Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification
Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification
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DOI：
10.1007/s10444-021-09889-0
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发表时间：
2020-05
影响因子：
1.7
通讯作者：
I. Graham;O. R. Pembery;E. Spence
中科院分区：
文献类型：
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作者：
I. Graham;O. R. Pembery;E. Spence
本文分析了非均质Helmholtz方程∇⋅(Aj∇uj) +k2njuj=−f的外部Dirichlet问题的有限元离散对应的伽辽金矩阵letAj，j= 1,2。对于任意大的k，应用于a2 (A1) - 1或a2 (A1) - 1的GMRES必须有多小（就k依赖性而言）才能在与ak无关的迭代次数中收敛？（换句话说，对于a2来说，fora1是一个好的左或右前置条件？）我们证明了回答这个问题的结果，给出了理论证据，并给出了支持估计的数值实验。我们解决这个问题的动机来自于计算带随机系数的亥姆霍兹方程的兴趣量。这样的计算可能需要解决许多确定性的亥姆霍兹问题，每个问题都有不同的andn，上面问题的答案决定了先前计算的一个伽辽金矩阵的逆在多大程度上可以用作其他伽辽金矩阵的前置条件。
This paper analyses the following question: letAj,j= 1,2, be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations ∇⋅ (Aj∇uj) +k2njuj= −f. How small mustandbe (in terms ofk-dependence) for GMRES applied to eitherorA2(A1)− 1to converge in ak-independent number of iterations for arbitrarily largek? (In other words, forA1to be a good left or right preconditioner forA2?) We prove results answering this question, give theoretical evidence for their sharpness, and give numerical experiments supporting the estimates. Our motivation for tackling this question comes from calculating quantities of interest for the Helmholtz equation withrandomcoefficientsAandn. Such a calculation may require the solution of many deterministic Helmholtz problems, each with differentAandn, and the answer to the question above dictates to what extent a previously calculated inverse of one of the Galerkin matrices can be used as a preconditioner for other Galerkin matrices.