Phase-space-inspired numerical methods for high-frequency wave scattering: from semiclassical analysis through numerical analysis to open-source software

Introduction

Designing fast and reliable algorithms to numerically simulate the behaviour of high-frequency acoustic and electromagnetic waves is a longstanding open problem in computational mathematics. Furthermore, these waves underpin a plethora of communication and imaging technologies; therefore any progress towards solving this problem will have wide impact.

Project Objectives

The project aims to design, analyse, and implement in open-source software new methods for the numerical solution of high-frequency acoustic and electromagnetic wave scattering problems.

Key Features

These methods will be:

• Faster
• More reliable
• More widely applicable than current state-of-the-art methods

This improvement will be achieved by exploiting techniques from pure mathematics specifically designed to study high-frequency problems. These techniques are from the area of semiclassical analysis and are based on studying functions in phase space (the combination of physical and frequency space).

Current State of Research

Since both numerical analysis and semiclassical analysis of high-frequency wave scattering problems are mature areas of mathematics, it is surprising that these two fields have operated largely in isolation from each other.

Lack of Integration

Essentially none of the deep insights about high-frequency waves from semiclassical analysis have been used in the design of numerical methods for these problems.

Implementation Plan

This project will exploit this untapped potential and implement the resulting methods in the open-source multiphysics finite-element software FreeFEM (https://freefem.org/) in easy-to-use packages.

Target Audience

These packages will be designed for researchers in the applied sciences and industry seeking to simulate the behaviour of high-frequency acoustic and electromagnetic waves.

Collaboration

These goals will be achieved by combining the expertise of the three PIs:

• JG, an expert in semiclassical analysis
• ES, an expert in the numerical analysis of high-frequency scattering
• PT, an expert in scientific computing and the lead developer of FreeFEM.