Efficient Spectral Stochastic Finite Element Methods for Helmholtz Equations with Random Inputs

Liao, Guanjie Wang &  Qifeng

doi:10.4208/eajam.140119.160219

Cited by 3 publications

(1 citation statement)

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“…The initial computational work on mean-based preconditioning for the equation was carried out by [30,52], and [42], with theory following from [58] and [20]. In the Helmholtz context, mean-based preconditioning has been used in [41] and [65] (see the literature review in [53, Sect. 4.7] for more discussion).…”

Section: Implications Of These Results For Uncertainty Quantification For the Helmholtz Equationmentioning

confidence: 99%

Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

2021

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This paper analyses the following question: let Aj, 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 must $\|A_{1} -A_{2}\|_{L^{q}}$ ∥ A 1 − A 2 ∥ L q and $\|{n_{1}} - {n_{2}}\|_{L^{q}}$ ∥ n 1 − n 2 ∥ L q be (in terms of k-dependence) for GMRES applied to either $(\mathbf {A}_1)^{-1}\mathbf {A}_2$ ( A 1 ) − 1 A 2 or A2(A1)− 1 to converge in a k-independent number of iterations for arbitrarily large k? (In other words, for A1 to be a good left or right preconditioner for A2?) 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.

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Section: Implications Of These Results For Uncertainty Quantification For the Helmholtz Equationmentioning

confidence: 99%

Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

2021

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Reduced basis stochastic Galerkin methods for partial differential equations with random inputs

Wang,

Liao

2024

Applied Mathematics and Computation

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Stochastic Finite Element Methods with the Euclidean Degree for Partial Differential Equations with Random Inputs

Huang

Wang

et al. 2020

2020 Chinese Control and Decision Conference (CCDC)

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Efficient Spectral Stochastic Finite Element Methods for Helmholtz Equations with Random Inputs

Cited by 3 publications

References 51 publications

Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

Analysis of a Helmholtz preconditioning problem motivated by uncertainty quantification

Reduced basis stochastic Galerkin methods for partial differential equations with random inputs

Stochastic Finite Element Methods with the Euclidean Degree for Partial Differential Equations with Random Inputs

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