Modul: MAT870 Zurich Colloquium in Applied and Computational Mathematics

## Hybrid compression of boundary element matrices for high-frequency Helmholtz problems

Prof. Dr. Steffen Börm talk

Speaker invited by: Prof. Dr. Stefan Sauter

**Date:** 10.10.18 **Time:** 16.15 - 17.45 **Room:** ETH HG E 1.2

Fast summation methods are well-established for non-local operators arising in the context for electrostatics, molecular dynamics, or linear elastostatics, where they can reduce the complexity from O(n²) to O(n) or O(n log n).

In the case of high-frequency Helmholtz equations, the situation is significantly more challenging: although the kernel function is still analytic, standard approximations, e.g., by polynomials, converge only slowly. This problem can be solved by splitting the kernel function into a smooth factor and a plane wave and approximating only the smooth factor, e.g., by interpolation. We obtain a fast summation scheme, but the storage requirements are fairly high: on one hand, multiple directions have to be handled simultaneously to reach a suitable accuracy. On the other hand, the computational domain has to be split into fairly small subdomains. Fortunately, we can combine the interpolation-based approximation with algebraic techniques to reduce the storage requirements, and this allows us to handle large problems efficiently. This talk gives an introduction to the directional interpolation approach, illustrates its properties in numerical examples, describes the algebraic re-compression, and demonstrates that it can significantly improve the overall performance of the method.