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通常是所得矩阵方程的有效解决方案。除了在复杂的应用程序中遇到固有的困难外，我们在处理时间谐波传播问题时通常会遇到额外的困难。这些困难源于所涉及的操作员的无限或非自我辅助性质。这需要对求解器的设计和分析进行范式转移。该项目的目的是基于普遍的特征值问题来构建和分析新一代的光谱预处理，从而使相对于培养基的物理特性具有稳健的行为。这需要组合数值分析和光谱分析工具。结果既是数学又是实用的，因为这将从根本上改变求解器的最新状态，并且结果将纳入开源软件中。目前，有一项大量的国际研究工作致力于有效的频域数值解决方案（取决于频率[小写的欧米茄]）或时机PDE，这是在许多应用程序中（包括EM散射）中驱动的，频率配方在时代域是可行的替代方案，可用于求职，可用于求解大型的系统，可用于求解大型的linise。在数学上求解该方程非常困难，尤其是对于高频问题。 [小写欧米茄]的自由度n数量的增长使实用的3D问题来自甚至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。但是，这些系统是无限期的，没有“良好”的预处理，迭代次数随[小写欧米茄]迅速增长。在这种情况下，“良好”意味着一个人希望理想地迭代数量独立于[小写欧米茄]，而预先调节器则大致说明了尽可能平行的。因此，我们希望达到同时的可伸缩性以及相对于波数的鲁棒性。 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对于不确定的波浪问题，Multigrid）在很大程度上是一个空旷的问题。粗网格允许在预处理中进行全局信息传递，并通过实现并行的可扩展性来增加对子域数量的鲁棒性。但是，用于频域波问题的实用粗空间的设计基本上是开放的（部分是由于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。但是，这些基础函数通常是对非自我辅助操作员的本征函数，因此从数学角度来看很难表征（即使从数值角度来看，即使它们的应用程序的应用似乎也成功了）。提出的工作计划包括： - 频谱非自动辅助问题的数学分析，特别是与dirichlet到neumann运营商有关； - 对光谱两级预处理的一般理论的设计； - 核对元素的频谱式评估和对平行性属性的质量和利用质量的geoplysical和geoplysical and geoplysical和geoply equall offeriation。