# Vorträge in 2016

# Kolloquiumsvortrag am Freitag, d. 9.12.2016

### H-matrices on many-core hardware with applications in parametric PDEs

**Peter Zaspel**(Interdisciplinary Center for Scientific Computing (IWR), Universität Heidelberg)

Hierarchical matrices approximate specific types of dense matrices, e.g., from discretized integral equations, kernel-based approximation and Gaussian process regression, leading to log-linear time complexity in dense matrix-vector products. To be able to solve large-scale applications, H-matrix algorithms have to be parallelized. A special kind of parallel hardware are many-core processors, e.g. graphics processing units (GPUs). The parallelization of H-matrices on many-core processors is difficult due to the complex nature of the underlying algorithms that need to be mapped to rather simple parallel operations.

We are interested to use these many-core processors for the full H-matrix construction and application process. A motivation for this interest lies in the well-known claim that future standard processors will evolve towards many-core hardware, anyway. In order to be prepared for this development, we want to discuss many-core parallel formulations of classical H-matrix algorithms and adaptive cross approximations.

In the presentation, the use of H-matrices is motivated by the model application of kernel-based approximation for the solution of parametric PDEs, e.g. PDEs with stochastic coefficients. The main part of the talk will be dedicated to the challenges of H-matrix parallelizations on many-core hardware with the specific model hardware of GPUs. We propose a set of parallelization strategies which overcome most of these challenges. Benchmarks of our implementation are used to explain the effect of different parallel formulations of the algorithms.

**Zeit und Ort: **14 Uhr c.t., Ort: Raum Ü2/K, Ludewig-Meyn-Straße 2

# Kolloquiumsvortrag am Freitag, d. 27.5.2016

### A Posteriori Error Majorant for Elliptic Partial Differential Equations with Applications to Homogenization

**Stefan Sauter**(Institut für Mathematik, Universität Zürich)

In our talk, we will present new two-sided estimates of modeling errors for linear elliptic boundary value problems with periodic coefficients solved by homogenization method. Our approach is based on the concept of functional a posteriori error estimation. The estimates are obtained for the energy norm and use solely the global flux of the non-oscillatory solution of the homogenized model and solutions of some cell problem. Numerical tests illustrate the efficiency of the estimates.

**Zeit und Ort: **14 Uhr c.t., Ort: Raum Ü2/K, Ludewig-Meyn-Straße 2

# Kolloquiumsvortrag am Freitag, d. 12.2.2016

### Stability and convergence of Galerkin discretizations of the Helmholtz equation

**Jens Markus Melenk**(Institut für Analysis und Scientific Computing,TU Wien)

We consider boundary value problems for the Helmholtz equation at large wave numbers **k**. In order to understand how the wave number **k** affects the convergence properties of discretizations of such problems, we develop a regularity theory for the Helmholtz equation that is explicit in **k**. At the heart of our analysis is the decomposition of solutions into two components: the first component is an analytic, but highly oscillatory function and the second one has finite regularity but features wavenumber-independent bounds. This new understanding of the solution structure opens the door to the analysis of discretizations of the Helmholtz equation that are explicit in their dependence on the wavenumber **k**. As a first example, we show for a conforming high order finite element method that quasi-optimality is guaranteed if

- the approximation order
**p**is selected as**p = O(log k)**and - the mesh size
**h**is such that**kh/p**is small. - The work is joint with
**Stefan Sauter**(Zürich).**Zeit und Ort:**14 Uhr c.t., Ort: Raum Ü2/K, Ludewig-Meyn-Straße 2# Kolloquiumsvortrag am Freitag, d. 29.1.2016

### Maxwell’s Equations and the Hodge-Laplacian

Prof. Dr.**Ralf Hiptmair**(Seminar of Applied Mathematics, ETH Zürich, Switzerland)The linear Maxwell equations in frequency domain can be recast in several different forms, as first-order system, as second-order equation involving the double-

**curl**operator, or a second-order equation featuring the Hodge-Laplacian**−∆ := curl curl − grad div**.Variational equations or boundary integral equations arising from these different formulations are equivalent, but pose very different challenges when it comes to discretization. For instance, there is the startling failure of smooth functions to be dense in

**H**for certain domains_{0}(curl, Ω) ∩ H(div, Ω)**Ω**. This foils direct Galerkin discretization of the Hodge-Laplacian and enforces the use of a mixed approach.Similarly, first-kind boundary integral operators associated with

**∆u = 0**display a strange lack of coercivity. The culprit is the failure of certain combinations of boundary conditions for**∆**; to ensure uniqueness of solutions of the related boundary value problems.**Zeit und Ort:**14 Uhr c.t., Ort: Raum Ü2/K, Ludewig-Meyn-Straße 2