Spectral Preconditioners for High Frequency Wave Propagation Problems

在为大规模物理应用开发现实的数学模型时，过程中的一个瓶颈通常是得到的矩阵方程的高效解。除了在复杂应用中可能遇到的固有困难外，在处理时谐波传播问题时，我们经常遇到额外的困难。这些困难源于所涉及的算子的不确定或非自伴随性质。这需要在求解器的设计和分析中进行范式转换。该项目的目的是建立和分析基于广义特征值问题的新一代频谱预调节器，允许对介质的物理性质具有稳健的行为。这需要数值分析和光谱分析工具的结合。结果将是数学和实用的，因为这将从根本上改变求解器的现状，结果将被纳入开源软件。目前，国际上有大量的研究工作致力于频域（取决于频率[小写ω]）或时谐微分方程的有效数值解，这是因为在许多应用（包括电磁散射）中，如果有合适的有效方法可用于解决出现的大型线性系统，那么频域公式是时域的可行替代方案。求解这个方程在数学上是困难的，特别是对于高频问题。随着[小写ω]自由度N的增加，实际的三维问题甚至超出了最先进的直接求解器的范围，因此必须使用(F)GMRES等迭代方法。然而，由于系统是不确定的，如果没有一个“好的”前置条件，迭代的次数会随着[小写的ω]迅速增长。在这种情况下，“好”意味着人们希望迭代的数量在理想情况下独立于[小写的ω]，并且对于前置条件，粗略地说，尽可能地并行化。因此，我们希望同时实现并行可扩展性和对波数的鲁棒性。域分解（DD）方法是一种有吸引力的预处理方法，因为它们本质上是并行的，并且已知对于自伴随强制标量椭圆偏微分方程具有可扩展性和鲁棒性。对于自伴随强制标量椭圆偏微分方程，有一个相当发达的DD方法理论，它允许非常一般的分解和粗网格，但是对不定波问题的DD方法（以及其他求解方法，如多重网格）的分析在很大程度上是一个开放的问题。粗网格允许在预条件中全局传递信息，并通过实现并行可扩展性来增加相对于子域数量的鲁棒性。然而，频域波问题的实际粗网格的设计在很大程度上仍然是开放的（部分原因是缺乏允许粗网格的理论框架）。一种获得实际粗糙空间的方法是使用振荡基函数。然而，这些基函数通常是非自伴随算子上的特征函数，因此很难从数学的角度来表征（即使从数值的角度来看，它们在给定构型上的应用似乎是成功的）。提出的工作计划包括：-谱非自伴随问题的数学分析，特别是与Dirichlet-to-Neumann算子有关的问题；-谱两级预调节器的一般理论设计；-在地球物理和电磁应用的非均质基准测试案例上的并行特性的数值评估和开发。

When developing realistic mathematical models for large-scale physical applications, one bottleneck in the procedure is often the efficient and effective solution of the resulting matrix equations. In addition to the inherent difficulties one can encounter in complex applications, we often experience extra difficulties when dealing with time-harmonic wave propagation problems. These difficulties stem from the indefinite or non-self-adjoint nature of the operators involved. This requires a paradigm shift in the design and analysis of solvers. The aim of this project is to build and analyse a new generation of spectral preconditioners based on generalised eigenvalue problems allowing a robust behaviour with respect to the physical properties of the medium. This requires a combination of numerical analysis and spectral analysis tools. The outcome will be both mathematical and practical, as this will fundamentally change the state of the art of solvers, and the results will be incorporated in open-source software. There is currently a large international research effort dedicated to the efficient numerical solution of frequency-domain (depending on the frequency [lowercase omega]) or time-harmonic PDEs, driven by the fact that in many applications (including EM scattering), the frequency-domain formulation is a viable alternative to the time domain, provided suitably efficient methods are available for solving the large linear systems that arise. Solving this equation is mathematically difficult especially for high-frequency problems. The growth of the number of degrees of freedom N with [lowercase omega] puts practical 3-d problems out of range of even state-of-the-art direct solvers, and so iterative methods such as (F)GMRES must be used. However, the fact that the systems are indefinite, without a "good" preconditioner, the number of iterations grows rapidly with [lowercase omega]. In this context, "good" means that one wants the number of iterations to ideally be independent of [lowercase omega], and for the preconditioner to be, roughly speaking, as parallelisable as possible. We therefore wish to achieve both parallel scalability together with the robustness with respect to the wave number. Domain decomposition (DD) methods are an attractive choice for preconditioners, since they are inherently parallel and known to be scalable and robust for self-adjoint coercive scalar elliptic PDEs.For self-adjoint coercive scalar elliptic PDEs there is a fairly well-developed theory for DD methods that allows very general decompositions and coarse grids, but the analysis of DD methods (and other solvers such as multigrid) for indefinite wave problems is largely an open problem. Coarse grids allow global transfer of information in the preconditioner, and increase robustness with respect to the number of the subdomains by achieving parallel scalability. The design of practical coarse spaces for frequency-domain wave problems, however, is still largely open (partly due to the lack of a theoretical framework that allows coarse grids). One approach to obtain practical coarse spaces is to use oscillatory basis functions. However, these basis functions are often eigenfunctions on non-self-adjoint operators and hence difficult to characterize from a mathematical point of view (even when their application to given configurations seems to be successful from a numerical point of view). The proposed plan of work includes:-mathematical analysis of spectral non-self-adjoint problems, in particular, such that arise in connection with Dirichlet-to-Neumann operators;-design of a general theory for a spectral two-level preconditioner;-numerical assessment and exploitation of the parallel properties on heterogenous benchmark test cases from geophysical and electromagnetic applications.