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

从数学上讲，工程师、经济学家和科学家感兴趣的许多模型都写成偏微分方程(PDE)。除了某些理想化的情形外，由此产生的偏微分方程组是不可能解析求解的。相反，我们依赖于数值近似。我特别感兴趣的是对二维或三维依赖时间的偏微分方程解的有效实现和自适应算法的分析，这些偏微分方程组的解表现出大的解变化、奇点形成或移动前沿。我们研究的方法试图通过将计算精力(在空间和时间上)集中在解具有有趣但难以跟踪行为的区域来有效地获得解。该策略使用移动的空间网格在时间上向前跟踪解的有趣特征。此外，还有机会和动机研究旨在利用不断发展的计算硬件的算法--具有数百或数千核的现成商用集群、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组的解映射到多核环境中，方法是将大问题分解成在单个核上计算的小块，然后利用区域分解(DD)算法进行重组以给出原始问题的解。这种策略将用于自适应网格的生成以及物理偏微分方程组的求解。可以通过计算同时预测和校正来增加时间上的小规模并行性。最终，我们将为时间依赖偏微分方程组的并行自适应解决方案提供一个新的、基于理论的、模块化的平台，适用于现有的和新兴的高性能计算硬件。这一研究程序为移动网格方法提供了一条影响路径，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算法的行为的了解。最后，它将为HQP提供数学专业知识和计算能力--雇主非常希望获得的可转移技能。

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.