Abstract In this Note, we formulate a sparse Krylov-based algorithm for solving large-scale linear systems of algebraic equations arising from the discretization of randomly parametrized (or stochastic) elliptic partial differential equations (SPDEs). We analyze the proposed sparse conjugate gradient (CG) algorithm within the framework of inexact Krylov subspace methods, prove its convergence and study its abstract computational cost. Numerical studies conducted on stochastic diffusion models show that the proposed sparse CG algorithm outperforms the classical CG method when the sought solutions admit a sparse representation in a polynomial chaos basis. In such cases, the sparse CG algorithm recovers almost exactly the sparsity pattern of the exact solutions, which enables accelerated convergence. In the case when the SPDE solution does not admit a sparse representation, the convergence of the proposed algorithm is very similar to the classical CG method.

中文翻译：

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随机伽辽金方程的稀疏近似解

摘要 在本笔记中，我们制定了一种基于稀疏 Krylov 的算法，用于求解由随机参数化（或随机）椭圆偏微分方程 (SPDE) 离散化产生的代数方程的大规模线性系统。我们在不精确 Krylov 子空间方法的框架内分析所提出的稀疏共轭梯度 (CG) 算法，证明其收敛性并研究其抽象计算成本。对随机扩散模型进行的数值研究表明，当寻求的解在多项式混沌基础上采用稀疏表示时，所提出的稀疏 CG 算法优于经典的 CG 方法。在这种情况下，稀疏 CG 算法几乎完全恢复了精确解的稀疏模式，从而可以加速收敛。