Effective preconditioners for linear systems in fractional diffusion

信号和粒子扩散的数学模型对于洞察当今英国面临的许多关键挑战非常重要。这些问题包括了解地下流体如何流动，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特征，这有助于我们了解心脏病；开发准确的金融流程模型，通过对金融市场提供更好的预测来改善我们的经济。这个项目专注于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程效应时就会发生。分数扩散存在于许多应用中，包括地下水流动、心脏电传播和上面所列的财务问题。求解分数扩散的数学模型是具有挑战性的，通常需要一种数值方法，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个、甚至数百万个相互依赖的线性方程。事实上，求解这一方程组所需的时间可能会如此之长，以至于我们无法模拟捕捉真实世界应用程序真正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，缩短求解时间是至关重要的。这个项目将开发新的方法来求解这些保证快速的巨大的方程组。我们将重点介绍迭代求解器，它非常适合我们关注的那类数值方法(计算机模拟)。方程组的迭代求解器在每一步计算一个新的近似值，如果只经过几次迭代就找到了一个好的近似值，那么速度会更快。然而，这通常只有在我们应用称为预条件的收敛加速器的情况下才可能实现，该加速器捕捉到线性系统的“本质”，但使用起来很便宜。对于许多分数阶扩散问题，这种预条件算子目前是启发式地选择的，即没有理论上的证明。因此，预处理器可能无法减少解线性系统所需的(非常大的)计算时间。这个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阶扩散问题的快速收敛。我们将开发新的软件，使有分数扩散问题的人能够轻松使用我们改进的求解器。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条件和迭代求解器。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，并识别可能的污染源。

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.