论文标题
分析由不确定性量化动机的Helmholtz预处理问题
Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification
论文作者
论文摘要
本文分析以下问题：让$ \ Mathbf {a} _J $，$ j = 1,2，$是与异构helmholtz方程$ \ nabla \ cdot（a_j \ cdot（a_j \ nabla ud $ kjj JJ）的dirichlet问题相对应的galerkin矩阵，对应于有限元元素离散问题。必须$ \ | | a_1 -a_2 \ | _ {l^q} $和$ \ | {n_1} - {n_2} \ | _ {l^q} $ be（以$ k $依赖性为$ k $ experence）适用于$ k $依赖性）或$ \ mathbf {a} _2（\ mathbf {a} _1）^{ - 1} $在$ k $ - 非依赖性的迭代次数中收敛，用于任意大的$ k $？ （换句话说，对于$ \ mathbf {a} _1 $，对于$ \ mathbf {a} _2 $？）。我们证明了回答这个问题的结果，提供理论证据证明其清晰度，并提供支持估计值的数值实验。 我们解决这个问题的动机来自用随机系数$ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $ a $。这样的计算可能需要解决许多确定性的Helmholtz问题的解决方案，每个问题都有不同的$ a $ and $ n $，并且对上述问题的答案决定了一个先前计算的Galerkin矩阵的倒数程度，可以用作其他Galerkin矩阵的预处理。
This paper analyses the following question: let $\mathbf{A}_j$, $j=1,2,$ be the Galerkin matrices corresponding to finite-element discretisations of the exterior Dirichlet problem for the heterogeneous Helmholtz equations $\nabla\cdot (A_j \nabla u_j) + k^2 n_j u_j= -f$. How small must $\|A_1 -A_2\|_{L^q}$ and $\|{n_1} - {n_2}\|_{L^q}$ be (in terms of $k$-dependence) for GMRES applied to either $(\mathbf{A}_1)^{-1}\mathbf{A}_2$ or $\mathbf{A}_2(\mathbf{A}_1)^{-1}$ to converge in a $k$-independent number of iterations for arbitrarily large $k$? (In other words, for $\mathbf{A}_1$ to be a good left- or right-preconditioner for $\mathbf{A}_2$?). 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 with random coefficients $A$ and $n$. Such a calculation may require the solution of many deterministic Helmholtz problems, each with different $A$ and $n$, 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.
