Beschreibung

Surface and volume integral equations are widespread modelling tools for many computational science and engineering applications, e.g., in computational electromagnetics and acoustic simulations, multi-component fluid flows, multi-phase problems in materials science, energy research, elasticity, other fields. The numerical solution of integral equations typically requires to solve large dense linear systems that are highly demanding of fast methods with reduced memory and algorithmic costs, and of large computational resources. For example, scattering of a plane wave by a perfectly conducting spherical object with a diameter of 1800 wavelengths modelled by surface integral equations gives rise to a fully populated matrix with more than three billion unknowns. Nearly all truly dense large linear systems arising from scientific applications come from the solutions of integral equations, and in emerging fields of statistical learning and machine/deep learning. Standard direct solution methods are not affordable to solve problems of this size even on modern parallel computers due to their large memory requirements.

The proposed research project will try to fill this gap by developing new matrix solvers based on the so-called hierarchical H2-matrices that provides a highly compact and accurate kernel independent representation of integral equations by fast compression of suitable matrix blocks. One task coordinated by the Free University of Bozen-Bolzano with Purdue University will produce highly efficient approximate factorizations and inversion algorithms in almost linear computational cost and memory footprint using the fast arithmetic of H2-matrices. The problems that this may cause, and the techniques that are required to be deployed to overcome these issues will be analysed. Another task, coordinated by the consortium CREATE will implement and test the proposed algorithmic solutions in CarMa0NL, a fast fusion research simulation software with the unprecedented capability of simultaneously considering three-dimensional effects of conductors surrounding the plasma and the inherent non-linearity of the plasma behaviour itself. The code is currently used worldwide to analyse fusion devices, like the Joint European Torus tokamak, UK and other devices, both existing (TCV, Switzerland; EAST, China), in the construction phase (ITER, France; JT-60SA, Japan) and under design (DEMO, Germany; DTT, Italy). Finally, a second real world case study arising from a boundary integral equations formulation of electromagnetics scattering problems with industrial applications will be considered.

Our goal is to develop numerical tools that can enable very fast simulation with the potential to be used by industry, filling a gap in current software to solve such problems.

Partner

Lead Partner Freie Universität Bozen, Fakultät für Ingenieurwesen