Young Researchers' Minisymposia 2020

Coupled multi-physics problems arise in many fields ranging from bio-medical to mechanical engineering applications. The nature of the resulting systems is therefore strongly influenced by the properties of each component and the enforced coupling conditions. This poses specific challenges to modeling, discretization, and solution approaches. The interplay of multiple physical fields and various time and length scales make these problems particularly demanding to solve. Stability, accuracy, and efficiency of the overall scheme are highly influenced by

(i) the choice of space and time discretization,

(ii) monolithic and partitioned coupling schemes,

(iii) linear and nonlinear solvers, as well as

(iv) preconditioning techniques.

This mini symposium will cover various aspects of complex multi-physics simulations to foster a fruitful discussion on the influence and interplay of modeling, discretization choices, and iterative solution methods of the resulting coupled nonlinear and linear systems of equations. Various physical phenomena and applications from different areas will be addressed. At the same time, aspects of multi-rate time integration, coupling of non-matching interface meshes, and efficient iterative solvers and preconditioners will be among the presented topics. We aim at providing a forum for young researchers working in the field of computational multi-physics to present their work, discuss promising developments and advances in the computational treatment of coupled multi-physics problems, and to give new impulses towards future research in this area.

Organizers: Alexander Heinlein (Universität zu Köln), Matthias Mayr (Universität der Bundeswehr München)

The continuous progress of applied mathematics and engineering highlights the need to solve problems of larger and larger dimension, generating an increasing demand of computational effort and memory requirements. These problems cannot be naively tackled; it is necessary to fully exploit the clear patterns of the data together with the possibly hidden structure of the solution. In this context, rank structures constitute
a powerful tool to reach this goal. Banded and low rank matrices represent a very accessible example of formats belonging to the vast family of rank structured matrices. In recent years, much more sophisticated matrix and tensor formats have been proposed in the literature: Hierachical matrices, HSS, CP, Tucker and tensor train. The theoretical analysis of these objects is currently an active branch of numerical linear algebra by itself and it is of fundamental importance to devise fast and effective algorithms for applications where the representation of rank structured data plays a crucial role. Many real-world problems can be analyzed by means of the timely algebraic techniques that exploit rank structures. An example comes from the numerical discretization of certain partial differential equations or integral equations. The discrete operator stemming from the discretization phase can be often represented by means of one of the aforementioned formats and the inversion of such an operator can include the solution of matrix and/or tensor equations and the computation of matrix functions. Therefore, the need for effective linear algebra procedures (e.g., rational Krylov subspace methods, fast multipole methods, and divide-and-conquer schemes) that efficiently deal with
rank structured objects has increased in recent years. The talks in this minisymposium illustrate some recent advances in this field.

Organizers: Stefano Massei (EPF Lausanne), Davide Palitta (Max Planck Institute for Dynamics of Complex Technical Systems)

Flow problems are of great importance in many engineering applications, e.g., water flow in turbines, pipes, and around constructions like ship hulls or bridge piers, blood flow in numerous medical applications, gas flow in pipeline networks, air flow in aeronautic engineering and many more.

The models in use are as various as the problems, reaching from viscous nearly incompressible (resp. compressible) flow described by the Navier-Stokes equations for e.g. water flows (resp. gas flows), over modified version for non-Newtonian fluids like blood or concrete, Darcy’s law for flows in porous media, to unsaturated flow modeled by Richards' equation.

In this mini-symposium, we will focus on spline-based discretizations in the context of Isogeometric Analysis (IGA), which was constituted in 2005. In this framework, the ansatz functions for the discretized problem are chosen to be some sort of B-splines or NURBS which are at the same time used to describe the computational domain. This results in very robust approximations with smooth functions but relatively few degrees of freedom, which turned out to be a very promising approach in a wide range of engineering applications and has motivated an impressively growing field of research in the last 15 years in both mathematics and engineering.

In order to address flow problems, various technical aspects have to be accounted for. With this mini-symposium, we want to bring together young mathematicians and engineers who investigate these problems to share their latest advances and pave the way to future collaborations.

Organizers: Andreas Apostolatos (TU München), Lutz Pauli (RWTH Aachen), Philipp Morgenstern (University of Hannover)

Realistic biomolecular systems are immensely challenging from a numerical, modelling, and data analysis perspective.

Due to vast differences in timescales between first principles of atomic motion and phenomena of biological interest, direct simulation of said phenomena is severely limited by computational cost. Where the simulation is indeed possible, it requires usage of specialized supercomputing hardware, generating huge amounts of raw data in the process. The high number of degrees of freedom, caused by the large number of individual atoms, renders a global analysis of this data practically impossible.

For these reasons, enhanced simulation techniques as well as efficient data analysis and model reduction methods are required to obtain biologically meaningful information. However, the construction of simplified models and accelerated integrators is often hindered by the characteristic multiscale behaviour of molecular systems, which makes it difficult to apply classical model reduction methods based on smallness parameters. In addition, no closed form of the original dynamics is available in most cases (only black-box numerical integrators), complicating a rigorous justification of proposed schemes.

