Spectral Preconditioners for High Frequency Wave Propagation Problems

用于高频波传播问题的频谱预处理器

• 批准号: 2595936
• 金额: --
• 依托单位: University of Strathclyde
• 依托单位国家: 英国
• 项目类别: Studentship
• 财政年份: 2021
• 资助国家: 英国
• 起止时间: 2021 至 无数据
• 项目状态: 未结题
• 来源: https://gtr.ukri.org/projects?ref=studentship-2595936
• 关键词: Spectral; Preconditioners; Frequency; Wave; Propagation

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.

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所得矩阵方程的高效且有效的解决方案。除了在复杂应用中遇到的固有困难之外，我们在处理时谐波传播问题时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性或非自伴性。这需要求解器的设计和分析进行范式转变。该项目的目的是基于广义特征值问题构建和分析新一代光谱预处理器，从而在介质的物理特性方面实现稳健的行为。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技术水平，并且结果将被纳入开源软件中。目前，国际上有大量研究工作致力于频域（取决于频率[小写欧米茄]）或时谐偏微分方程的有效数值求解，这是由于在许多应用（包括电磁散射）中，只要有适当有效的方法可用于求解出现的大型线性系统，频域公式是时域的可行替代方案。求解这个方程在数学上很困难，尤其是对于高频问题。 [小写 omega] 自由度 N 数量的增长使得实际的 3-d 问题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，事实上，系统是不确定的，如果没有“好的”预处理器，迭代次数会随着[小写欧米茄]而迅速增长。在这种情况下，“好”意味着人们希望迭代次数理想地独立于[小写欧米茄]，并且粗略地说，预处理器尽可能可并行化。因此，我们希望实现并行可扩展性以及波数方面的鲁棒性。域分解 (DD) 方法对于预处理器来说是一个有吸引力的选择，因为它们本质上是并行的，并且对于自伴强制标量椭圆偏微分方程具有可扩展性和鲁棒性。对于自伴强制标量椭圆偏微分方程，有一个相当完善的 DD 方法理论，允许非常一般的分解和粗网格，但是 DD 方法的分析（以及其他求解器，例如 多重网格）对于不定波问题在很大程度上是一个开放问题。粗网格允许在预处理器中进行全局信息传输，并通过实现并行可扩展性来提高子域数量的鲁棒性。然而，频域波问题的实际粗空间设计仍然在很大程度上是开放的（部分原因是缺乏允许粗网格的理论框架）。获得实用粗糙空间的一种方法是使用振荡基函数。然而，这些基函数通常是非自伴算子的本征函数，因此很难从数学角度来表征（即使从数值角度来看它们对给定配置的应用似乎是成功的）。拟议的工作计划包括： - 光谱非自伴随问题的数学分析，特别是与狄利克雷到诺依曼算子相关的问题； - 光谱两级预处理器的一般理论的设计； - 来自地球物理和电磁应用的异质基准测试用例的并行属性的数值评估和开发。