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深入了解当今英国面临的许多关键挑战非常重要。这些问题包括了解流体如何在地下流动，这有助于确保我们有安全的饮用水；描述电脉冲在心脏中的传播，这有助于我们对心脏病的理解；开发准确的金融过程模型，通过更好地预测金融市场来改善我们的经济。这个项目的重点是所谓的分数扩散问题，当扩散过程涉及许多不同的流速或长期影响时，就会发生这种问题。分数扩散发生在许多应用中，包括地下水流动、心脏电传播和上面列出的财务问题。解决分数扩散的数学模型是具有挑战性的，通常需要数值方法，即计算机模拟。通常，这种模拟最耗时的部分是在计算机上求解数千甚至数百万个相互依赖的线性方程。事实上，解决这个方程组所需的时间可能太长，以至于我们无法模拟分数扩散问题，而这些问题能够捕捉到真实世界应用的真正复杂性。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，减少这个求解时间是至关重要的。这个项目将开发新的方法来解决这些巨大的方程组，保证速度快。我们将专注于迭代求解，它非常适合于我们所关注的数值方法（计算机模拟）。方程组的迭代求解器在每一步都计算一个新的近似解，因此如果在几次迭代之后就找到了一个好的近似解，那么求解速度很快。然而，这通常只有在我们使用收敛加速器（称为前置条件）时才有可能实现，它捕获了线性系统的“本质”，但使用成本较低。对于许多分数扩散问题，这个预条件目前是启发式地选择的，即没有理论依据。因此，预条件可能无法减少求解线性系统所需的（非常大的）计算时间。该项目的目标是提出新的预条件和迭代方法，这些方法在理论上是合理的，因此保证了对一系列分数扩散问题的快速收敛。我们将开发新的软件，使人们能够轻松地使用我们改进的求解器来解决分数扩散问题。此外，我们将在英国一个重要含水层的地下水流动的分数扩散模型中应用这些快速预调节器和迭代求解器。快速解决这个模型将使我们能够更好地跟踪我们的饮用水，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

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

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