Effective preconditioners for linear systems in fractional diffusion

分数扩散线性系统的有效预处理器

• 批准号: EP/R009821/1
• 负责人: Jennifer Pestana
• 金额: $ 11.82万
• 依托单位: University of Strathclyde
• 依托单位国家: 英国
• 项目类别: Research Grant
• 财政年份: 2018
• 资助国家: 英国
• 起止时间: 2018 至 无数据
• 项目状态: 已结题
• 来源: https://gtr.ukri.org/projects?ref=EP%2FR009821%2F1
• 关键词: Effective; preconditioners; linear; systems; fractional

Mathematical models of the diffusion of signals and particles are important for gaining insights into many key challenges facing the UK today. These problems include understanding how fluid flows through the ground, which helps to ensure that we have safe drinking water; characterising the propagation of electrical impulses through the heart, which aids our understanding of heart disease; and developing accurate models of financial processes, which improve our economy by providing better predictions of financial markets. This project focuses on so-called fractional diffusion problems, which occur when the diffusion process involves a number of different flow rates or long-range effects. Fractional diffusion occurs in many applications, including the groundwater flow, cardiac electrical propagation, and finance problems listed above. Solving mathematical models of fractional diffusion is challenging, and typically requires a numerical method, i.e. a computer simulation. Usually, the most time-consuming part of this simulation is solving thousands, or even millions, of interdependent linear equations on a computer. Indeed, the time required to solve this system of equations may be so large that we are prevented from simulating fractional diffusion problems that capture the true complexity of real-world applications. Reducing this solve time is thus crucial if we are to generate new scientific insights in important applications involving fractional diffusion. This project will develop new methods for solving these huge systems of equations that are guaranteed to be fast. We will focus on iterative solvers, which are well suited to the class of numerical methods (computer simulations) on which we focus. Iterative solvers of systems of equations compute a new approximation to the solution at each step, and so are fast if a good approximation is found after only a few iterations. However, this is generally only possible if we apply a convergence accelerator, called a preconditioner, which captures the 'essence' of the linear system, but is cheap to use. For many fractional diffusion problems, this preconditioner is currently chosen heuristically, i.e. without theoretical justification. Consequently, the preconditioner may fail to reduce the (very large) computation time needed to solve the linear system. The goal of this project is to propose new preconditioners and iterative methods that are theoretically justified, and hence guaranteed to converge quickly, for a range of fractional diffusion problems. We will develop new software that will enable people with fractional diffusion problems to easily use our improved solvers. Additionally we will apply these fast preconditioners and iterative solvers in a fractional diffusion model of groundwater flow of an important UK aquifer. Solving this model quickly will enable us to better track our drinking water, and identify possible sources of contamination.

信号和粒子扩散的数学模型对于了解当今英国面临的许多关键挑战至关重要。这些问题包括了解流体如何流过地面，这有助于确保我们拥有安全的饮用水；表征通过心脏电脉冲传播的，这有助于我们对心脏病的理解；并开发准确的财务流程模型，通过提供更好的金融市场预测来改善我们的经济。该项目着重于所谓的分数扩散问题，这些问题是在扩散过程涉及多种流量或远距离效应时发生的。分数扩散发生在许多应用中，包括地下水流，心脏电气传播以及上面列出的金融问题。求解分数扩散的数学模型具有挑战性，通常需要一种数值方法，即计算机模拟。通常，此模拟最耗时的部分是在计算机上求解数千甚至数百万的相互依赖线性方程。实际上，解决该方程系统所需的时间可能是如此之大，以至于我们无法模拟捕获现实世界应用的真实复杂性的分数扩散问题。因此，如果我们要在涉及分数扩散的重要应用中产生新的科学见解，那么减少解决时间至关重要。该项目将开发新的方法来解决这些庞大的方程式系统，这些方程式可以保证很快。我们将重点关注迭代求解器，这些迭代求解器非常适合我们关注的数值方法（计算机模拟）类别。方程系统的迭代求解器在每个步骤中计算新的溶液的新近似值，因此，如果仅在少量迭代后发现了良好的近似值，则很快。但是，只有当我们应用一种称为预处理的融合加速器（即捕获线性系统的本质”，但使用便宜的情况下，这通常才有可能。对于许多分数扩散问题，目前是启发式的，即没有理论上的理由。因此，预处理可能无法减少解决线性系统所需的（非常大的）计算时间。该项目的目的是提出理论上有理由的新预设者和迭代方法，因此对于一系列分数扩散问题，保证会迅速融合。我们将开发新的软件，以使具有分数扩散问题的人可以轻松使用我们改进的求解器。此外，我们将在重要的英国含水层的地下水流量的分数扩散模型中应用这些快速的预处理和迭代求解器。快速解决该模型将使我们能够更好地跟踪饮用水，并确定可能的污染源。

项目成果

The asymptotic spectrum of flipped multilevel Toeplitz matrices and of certain preconditionings

• DOI: 10.1137/20m1379666
• 发表时间: 2020-11
• 期刊: ArXiv
• 作者: M. Mazza;J. Pestana

Spectral properties of flipped Toeplitz matrices and related preconditioning

• DOI: 10.1007/s10543-018-0740-y
• 发表时间: 2018-12
• 期刊: BIT Numerical Mathematics
• 影响因子: 1.5
• 作者: M. Mazza;J. Pestana

Preconditioners for Symmetrized Toeplitz and Multilevel Toeplitz Matrices

• DOI: 10.1137/18m1205406
• 发表时间: 2018-12
• 期刊: SIAM J. Matrix Anal. Appl.
• 作者: J. Pestana

Analysis of parallel Schwarz algorithms for time-harmonic problems using block Toeplitz matrices

• DOI: 10.1553/etna_vol55s112
• 发表时间: 2020-06
• 期刊: ArXiv
• 作者: N. Bootland;V. Dolean;Alexandros Kyriakis;J. Pestana

Preconditioners for Krylov subspace methods: An overview

• DOI: 10.1002/gamm.202000015
• 发表时间: 2020-06
• 期刊: GAMM‐Mitteilungen
• 作者: J. Pearson;J. Pestana