Parallel Space-Time Approaches for the Numerical Solution of Partial Differential Equations

在数学上，许多工程师、经济学家和科学家感兴趣的模型都被写成偏微分方程（PDEs）。除某些理想情况外，产生的偏微分方程是不可能用分析方法求解的。相反，我们依靠数值近似。我特别感兴趣的是研究有效的实现和自适应算法的分析，以解决两个或三个空间维度的时变偏微分方程，其解表现出很大的解变化，奇点形成或移动锋。我们研究了一些方法，这些方法试图通过在解具有有趣但难以跟踪的行为的区域集中计算（在空间和时间上）来有效地获得解。该策略通过使用移动空间网格及时跟踪解决方案的有趣特征来工作。此外，还有机会和动机来研究算法，这些算法旨在利用不断发展的计算硬件——随时可用的具有数百或数千核的商品集群，CPU-GPU混合系统，甚至具有4-24核的台式机。我们提出将时间相关偏微分方程的解映射到多核环境中，方法是将大问题划分为在单个核上计算的小问题，并使用域分解（DD）算法重新组合以给出原始问题的解。这种策略既可用于自适应网格的生成，也可用于物理偏微分方程的求解。可以通过计算同时预测和修正来增加时间上的小规模并行性。最终，我们将提供一个新的、基于理论的模块化平台，用于适用于现有和新兴HPC硬件的时间相关pde的并行自适应解决方案。本研究项目为移动网格方法的影响提供了一条途径，为计算科学家解决复杂问题提供了一个软件工具。从理论上讲，它将增强我们对非线性问题的DD算法行为的认识。最后，它将为HQP提供数学专业知识和计算能力-雇主高度寻求的可转移技能。

Mathematically, many models of interest to engineers, economists and scientists are written as partial differential equations (PDEs). Except for certain idealized situations, the PDEs which result are not possible to solve analytically. Instead we rely on numerical approximations. I am particularly interested in the study of efficient implementations and analysis of adaptive algorithms for the solution of time-dependent PDEs in two or three spatial dimensions whose solutions exhibit large solution variation, singularity formation or moving fronts. We study methods which attempt to obtain a solution efficiently by concentrating computational effort (in both space and time) in regions where the solution has interesting but difficult to track behavior. The strategy works by using moving spatial meshes to track interesting features of the solution forward in time. Moreover, there is an opportunity and motivation to study algorithms designed to take advantage of the continually evolving computing hardware - readily available commodity clusters with hundreds or thousands of cores, hybrid CPU-GPU systems and even desktop machines with 4-24 cores. We propose mapping the solution of time dependent PDEs to multi-core environments by dividing the large problem into small pieces computed on individual cores and recombined to give a solution of the original problem using domain decomposition (DD) algorithms. Such a strategy will be used for both the generation of the adaptive grids, as well as the solution of the physical PDE. Small scale parallelism in time may be added by computing simultaneous predictions and corrections. Ultimately, we will provide a new, theoretically based, modular platform for the parallel adaptive solution of time dependent PDEs suitable for existing and emerging HPC hardware. This research program provides a route to impact for moving mesh methods, providing a software tool for computational scientists for the solution of complex problems. Theoretically it will enhance our knowledge of the behaviour of DD algorithms for nonlinear problems. Finally, it will provide HQP with mathematical expertise and computational competency - transferable skills highly sought by employers.