Workshop on Applied Problems with Partial Differential Equations

The purpose of the workshop is to bring together experts from the mathematical, computational, physical, and engineering communities from Vienna, who work with Partial Differential Equations, and to provide a platform for the exchange of new ideas. The meeting aims at identifying scientific contact points and stimulating joint research between the different communities, for instance in the framework of joint projects or joint bachelor, master, and PhD theses. The workshop welcomes both junior as well as senior participants.

Time: December 15, 2015, 9:00h - 17:30h

Location: TUtheSky, 11th floor, Getreidemarkt 9, 1060 Wien

Organizers: Anton Arnold (Vienna), Ansgar Jüngel (Vienna)

Speakers:

• Lukas Jadachowski (Automation and Control, TU Wien)
  Backstepping observers for some classes of parabolic PDEs
  This talk is mainly concerned with the state estimation problem for some classes of parabolic PDEs with boundary sensing. For this, a Luenberger-type state observer is introduced with observer corrections entering both the PDE and the boundary conditions. The backstepping method for PDEs is employed and suitably extended to systematically determine the output injection gains, which ensures the exponential decay of the observer error dynamics. Additionally, an efficient approach for the solution of the arising kernel-PDE of the backstepping transformation is proposed. Finally, some extensions of the backstepping-based observer design for hyperbolic PIDEs are presented.
• Georg Kresse (Computational materials Physics, Uni Wien)
  Schrödinger equation for many electrons: an unsolved or unsolvable problem
  The solution of the many particle Schrödinger equation scales combinatorial with the basis set size (M) and the number of electrons N, specifically the scaling is (M over N). This makes an exact solution of the many particle Schrödinger equation practically impossible for all but the simplest few electron systems. Finding approximate solutions to the Schrödinger equation has been an active field of research for almost 50 years in physics and chemistry, but progress has been slow and at best incremental. This talk summarises successful approaches to the solution of the Schrödinger equation. Usually the approaches rely on second quantization (a compact algebra for Fermions), and either Monte-Carlo procedures or an exponential ansatz for the many body wave function. Whereas the Monte-Carlo procedures retains exponential scaling with the particle number, the exponential ansatz yields algebraic scaling with respect to the basis set size, which is quite remarkable considering that the original problem is generally considered to be NP-hard. Alternative approaches, such as tensor networks will be also briefly mentioned.
• Dirk Praetorius (Analysis and Scientific Computing, TU Wien)
  Numerical integration of the Landau-Lifshitz-Gilbert equation
  The nonlinear Landau-Lifshitz-Gilbert equation (LLG) describes time-dependent micromagnetic phenomena in terms of a magnetization field m which solves LLG. Numerical challenges arise from an inherent non-convex constraint |m| = 1 and a possibly nonlinear coupling with other partial differential equations to describe, e.g. magnetostrictive effects or the interaction of m with spin-polarized currents. In our talk, we will discuss numerical integrators which are proved to be unconditionally convergent in the sense that any CFL-type coupling of the space discretization and the time stepping is avoided. A particular focus will be on IMEX-type integrators which also decouple the time stepping of LLG and the coupled second PDE.
• Stefan Rotter (Theoretical Physics, TU Wien)
  Non-Hermitian physics close to exceptional points
  
I will discuss our recent work on non-Hermitian systems with gain or loss in which so-called "exceptional points" appear. These are non-Hermitian degeneracies at which two eigenvalues coalesce in both their real and imaginary parts. As we could recently demonstrate in collaboration with several experimental groups, such exceptional points are a veritable source of non-trivial phenomena. I will discuss our recent insights on this matter as well as open questions and possible future directions.
• Karl Rupp (Microelectronics, TU Wien)
  Massively parallel numerical solution of PDEs: challenges and opportunities
  The efficient numerical solution of PDEs on current supercomputers equipped with thousands of cores requires algorithms exposing fine-grained parallelism. Also, all-to-all communication between processors becomes increasingly expensive, hence novel algorithms employ so-called pipelining techniques and avoid communication whenever possible. This talk provides an overview of the current status of research and outlines promising directions for scaling the numerical solution of PDEs to even larger machines.
• Christian Schmeiser (Mathematics, Uni Wien)
  Cell membrane interaction with membrane proteins and with the cytoskeleton
  Biological cells and intracellular compartments are enclosed by the cell membrane, a lipid bilayer, which resists bending and has the tangential behavior of a two-dimensional incompressible fluid. It strongly interacts with other cellular compartments and it is important for the regulation of various cellular processes. A model for the interaction of the cell membrane with the cytokeleton in lamellipodia will be presented, and various modeling and computational issues will be discussed. These are results of cooperations with experimental biologists at IMBA and at IST.
• Jörg Schmiedmayer (Vienna Center for Quantum Science and Technology, Atominstitut, TU-Wien)
  Bose-Einstein condensates described by the Gross-Pitaevskii equation - numerics helps experiment
  In our experiments we study the equilibrium and out of equilibrium properties of ultra-cold one-dimensional degenerate quantum fluids. To a good approximation our experiments are described by the Gross-Pitaevskii equation (GPE), and simulating the finite temperature states and their dynamics guides the experiments. I will give a few examples ranging from relaxation in 1d systems, to dynamics and optimal control. Finally I will discuss physical situations when the stochastic GPE is not sufficient enough and one needs to apply multi-configurational methods like MCTDHB to describe the many body dynamics.
• Herbert Steinrück (Fluid Mechanics and Heat Transfer, TU Wien)
  Marginal separation, the Fisher equation and blow-ups
  Problem formulation: In flows around an airfoil, small separation bubbles may occur near the leading edge. The structure of the flow near these bubbles can be analyzed using an asymptotic expansion with respect to large Reynolds numbers and small deviations of the angle of attack from a critical value. An integro-differential equation can be obtained as a solvability condition which describes the local flow behavior. For two-dimensional, stationary flows this equation has two, one or no solution depending on the scaled angle of attack. Near the bifurcation point a further analysis for weakly three-dimensional, slowly time-varying flows is possible yielding the so-called Fisher-Kolmogoroff-Petrovsky-Piscounoff (FKPP) equation.
  Mathematical Problem: The FKPP-equation has solutions with a finite time blow-up. From a physical point of view, these blow-ups have an interpretation as bubble bursts when fluid is rapidly ejected from the boundary in a very short event. After that, the boundary-layer flow calms down again. Thus, the question arises if there is a meaningful mathematical concept to continue the solution after the (local) blow-ups and how to handle the blow-ups numerically.

Schedule:

09:00 - 09:45	Georg Kresse
09:50 - 10:20	Coffee break
10:20 - 11:05	Stefan Rotter
11:10 - 11:55	Dirk Praetorius
12:00 - 13:30	Lunch buffet
13:30 - 14:15	Christian Schmeiser
14:20 - 15:05	Jörg Schmiedmayer
15:10 - 15:40	Coffee break
15:40 - 16:10	Herbert Steinrück
16:15 - 16:45	Lukas Jadachowski
16:50 - 17:20	Karl Rupp