Advances in Computational Scattering (06w5065)

Objectives

Scattering is the study of the interaction of waves with obstacles. These obstacles could be anything from gratings, tumours, to ships; the waves could be electromagnetic, elastic, or acoustic. This is a very well-established field of study, mathematically, but only a limited number of exterior scattering problems can be solved analytically.

In recent years, many engineers, computational scientists and numerical analysts have investigated numerical algorithms to simulate scattering, including (but not limited to) the use of integral equations, finite element methods, series methods, geometrical optics, absorbing layers and spectral methods. These developments have helped to make computational scattering algorithms indispensible in several industries for design purposes. Many deep mathematical questions have also been raised as a consequence of this development. The overarching principle and the central challenge in computational scattering is to approximate the scattered wave as accurately and efficiently as possible. Indeed, one may identify the major open problems in the field as the development of high-frequency, high-accuracy algorithms, efficient and accurate absorbing boundary conditions, and preconditioners for discretizations of exterior scattering problems. Regardless of the specific algorithms one may use to study scattering, these issues must be confronted head-on.

By bringing together experts on different aspects of computational scattering, this workshop aims to identify specific common challenges, and to make progress towards resolving these. The major themes of the workshop will be:
1. Absorbing boundary conditions: When finite difference, finite element or spectral methods are used to resolve the scattered wave near the obstacle, the computational region must be restricted to be finite. This truncation is achieved by means of absorbing or exact boundary conditions. These conditions can be implemented in various ways, e.g. by using boundary integral equations, series implementations, or the perfectly matched layer of Berenger. No matter which techniques are used, the goal is to obtain as accurate an approximation to solutions of the original scattering problem, as efficiently as possible. Unfortunately, these are competing requirements. There are significant implementation and/or accuracy issues which remain open problems.
2. High-frequency methods: The conditioning and accuracy of most discretization techniques for scattering problems depend crucially on the wave number of the incident wave. In addition, there are algorithms suitable for moderate frequency scattering, and others appropriate for geometrical optics. A key challenge in this field remains the development and analysis of an algorithm which works over a large range of frequencies, and whose performance can be controlled independant of the frequency. In this workshop developments of new high-accuracy methods suitable for a large range of wavenumbers will be discussed.
3. Preconditioning: The efficient solution of the linear systems obtained as a consequence of the discretization of exterior scattering problems is an open problem, since these systems are typically dense. �Canned� preconditioning techniques have been rather unsuccessful. Preconditioning techniques have recently been developed for boundary integral equations used in this context; there is a pressing need for a systematic preconditioning strategy for other algorithms as well.

There are many disparate groups of numerical analysts and engineers who are independently developing algorithms in this direction. This workshop aims to bring together experts from the mathematics and engineering communities, with a view to cross-fertilization and communication. The format includes a few overview-style talks each day, followed by informal discussion periods. These discussions are intended to facilitate the exchange of ideas between experts from different fields. This is particularly important in an area where engineers and mathematicians appear to be evolving a subject in seemingly parallel directions, without much interaction. Participants would learn about other techniques being used to study exterior scattering, discuss common issues and open problems, and hopefully form cross-discipline collaborations. This forum would provide a particularly suitable place for students entering the field to gain an overview of the subject area.