A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

Lee, Kookjin; Elman, Howard C.

doi:10.1137/16m1075582

Cited by 25 publications

(33 citation statements)

References 26 publications

(39 reference statements)

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“…This leads to linear systems in Kronecker formulation [10,41,43]. We note that for stochastic Galerkin linear systems, various efficient iterative solvers such as mean-based preconditioning methods [41,43], hierarchical preconditioners [47,48], preconditioned low-rank projection methods [36] are vigorously studied. Nevertheless, to the best of our knowledge, in the case of stochastic Helmholtz problems these methods have not been analysed.…”

Section: Introductionmentioning

confidence: 99%

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

Liao¹

2019

EAJAM

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The implementation of spectral stochastic Galerkin finite element approximation methods for Helmholtz equations with random inputs is considered. The corresponding linear systems have matrices represented as Kronecker products. The sparsity of such matrices is analysed and a mean-based preconditioner is constructed. Numerical examples show the efficiency of the mean-based preconditioners for stochastic Helmholtz problems, which are not too close to a resonant frequency.

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Section: Introductionmentioning

confidence: 99%

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

Liao¹

2019

EAJAM

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“…To the best of our knowledge, there is little work on merging multilevel and tensor approximation techniques in uncertainty quantification. Recently, Lee and Elman [35] proposed a two-level scheme in the context of the Galerkin method for PDEs with random data. This scheme uses the solution from the coarse level to identify a dominant subspace in the domain of the random parameter, which in turn is used to speed up the solution on the fine level by avoiding costly low-rank truncations.…”

Section: Introductionmentioning

confidence: 99%

Multilevel tensor approximation of PDEs with random data

Ballani

Kreßner

Peters

2017

Stoch PDE: Anal Comp

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In this paper, we introduce and analyze a new low-rank multilevel strategy for the solution of random diffusion problems. Using a standard stochastic collocation scheme, we first approximate the infinite dimensional random problem by a deterministic parameter-dependent problem on a high-dimensional parameter domain. Given a hierarchy of finite element discretizations for the spatial approximation, we make use of a multilevel framework in which we consider the differences of the solution on two consecutive finite element levels in the collocation points. We then address the approximation of these high-dimensional differences by adaptive low-rank tensor techniques. This allows to equilibrate the error on all levels by exploiting analytic and algebraic properties of the solution at the same time. We arrive at an explicit representation in a low-rank tensor format of the approximate solution on the entire parameter domain, which can be used for, e.g., the direct and cheap computation of statistics. Numerical results are provided in order to illustrate the approach. arXiv:1606.05505v1 [math.NA]

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“…Combined with rank compression techniques, the iterates are forced to stay in low-rank format. This idea has been used with Krylov subspace methods [2,5,20,23] and multigrid methods [9,15]. The low-rank solution methods solve the linear systems to a certain accuracy with much less computational effort and facilitate the treatment of larger problem scales.…”

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confidence: 99%

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

Elman¹,

Su²

2019

SIAM J. Sci. Comput.

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We study efficient solution methods for stochastic eigenvalue problems arising from discretization of self-adjoint partial differential equations with random data. With the stochastic Galerkin approach, the solutions are represented as generalized polynomial chaos expansions. A low-rank variant of the inverse subspace iteration algorithm is presented for computing one or several minimal eigenvalues and corresponding eigenvectors of parameter-dependent matrices. In the algorithm, the iterates are approximated by low-rank matrices, which leads to significant cost savings. The algorithm is tested on two benchmark problems, a stochastic diffusion problem with some poorly separated eigenvalues, and an operator derived from a discrete stochastic Stokes problem whose minimal eigenvalue is related to the inf-sup stability constant. Numerical experiments show that the low-rank algorithm produces accurate solutions compared to the Monte Carlo method, and it uses much less computational time than the original algorithm without low-rank approximation.

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A Preconditioned Low-Rank Projection Method with a Rank-Reduction Scheme for Stochastic Partial Differential Equations

Cited by 25 publications

References 26 publications

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

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

Multilevel tensor approximation of PDEs with random data

Low-Rank Solution Methods for Stochastic Eigenvalue Problems

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