Nevertheless, over the last decade, theory from many different mathematical fields, including dynamical systems theory, statistics, machine learning, and approximation theory, has been successfully applied to address these tasks. This has led to the construction of powerful discrete models for metastable molecular processes, the formulation of highly efficient accelerated sampling schemes that retain accuracy thanks to re-weighting techniques, and adaptive algorithms for finding optimal low-dimensional representations of molecular data, among countless other advances in the field. More recently, various methods to learn reduced models and the associated dynamical laws directly from bulk data have emerged, including methods based on deep neural networks.

Multiscale systems are increasingly becoming the focus of attention, which require the inclusion of memory effects in derived models. Another emerging area of application concerns non-equilibrium and non-stationary molecular processes, for which many concepts developed originally for stationary processes must be reconsidered.

The proposed minisymposium is aimed at bringing together young researchers in order to discuss upcoming trends, challenges and novel ideas in the mathematical and computational treatment of systems at atomistic scales. In that, it is strongly aligned with and devoted to highlight the topics of the recently-founded MOANSI GAMM activity group (https://moansi.wixsite.com/gamm).

Organizers: Andreas Bittracher (Freie Universität Berlin), Feliks Nüske (Rice University)

Optimal control problems occur in a wide range of applications, e.g. shape optimization of vehicles, material sciences, optimal design of machine tools, medical imaging or intraday trading of electricity. In order to obtain a solution of these optimal control problems, a system of equations is solved repeatedly in order to find a minimum (or maximum) of a given cost functional. This procedure can be very costly, especially when the underlying equations are expensive to solve. Moreover, in many settings the underlying system has to be solved for different parameter sets or might contain some unknown or uncertain quantities, which increases the level of computational complexity. To overcome these difficulties, suitable model order reduction techniques are applied. The key idea is to replace the high-fidelity systems by surrogate models of small size which approximate the original solution sufficiently well and at the same time can be computed with less computational effort.

In this minisymposium, we provide a survey over different applications of complex optimal control problems, for which it is too costly or even infeasible to solve the discretized model directly. For these problems, different model order reduction approaches are presented and novel strategies and mathematical and numerical concepts are proposed. In particular, it is analyzed how they speed up the computation of a solution.

Complexity reduction in optimal control combines different disciplines: First, it requires deep under­standing of mathematical areas like functional analysis, linear algebra and theory of simulation and control of differential equations. Second, numerical algorithms of reducing large-scale systems by model reduction methods are developed, investigated and tested numerically. Third, the developed methods are applied to a vast range of industrial applications. We follow the guiding principle math­ematics drives application while being inspired by applications: the advancement of mathematical concepts makes computational issues implementable whereas real-world applications lead to interest­ing mathematical questions. In this sense, the aim of the minisymposium is to contribute and benefit from the interdisciplinary nature of the GAMM sections in the topic of complexity reduction in optimal control.

Organizers: Carmen Gräßle (Max Planck Institute for Dynamics of Complex Technical Systems), Silke Glas (Cornell University)

Problems involving the modeling of cracking in complex materials require advanced techniques in mathematical modeling. Recently, the phase-field modeling approach has gained enormous popularity. The general phase-field approach has been applied in the simulation of a large range of physical phenomena ranging from liquid-solid phase transformations and cracking. The modeling approach to cracking requires the introduction of an order parameter which tracks the evolution or the state of an interface, e.g. intact and cracked material. An advantage of the phase-field model is that interfaces are smeared out, i.e. no discontinuity is present and cracking is modeled explicitly. The phase-field evolution equation is coupled to the governing equations of the problem. To be solved efficiently, the obtained system of equations requires a special and reliable numerical treatment. In the context of crack propagation, especially brittle fracture, reliable error estimates should be further developed.

This minisymposium aims at exploring novel techniques for the numerical treatment and discretization methods for phase-field models in fracture mechanics and is directly correlated to the objectives of the SPP 1748. The topics cover error estimates, non-standard mixed finite element schemes and the implications and interactions of these on the phase-field modeling of fracture. Furthermore, we cover a range of applications, such as fractures in porous or incompressible materials, pressure driven fractures and the influence of other state variables on cracking, e.g. temperature. Conducting this minisymposium at the annual GAMM meeting would allow the scientific exchange on novel numerical simulation techniques and mixed schemes. Furthermore, this topic-based session and the possibility of presentation and discussion would endorse the network between engineering and mathematical groups and link the wide field of modeling and applications. In conclusion, such a minisymposium aims to visualize the scientific advances and interactions, to disseminate recent developments through invited speakers associated with the SPP 1748 to broader attention and to create the needed synergies between the involved researchers.

Organizers: Tuanny Cajuhi (TU Braunschweig), Fleurianne Bertrand (HU Berlin)