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Scalable domain decomposition solvers for stochastic PDEs in high performance computing

机译：高性能计算中随机PDE的可扩展域分解求解器

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摘要

Stochastic spectral finite element models of practical engineering systems may involve solutions of linear systems or linearized systems for non-linear problems with billions of unknowns. For stochastic modeling, it is therefore essential to design robust, parallel and scalable algorithms that can efficiently utilize high-performance computing to tackle such large-scale systems. Domain decomposition based iterative solvers can handle such systems. Although these algorithms exhibit excellent scalabilities, significant algorithmic and implementational challenges exist to extend them to solve extreme-scale stochastic systems using emerging computing platforms. Intrusive polynomial chaos expansion based domain decomposition algorithms are extended here to concurrently handle high resolution in both spatial and stochastic domains using an in-house implementation. Sparse iterative solvers with efficient preconditioners are employed to solve the resulting global and subdomain level local systems through multi-level iterative solvers. Parallel sparse matrix-vector operations are used to reduce the floating-point operations and memory requirements. Numerical and parallel scalabilities of these algorithms are presented for the diffusion equation having spatially varying diffusion coefficient modeled by a non-Gaussian stochastic process. Scalability of the solvers with respect to the number of random variables is also investigated. (C) 2017 Elsevier B.V. All rights reserved.
机译：实际工程系统的随机频谱有限元模型可能涉及线性系统或线性系统的解，这些非线性系统具有数十亿未知数的非线性问题。因此，对于随机建模，设计稳健，并行和可扩展的算法至关重要，这些算法可以有效地利用高性能计算来解决此类大规模系统。基于域分解的迭代求解器可以处理此类系统。尽管这些算法具有出色的可扩展性，但是存在巨大的算法和实现挑战，需要扩展它们以使用新兴的计算平台来解决极端规模的随机系统。在此扩展了基于侵入式多项式混沌扩展的域分解算法，以使用内部实现同时处理空间域和随机域中的高分辨率。具有有效预处理器的稀疏迭代求解器用于通过多级迭代求解器求解生成的全局和子域级本地系统。并行稀疏矩阵矢量运算用于减少浮点运算和内存需求。针对具有通过非高斯随机过程建模的空间变化扩散系数的扩散方程，给出了这些算法的数值和并行可扩展性。还研究了求解器相对于随机变量数量的可伸缩性。 （C）2017 Elsevier B.V.保留所有权利。

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