Talks

All existing QR decompositions for hierarchical matrices suffer from numerical drawbacks that limit their use in many applications. In this talk, I will present a new method based on the recursive WY-based QR decomposition by Elmroth and Gustavson. It is an extension of an already existing method for a subclass of hierarchical methods developed by Kressner and Susnjara.

I will try to keep things as simple as possible and give a short introduction to hierarchical matrices as well. Previous knowledge of hierarchical matrices is not necessary to understand the basic ideas and main obstacles of the new algorithm.

In the talk, we shall develop a general framework for the treatment of both deterministic and stochastic homogenisation problems for evolutionary equations. The versatility of the methods allow the unified treatment of static, dynamic as well as mixed type problems.

The efficient use of modern supercomputers has become one of the key challenges in computational science. New mathematical concepts are needed to fully exploit massively parallel architectures. For the numerical solution of time-dependent processes, time-parallel methods have opened new ways to overcome scaling limits. With the "parallel full approximation scheme in space and time" (PFASST), multiple time-steps can be integrated simultaneously. Based on spectral deferred corrections (SDC) methods and nonlinear multigrid ideas, PFASST uses a space-time hierarchy with various coarsening strategies to maximize parallel efficiency. In numerous studies, this approach has been used on up to 448K cores and coupled to space-parallel solvers with finite differences, spectral methods or even articles for discretization in space. Yet, since the integration of SDC or PFASST into an existing application code is not straightforward and the potential gain is typically uncertain, we will present in this talk our Python prototyping framework pySDC. It allows to rapidly test new ideas and to implement first toy problems more easily. We will also discuss the transition from pySDC to application-specific implementations and show recent use cases.

The plasma in a fusion reactor is heated by neutral beam injection: injecting high energy neutrons which quickly ionize and swirl around in the reactor's magnetic fiel. Modelling this process requires solving the Lorentz equations numerically over long times (up to a second) with very small time steps (order of nanoseconds), which means very many time steps and thus long simulation times (from days up to a week). The talk will introduce GMRES-Boris-SDC (GBSDC), a new time stepping algorithm that can reduce computational cost compared to the currently used Boris method. The method is a potpourri of various numerical techniques, including the GMRES linear solver, spectral deferred corrections, the velocity Verlet scheme and the Boris trick. I will describe the algorithm and show examples of its performance for benchmarks with varying degree of realism.

This is joint work with Dr Krasymyr Tretiak, School of Mathematics, University of Leeds.

During their work-term at TUHH the two Canadian students worked on projects relating to current research in the institute.
As their term comes to an end they will present their ongoing work in short talks.

In this talk we study sufficient conditions for obserability of systems in Banach spaces. In an abstract Banach space setting we show that an uncertainty relation together with a dissipation estimate implies an observability estimate with explicit dependence on the model parameters. Our approach unifies and generalizes the respective advantages from earlier results obtained in the context of Hilbert spaces. As an application we consider elliptic operators on Lp spaces. Combined with the well-known relation between observability and controllability we derive sufficient conditions for null-controllability and bounds on the control cost.
The talk is based on joint work with Christian Seifert and Martin Tautenhahn.

We give a unified approach to handle the problem of extending functions with values in a locally convex Hausdorff space $E$ over the field $\mathbb{K}=\mathbb{R}$ or $\mathbb{C}$, which have weak extensions in a space $\mathcal{F}(\Omega,\mathbb{K})$ of scalar-valued functions on a set $\Omega$, to functions in a vector-valued counterpart $\mathcal{F}(\Omega,E)$ of $\mathcal{F}(\Omega,\mathbb{K})$. The main tool is the representation of vector-valued functions as linear continuous operators.