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气候变化、个性化医疗保健、可再生能源和高价值制造等紧迫挑战的洞察和解决方案。其中许多模型使用相互关联的“偏微分方程组”(称为偏微分方程组)来描述这些现象。这些方程通过抽象地将相关的科学兴趣数量以及它们如何在空间和时间上相互变化来描述这些现象。这些方程式是人类可读的，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方式近似这些方程。科学现象通常被模拟为发生在特定的空间和特定的时间跨度内。这一过程被称为离散化，在一组离散的点上对物理空间和时间进行采样。我们不考虑整个空间和时间上的PDE，而是仅在这些离散的点处近似由PDE抽象地传递的关系。这将抽象的、人类可读的偏微分方程组转化为一组代数方程，其未知数是仅在这些点上感兴趣的量的近似值。为了使这些方程的解足够好地逼近偏微分方程组的解，离散化通常必须具有高分辨率，这意味着通常存在数亿个或更多的未知数。因此，这些代数方程是大规模的，必须用有效的计算机程序来处理。由于方程本身通常能够以压缩方式存储，因此不需要直接表示方程的迭代方法通常是最有吸引力的。这些方法产生一系列近似解，当所讨论的模型的精度令人满意时，这些方法停止。本提案中的工作涉及分析、预测和加速这些迭代方法，以便它们更快地产生令人满意的精确解。由上述离散化产生的代数方程具有称为“Toeplitz”的附加结构，这是很常见的。为了理解应用于这些Toeplitz结构问题的迭代方法的行为，已经进行了大量的工作。在这项建议中，我们将进一步扩大这一认识，并开发新的加速方法来处理这些问题。此外，一类更广泛的被称为广义局部Toeplitz(GLT)问题的结构化问题可以用来描述从更大类别的数学模型中产生的方程。我们将把Toeplitz问题的大部分分析扩展到GLT设置。这项建议中的工作将导致以较低的能源成本更快、更准确地对现象进行建模，因为它们不需要在大型超级计算机上运行那么多时间。我们的建议涵盖了新的数学发展、高效迭代方法的提出、它们在波浪传播和风力涡轮机模型中的应用，以及为最终用户生产软件。

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.