# Research: Overview

Currently our research activities are focused on the following topics:

- H²-matrix arithmetics: Certain densely populated matrices appearing, e.g., when treating integral or differential equations, can be approximated efficiently by H²-matrices. This project is dedicated to developing algorithms for carrying out fundamental operations like multiplication, inversion, or factorization of H²-matrices without sacrificing the efficiency afforded by matrix compression.
- Approximation of integral equations: Discretization schemes for integral equations, e.g., the boundary integral method for elliptic partial differential equations, lead to large densely populated matrices. Fortunately, the corresponding kernel functions typically have properties that can be used to construct compression schemes. We are currently focusing on an approach that applies quadrature to Green's representation formula in order to obtain a low-rank approximation that can be further improved by algebraic algorithms.
- Approximation of the Helmholtz equation: Applying the boundary integral method to the wave equation leads to a particularly interesting integral equation involving the Helmholtz kernel function. This kernel function oscillates at the same frequency as the wave, so polynomial approximations are fairly unattractive in the high-frequency setting. We are investigating approximation techniques that combine polynomials and plane waves to obtain significantly improved approximation results.
- Matrix-Galerkin methods: The solutions of certain matrix equations appearing, e.g., in the context of stochastic partial differential equations or control theory, frequently have properties that make them accessible to H²-matrices. We have developed algorithms that can construct H²-matrix approximations of these solutions directly by Galerkin's technique.