameter. Simulation results show that differences in working memory capacity distributions among the agents influence belief polarization. Future steps towards simulating the cognitive process of each agent in a more realistic way by merging agent-based modeling with our mathematical model of the ACT-R declarative memory module (Said et al. 2016) will be discussed.

##### Modeling and Numerical Simulation of Rarefied Gases
###### Julian Koellermeier (Freie Universität Berlin)

Rarefied gases occur in high-altitude flights like supersonic atmospheric reentry or in micro-channels and pose challenges for both mathematical modeling and numerical simulations. The setting requires models beyond Euler or Navier-Stokes equations, as these are no longer valid under the extreme conditions of rarefied gases. I will present an approach to derive extended equations by means of so-called moments that leads to a speed up with respect to conventional particle-based models while still resulting in high accuracy. The numerical solution of these moment models is particularly interesting as dedicated, non-conservative numerical schemes have to be applied. Some examples for test cases and real applications will be shown to demonstrate the capability of the new extended moment models.

##### Algebraic Dynamic Multilevel Method for Flow in Heterogeneous Geothermal Reservoirs
###### Mousa HosseiniMehr (TU Delft)

Accurate numerical simulation of coupled fluid flow and heat transfer in heterogeneous geothermal reservoirs demand for high resolution computational grids. The resulting fine-scale discrete systems--though crucial for accurate predictions--are typically upscaled to lower resolution systems due to computational efficiency concerns. Therefore, advanced scalable methods which are efficient and accurate for real-field applications are more than ever on demand. To address this need, we present an algebraic dynamic multilevel method for flow and heat transfer in heterogeneous formations, with non-isothermal equilibrium. The fine-scale fully-implicit discrete system is mapped to a dynamic multilevel grid, the solution at which are connected through local basis functions. These dynamic grid cells are imposed such that the sub-domain of sharp gradients are resolved at fine-scale, while the rest of the domain remains at lower (coarser) resolutions. In order to guarantee the quality of the local (heat front) components, advanced multiscale basis functions are employed for global (fluid pressure and rock temperature) unknowns at coarser grids. Numerical test cases are presented for homogeneous and heterogeneous domains, where ADM employs only a small fraction of the fine-scale grids to find accurate complex nonlinear thermal flow solutions. As such, it develops a promising scalable framework for field-scale geothermal simulations.

##### Multi-Scale Structural Modeling of Arterial Wall
###### Marek Netu\v{s}il (Charles University)

Cardiovascular diseases are the most common cause of death worldwide. Moreover, a structure of a vessel wall tissue is nothing like other usually studied and modeled materials, such as concrete or rubber, and is far more complex. All that makes mathematical modeling of arterial walls an important and highly interesting issue. On the other hand, most works in this field deal only with very simple models trying to tackle just the most obvious of the mechanical properties. This often leads to confusion and misinterpretation of mathematical results.
The presentation will introduce an attempt to model the mechanical properties of an arterial wall in a greater detail, utilizing the theories of asymptotic homogenization and non-linear poroelasticity.

##### Solving Large (Skew-)Symmetric Eigenvalue Problems on High Performance Computers
###### Carolin Penke (Max Planck Institute for Dynamics of Complex Technical Systems)

Eigenvalue problems have applications in many areas. For example they come up in electronic structure theory which aims to compute interesting properties of molecular systems using quantum mechanical principles. There the arising matrices are typically symmetric and can be dense. They easily become very large when more complex systems are considered. In this talk we review the state-of-the art method to solve these problems on high performance compute clusters. It consists of a two-step band reduction and a tridiagonal solver such as bisection and inverse iteration or divide-and-conquer. It is available in the form of high performing libraries, such as ELPA [1]. We extend this method to skew-symmetric matrices.

