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都是以偏微分方程(PDE)的数学形式写成的。除了某些理想化的情形外，所得到的偏微分方程解是不可能的。相反，我们依赖于数值近似。特别令人感兴趣的是对二维或三维空间中依赖时间的偏微分方程解的高效实现和自适应算法的分析，这些偏微分方程组可能生活在非平坦表面上，其解表现出很大的解变化、奇点形成或移动前沿。我和我的学生学习的方法是，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，该PDE取决于未知的解，并因此耦合到物理PDE。我们希望研究专门设计的算法，以利用现成的具有数百或数千个核心的计算集群、CPU-GPU混合系统，甚至是具有多个核心的台式机。我们提出通过将大问题分成小块，在单个核上计算，然后使用区域分解(DD)算法和预条件子进行重组，从而将曲面上的时变偏微分方程组的自适应解映射到多核环境中。我们将提供交替或整体(一次性)框架中自适应网格的生成和物理偏微分方程组的求解的实现和分析算法。为了使非常多的核饱和，将通过计算同时预测和校正来增加时间上的小规模并行度。所产生的软件将应用于与自然资源勘探有关的情景中的地球物理电磁问题。将我们的求解器扩展到曲面上定义的偏微分方程组，将允许对计算科学家和工程师感兴趣的更真实的几何图形进行快速、可扩展的模拟。*最终，我们将提供一个新的、基于理论的、模块化的平台，用于适用于现有和新兴的高性能计算硬件的通用曲面上的时间相关偏微分方程组的并行自适应解决方案。该研究项目为基于偏微分方程的网格生成方法提供了一个影响链，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算法的行为的了解。最后，它将为HQP提供数学专业知识、计算能力和雇主热衷于寻求的关键可转让技能。

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.