BOUNDARY INTEGRAL EQUATION METHODS FOR HIGH FREQUENCY SCATTERING PROBLEMS

该项目涉及用于计算模拟高频波散射问题的新的数值方法的发明、分析和实现。这些问题有着广泛的应用，例如，在模拟雷达、声纳、声屏障、医学超声和大气粒子的辐射散射方面。我们的研究得到了四个工业/研究组织的支持，他们组成了一个指导委员会，将为该项目提供激励性的物理应用。我们在该项目中面临的主要技术困难是准确计算高度振荡(即变化非常快)的波解。标准的近似技术通常将问题的领域分解成小的“元素”，并对每个元素使用简单的(例如多项式)近似。众所周知，每个波长需要大约5-10个元素才能达到可接受的精度，因此所需的计算工作至少与波的频率成比例增长(通常比这个速度更快)。在这个意义上，这些方法被称为随着频率的增加而不稳健。我们将设计、分析和实现新的稳健方法，对于这些方法，获得固定精度的成本是有界的(或者在最坏的情况下，随着频率的增加而增长非常缓慢)。该程序不仅涉及解决高度振荡解的近似问题，而且还涉及(这通常更难)分析支配它们的方程的稳定性和条件性(即敏感性)。我们将用来实现我们的目标的主要工具是高频波现象的大量渐近技术，其中一些在数学和物理界是众所周知的，但到目前为止在数值计算中很少使用。我们项目的一部分将涉及新的渐近分析的推导，并将它们以适合于数值分析的形式加以应用。散射问题将以这样一种方式重新表述，即显式地(从而精确地)处理解的高频部分，只留下可以以低成本完成的低频分量的近似。这种方法在一致性、稳定性、条件性和数值积分方面带来了新的、具有挑战性的深层次问题，这些问题必须在严格确定方法的稳健性之前得到解决。我们面临的一些问题需要应用技术，因为我们以前的研究知道这些技术会奏效；另一些问题需要重大的风险因素和冒险精神。该项目将涉及四名调查人员和两名PDRA，一名主要从事分析，另一名主要从事科学计算。两者还将致力于与我们的合作者相关的应用程序。

This project concerns the invention, analysis and implementation of new numerical methods for computationally simulating high frequency wave scattering problems. These problems have diverse applications, for example in modelling radar, sonar, acoustic noise barriers, medical ultrasound, and scattering of radiation by atmospheric particles. Our research is supported by four industrial/research organisations who comprise a steering committee and will provide motivating physical applications for the project.The chief technological difficulty which we face in the project is that of computing accurately wave solutions which are highly oscillatory (i.e. varying very quickly). Standard approximation techniques usually break the domain of the problem up into small ''elements'', and use simple (e.g. polynomial) approximations on each element. Then it is known that about 5-10 elements are required in each wavelength to achieve acceptable accuracy, and so the computational work required grows at least in proportion to the frequency of the wave (and often faster than this). In this sense the methods are termed ''not robust'' as frequency increases.We are going to devise, analyse and implement new robust methods for which the cost to obtain a fixed accuracy is bounded (or at worst grows very slowly) as the frequency increases. The programme involves solving problems not only of approximation of highly oscillatory solutions, but also (and this is often harder) analysing the stability and conditioning (i.e. sensitivity ) of the equations which govern them.The chief device which we will use to achieve our objective is the great body of asymptotic techniques for high frequency wave phenomena, some of which which are well-known in the mathematics and physics communities but which have so far been very little used in numerical computation. Part of our project will involve the derivation of new asymptotic analyses and putting them in a form suitable for use in numerical analysis. Scattering problems will be reformulated in such a way that high frequency parts of the solution are handled explicitly (and thus exactly), leaving only the approximation of low frequency components which can be done with low cost. This approach leads to new, challenging and deep problems in consistency, stability, conditioning and numerical integration which must be solved before the robustness of the methods can be rigorously determined. Some of the problems which we face require applying technology which we know will work because of our earlier studies; others require a significant element of risk and a spirit of adventure.The project will involve four investigators and two PDRAs, one involved primarily in analysis and one primarily in scientific computation. Both will also work on applications of relevance to our collaborators.