The Bethe-Salpeter eigenvalue problem results from a more sophisticated approach in electronic structure theory that takes electron-hole interaction in excited states into account. It has the potential for more accurate prediction of molecular properties. The matrix has a block structure which implies an additional symmetry in the spectrum. There are algorithms for solving the Bethe-Salpeter eigenvalue problem that preserve this structure and run on high performance computers in parallel. They rely on the solution of skew-symmetric eigenvalue problems. A higher performance can be achieved using the presented extension of the two-step band reduction method.
[1] https://elpa.mpcdf.mpg.de

##### Hierarchical control theory in power grids based on a behavioral approach
###### Lia Strenge (TU Berlin)

The energy transition towards renewable energy sources is
posing challenges and opportunities for the structure and control of the
power grid. Challenges are mainly caused by the intermittency of
renewable infeed and bidirectional power flow in a system designed for
unidirectional power distribution (power plants $\rightarrow$ households/loads). At
the same time the reduced distance between local electricity production
and consumption by renewable generation can be an opportunity. The
theoretical analysis and partition of the overall system, however, will
be more complex and be based on high-dimensional systems. Since the
characteristics of a future power grid are highly unknown, former
complexity reduction methods need to be revisited. We investigate if
hierarchical control theory based on a behavioral approach can be used
to reduce complexity when modeling future power grids and formalize
multi-layered hybrid (AC/DC) power grid control. The framework will be
applied to an aggregated AC toy model to demonstrate its suitability for
the application in power grids. We will present work in progress rather
than validated results.

##### Computation of Effective Coefficients by Numerical Homogenization
###### Dora Varga (Universität Augsburg)

The presentation is based on numerical homogenization by the local orthogonal decomposition method and has the aim of providing a link between numerical and analytical homogenization. A new, corrector-dependent coefficient is considered, defined as a piecewise constant on the underlying triangulation of the domain. We show that in periodic homogenization, under certain assumptions on the geometry of the mesh, the proposed coefficient coincides with the effective coefficient from classical homogenization.

##### Periodic Solutions for Microscopic Traffic Models
###### Hannes von Allwörden (Universität Hamburg)

Microscopic models, describing the motion of each vehicle individually by means of ordinary or delay differential equations, are a popular tool for the investigation of traffic flow phenomena.
Even simple models show effects that appear well-known from everyday experience, such as the loss of stability of uniform flow solutions for certain parameters while at the same time periodic \glqq stop-and-go\grqq-solutions emerge. We investigate how these solutions depend on the number of cars and the design of the road and discuss their stability.

##### MORLAB - A Model Reduction Framework in MATLAB & Octave
###### Steffen W. R. Werner (Max Planck Institute for Dynamics of Complex Technical Systems)

Many different real-world applications, like chemical processes, electrical circuits, or computational fluid dynamics, can be modeled by dynamical systems. Since experiments can be very expensive and time-consuming, these models are used for simulations and the design of controllers. Often these original models are very complex and quickly reach the limits of computational resources. The aim of model reduction is to compute a surrogate model, for replacing the original one, that is much easier to evaluate. The MORLAB (Model Order Reduction Laboratory) toolbox is a software solution in MATLAB and Octave for the model reduction of linear time-invariant continuous-time systems. Also, the toolbox provides efficient subroutines for computations with linear dynamical systems as well as matrix equation solvers.

##### Coupled Advection-Reaction-Diffusion Processes on an Evolving Microstructure: Analysis and Homogenization
###### David Wiedemann (Universität Augsburg)

We consider a porous medium composed of solid matrix and pore space, which is completely saturated with a fluid. A dissolved concentration is present in the fluid, which is affected by diffusion and advection as well as reaction at the surface of the solid matrix. This reaction causes the solid matrix to grow or shrink locally. Thus, the microstructure of the porous medium changes, which affects the transport of the concentration.

In order to upscale this problem, we consider an advection-reaction-diffusion problem coupled with the Stokes equation in a domain with an evolving microstructure. The homogenization of this problem is performed utilising a transformation to a periodic reference domain and the macroscopic limit problem is determined using two-scale convergence.

##### Model reduction for switched systems: Dos and Don'ts
###### Benjamin Unger (TU Berlin)

The simulation-based development that is present in many real-world applications is mainly driven by two cost factors: the time to develop the model equations and the computational cost involved in solving the model equations. A remedy for the second issue is model order reduction (MOR), which aims in approximating the usually high-dimensional model with a low-dimensional model in an automated and reliable fashion. Although the theory has reached a certain level of maturity for standard state-space systems, the investigation of the approximation of so-called switched systems is still far from complete. In this talk, we emphasize important aspects in the reduction of switched systems, which translate to scenarios where approximation should be avoided. Based on the gained insight, we propose a simple reduction strategy that allows the usage of standard methods. This talk describes joint work with Philipp Schulze (TU Berlin).