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

在数学上，工程师、经济学家和科学家感兴趣的许多模型都被写成偏微分方程（PDE）。 除了某些理想化的情况下，偏微分方程的结果是不可能解决的分析。相反，我们依赖于数值近似。 我特别感兴趣的是研究有效的实施和分析的自适应算法的解决方案的时间依赖的偏微分方程在两个或三个空间维度的解决方案表现出大的解决方案的变化，奇点形成或移动的前线。我们研究的方法，试图通过集中计算工作（在空间和时间）的解决方案具有有趣的，但难以跟踪的行为的区域，以获得有效的解决方案。该策略的工作原理是使用移动的空间网格来跟踪解决方案的有趣功能。 此外，还有机会和动机来研究旨在利用不断发展的计算硬件的算法-具有数百或数千个核心的现成商品集群，混合CPU-GPU系统，甚至具有4-24个核心的台式机。我们建议映射的时间相关的偏微分方程的解决方案，多核环境划分成小块计算的个别核心和重组的大问题，给一个解决方案的原始问题，使用域分解（DD）算法。 这种策略将用于自适应网格的生成，以及物理PDE的解决方案。 可以通过计算同时的预测和校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，模块化的平台，适用于现有的和新兴的HPC硬件的时间依赖PDE的并行自适应解决方案。该研究计划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，它将提高我们的知识的行为DD算法的非线性问题。 最后，它将为HQP提供数学专业知识和计算能力-雇主高度追求的可转移技能。

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.