Analysis and Implementation of Parallel Solvers for PDE Based Mesh Generation and Coupled Systems

工程师和科学家感兴趣的许多模型在数学上被写成偏微分方程（PDEs）。除了某些理想化的情况，产生的偏微分方程是不可能解析解决的。相反，我们依靠数值近似。特别令人感兴趣的是有效实现的发展和自适应算法的分析，用于在二维或三维空间中解决时变偏微分方程，可能存在于非平面上，其解决方案表现出很大的解决变化，奇点形成或移动锋。我和我的学生们研究的方法是，通过自动将计算精力集中在解决方案具有这种有趣但难以跟踪的行为的空间和时间区域来获得有效的解决方案。该策略通过使用自动生成的移动空间网格来适应所需的特征。这是通过为网格制定一个（可能是非线性的）PDE来实现的，它取决于未知解，因此与物理PDE相耦合。我们希望研究专门设计的算法，以利用随时可用的具有数百或数千核的计算集群，CPU-GPU混合系统甚至具有多核的桌面机器。我们提出将表面上的时间相关偏微分方程的自适应解映射到多核环境中，方法是将大问题分成小块，在单个核上计算，然后使用域分解（DD）算法和前置条件重新组合以给出原始问题的解。我们将提供自适应网格生成的实现和分析算法，以及在交替或单片（一次性）框架中物理PDE的解决方案。为了使大量的核饱和，将通过计算同时预测和修正来增加时间上的小规模并行性。由此产生的软件将应用于与自然资源勘探有关的情况下的地球物理电磁问题。将我们的求解器扩展到定义在表面上的偏微分方程，将允许对计算科学家和工程师感兴趣的更现实的几何形状进行快速，可扩展的模拟。******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高性能计算硬件的一般表面上的时间相关偏微分方程的并行自适应解决方案。本研究项目为基于PDE的网格生成方法提供了影响链，为需要解决复杂问题的计算科学家提供了一个软件工具。从理论上讲，该研究计划将增强我们对非线性耦合系统的DD算法行为的认识。最后，它将为HQP提供雇主高度寻求的数学专业知识、计算能力和关键的可转移技能。

Many models of interest to engineers and scientists are written mathematically as partial differential equations (PDEs). Except for certain idealized situations, the resulting PDEs are not possible to solve analytically. Instead we rely on numerical approximations. Of particular interest is the development of efficient implementations and the analysis of adaptive algorithms for the solution of time-dependent PDEs in two or three spatial dimensions, possibly living on non-flat surfaces, whose solutions exhibit large solution variation, singularity formation or moving fronts. My students and I study methods which obtain an efficient solution by automatically concentrating computational effort in space and time regions in which the solution has this interesting, but difficult to track behaviour. The strategy works by using automatically generated moving spatial meshes to adapt to the desired features. This is achieved by formulating a (possibly nonlinear) PDE for the mesh, which depends on the unknown solution, and hence is coupled to the physical PDE. We wish to study algorithms specifically designed to take advantage of readily available compute clusters with hundreds or thousands of cores, hybrid CPU-GPU systems and even desktop machines with multiple cores. We propose mapping the adaptive solution of time dependent PDEs on surfaces to multicore environments by dividing the large problem into small pieces, computing on individual cores and then recombining to give a solution of the original problem using domain decomposition (DD) algorithms and preconditioners. We will provide implementations and analyze algorithms for the generation of the adaptive grids and the solution of the physical PDE in either an alternating or monolithic (one-shot) framework. To saturate very large numbers of cores, small scale parallelism in time will be added by computing simultaneous predictions and corrections. The resulting software will be applied to geophysical electromagnetic problems in scenarios relevant to natural resource exploration. Extending our solvers to PDEs defined on surfaces will allow for fast, scalable simulations on more realistic geometries of interest to computational scientists and engineers.******Ultimately, we will provide a new, theoretically based, modular platform for the parallel adaptive solution of time dependent PDEs on general surfaces suitable for existing and emerging high performance computing hardware. This research program provides an impact chain for PDE based mesh generation methods, providing a software tool for computational scientists requiring the solution of complex problems. Theoretically, the research program will enhance our knowledge of the behaviour of DD algorithms for nonlinear coupled systems. Finally, it will provide HQP with mathematical expertise, computational capability, and key transferable skills highly sought by employers.