EPSRC-SFI: Krylov subspace methods for non-symmetric PDE problems: a deeper understanding and faster convergence

科学现象的精确数学模型提供了对紧迫挑战的见解和解决方案，例如，气候变化、个性化医疗、可再生能源和高价值制造业。这些模型中的许多使用相互关联的“偏微分”方程组（称为偏微分方程）来描述现象。这些方程通过抽象地将科学感兴趣的相关量以及它们在空间和时间中的变化相互联系起来来描述这些现象。这些方程是人类可读的，但由于它们是抽象的，计算机无法解释它们。由于计算机的使用是有效和准确建模的基础，因此必须进行“离散化”过程，以便用计算机可以理解的东西来近似这些方程。科学现象通常被建模为发生在特定空间和特定时间跨度内。称为离散化的过程在一组离散点上对物理空间和时间进行采样。而不是考虑整个空间和时间上的偏微分方程，而是近似的偏微分方程只在这些离散点抽象传达的关系。这将抽象的、人类可读的偏微分方程转换为一组代数方程，其未知数仅是这些点上感兴趣的量的近似值。为了使这些方程的解足够接近偏微分方程的解，离散化通常必须具有高分辨率，这意味着通常有数亿个或更多的未知数。因此，这些代数方程是大规模的，必须通过有效的计算机程序来处理。由于方程本身通常能够以压缩的方式存储，因此不需要直接表示方程的迭代方法通常最有吸引力。这些方法产生一系列的近似解，并停止时，精度是令人满意的模型在question.The工作建议关注的分析，预测和加速这些迭代方法，使他们产生一个令人满意的准确的解决方案更快。很常见的是，由上述离散化产生的代数方程具有称为“Toeplitz”的附加结构。大量的工作已经进入了解的行为迭代方法应用到这些Toeplitz结构的问题。在本提案中，我们将进一步扩展这种理解，并开发新的加速方法来处理这些问题。此外，更广泛的一类结构化问题，称为广义局部Toeplitz（GLT）问题，可以用来描述方程所产生的一个更大的类的数学模型。我们将把Toeplitz问题的分析扩展到GLT环境。该提案中的工作将导致更快、更准确的现象建模，同时降低能源成本，因为它们不需要在大型超级计算机上运行那么多时间。我们的提案涵盖了新的数学发展、高效迭代方法的提案、它们在波浪传播和风力涡轮机模型中的应用，以及为最终用户制作软件。

Accurate mathematical models of scientific phenomena provide insights into, and solutions to, pressing challenges in e.g., climate change, personalised healthcare, renewable energy, and high-value manufacturing. Many of these models use groups of interconnected "partial differential" equations (called PDEs) to describe the phenomena. These equations describe the phenomena by abstractly relating relevant quantities of scientific interest, and how they change in space and time, to one another. These equations are human-readable, but since they are abstract, computers cannot interpret them. As the use of computers is fundamental to effective and accurate modeling, a process of "discretisation" must be undertaken to approximate these equations by something understandable to the computer.Scientific phenomena are generally modelled as occurring in a particular space and during a particular time-span. The process called discretisation samples both the physical space and time at a discrete set of points. Instead of considering the PDEs over the whole space and time, we instead approximate the relationships communicated abstractly by the PDEs only at these discrete points. This transforms abstract, human-readable PDEs into a set of algebraic equations whose unknowns are approximations of the quantities of interest only at these points. In order that the solution to these equations approximates the solution to the PDEs well enough, the discretisation generally must have a high resolution, meaning there are often hundreds of millions of unknowns or more. These algebra equations are thus large-scale and must be treated by efficient computer programs. As the equations themselves are often able to be stored in a compressed manner, iterative methods that do not require direct representation of the equations are often most attractive. These methods produce a sequence of approximate solutions and are stopped when the accuracy is satisfactory for the model in question.The work in this proposal concerns analysing, predicting, and accelerating these iterative methods so they produce a satisfactorily accurate solution more rapidly. It is quite common that the algebraic equations arising from the aforementioned discretisation have an additional structure known as "Toeplitz". A great deal of workhas gone into understanding the behaviour of iterative methods applied to these Toeplitz-structured problems. In this proposal, we will extend this understanding further and develop new accelerated methods to treat these problems. Furthermore, a wider class of structured problems called Generalised Locally Toeplitz (GLT) problems can be used to describe the equations arising from an even larger class of mathematical models. We will extend much of the analysis of Toeplitz problems to the GLT setting. The work in this proposal will lead to faster, more accurate modelling of phenomena with lower energy costs, as they will not require as much time running on large supercomputers.Our proposal spans new mathematical developments, the proposal of efficient iterative methods, their application to models of wave propagation and wind turbines, and the production of software for end-users.