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，该过程中的一个瓶颈往往是所得到的矩阵方程的高效和有效解。除了在复杂的应用中可能遇到的固有困难之外，我们在处理时间谐波传播问题时还经常遇到额外的困难。这些困难源于所涉及的操作员的不确定或非自伴性质。这需要在求解器的设计和分析方面进行范式转换。这个项目的目的是建立和分析基于广义特征值问题的新一代光谱预处理器，允许关于介质的物理属性的稳健行为。这需要结合数值分析和光谱分析工具。结果将是数学的和实际的，因为这将从根本上改变解算器的技术状态，结果将被合并到开源软件中。目前，国际上有大量的研究致力于频域(取决于频率[小写欧米伽])或时间调和偏微分方程组的有效数值解，这是因为在许多应用(包括电磁散射)中，频域公式是一种可行的替代时间域的方法，前提是有适当有效的方法来求解出现的大型线性系统。求解这个方程在数学上是困难的，特别是对于高频问题。自由度数N与[小写欧米伽]的增长使实际的三维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，由于系统是不确定的，没有“好”的预条件，迭代次数会随着[小写欧米伽]的增加而迅速增长。在这种情况下，“良好”意味着人们希望迭代的次数理想地独立于[小写omega]，并且粗略地说，预条件函数尽可能地可并行化。因此，我们希望实现并行可伸缩性以及关于波数的稳健性。对于自伴的强制标量椭圆型方程，区域分解(DD)方法是一个很有吸引力的预条件选择，因为它们本质上是并行的，并且具有可伸缩性和健壮性。对于自伴的强制标量椭圆型偏微分方程组，已经有了相当成熟的DD方法理论，它允许非常一般的分解和粗网格，但对于不定波动问题的DD方法(以及其他求解器，如多重网格)的分析在很大程度上是一个开放的问题。粗网格允许在预处理器中全局传输信息，并通过实现并行可伸缩性来增加相对于子域数量的稳健性。然而，用于频域波动问题的实用粗空间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)。获得实用粗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上的本征函数，因此很难从数学的角度来描述(即使从数值的角度来看，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：-对频谱非自伴问题进行数学分析，特别是与Dirichlet-to-Neumann算子有关的问题；-设计频谱两级预处理器的一般理论；-对来自地球物理和电磁应用的不同基准测试用例的并行特性进行数值评估和开发。