Adaptive algorithm for stochastic Galerkin method

Pultarová, Ivana

doi:10.1007/s10492-015-0111-9

Cited by 8 publications

(7 citation statements)

References 25 publications

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“…The eigenvalue estimates for the preconditioned SGFE Laplacian A −1 mg A can be derived as in [13], where the procedure is carried out for a preconditioned SGFE matrix in the context of a mixed diffusion problem with uncertain coefficient. The main difference between our approach and the one in [13,Lemma 4.6] is that we use a product estimate of the Rayleigh quotient similar to [28] to derive a lower bound for the eigenvalues of the preconditioned SGFE Laplacian. This bound is tighter than the one we would get by following the steps in [13].…”

Section: Matrix Formulationmentioning

confidence: 99%

A Bramble--Pasciak Conjugate Gradient Method for Discrete Stokes Equations with Random Viscosity

Müller¹,

Ullmann²,

Lang³

2019

SIAM/ASA J. Uncertainty Quantification

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We study the iterative solution of linear systems of equations arising from stochastic Galerkin finite element discretizations of saddle point problems. We focus on the Stokes model with random data parametrized by uniformly distributed random variables. We introduce a Bramble-Pasciak conjugate gradient method as a linear solver. This method is associated with a block triangular preconditioner which must be scaled using a properly chosen parameter. We show how the existence requirements of such a conjugate gradient method can be met in our setting. As a reference solver, we consider a standard MINRES method, which is restricted to symmetric preconditioning. We analyze the performance of the two different solvers depending on relevant physical and numerical parameters by means of eigenvalue estimates. For this purpose, we derive bounds for the eigenvalues of the relevant preconditioned sub-matrices. We illustrate our findings using the flow in a driven cavity as a numerical test case, where the viscosity is given by a truncated Karhunen-Loève expansion of a random field. In this example, a Bramble-Pasciak conjugate gradient method with a block triangular preconditioner converges faster than a MINRES method with a comparable block diagonal preconditioner in terms of iteration counts.

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Section: Matrix Formulationmentioning

confidence: 99%

A Bramble--Pasciak Conjugate Gradient Method for Discrete Stokes Equations with Random Viscosity

Müller¹,

Ullmann²,

Lang³

2019

SIAM/ASA J. Uncertainty Quantification

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“…The design of error estimators for SGFEMs for parameter-dependent PDEs is still under development. However, there have been a few recent works (see [8][9][10][11][12][16][17][18]29]) for the model problem (1a)-(1b). In [12] and [11], algorithms constructing so-called sparse SGFEM approximations are driven by a priori error analysis, where the error associated with each discretisation parameter is balanced against the total number of degrees of freedom.…”

Section: Theorem 1 Let H Be a Hilbert Space Equipped With Inner Produmentioning

confidence: 99%

“…In particular, due to cost restrictions imposed to avoid high-dimensional detail spaces, we investigate non-standard choices that aren't typically considered in the deterministic setting. The error estimation strategy in [29] also relies on a CBS constant, but in a different setting. Enrichment of the finite element space is not considered.…”

Section: Theorem 1 Let H Be a Hilbert Space Equipped With Inner Produmentioning

confidence: 99%

CBS Constants & Their Role in Error Estimation for Stochastic Galerkin Finite Element Methods

Crowder

Powell

2018

J Sci Comput

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Stochastic Galerkin finite element methods (SGFEMs) are commonly used to approximate solutions to PDEs with random inputs. However, the study of a posteriori error estimation strategies to drive adaptive enrichment of the associated tensor product spaces is still under development. In this work, we revisit an a posteriori error estimator introduced in Bespalov and Silvester (SIAM J Sci Comput 38(4):A2118-A2140, 2016) for SGFEM approximations of the parametric reformulation of the stochastic diffusion problem. A key issue is that the bound relating the true error to the estimated error involves a CBS (Cauchy-Buniakowskii-Schwarz) constant. If the approximation spaces associated with the parameter domain are orthogonal in a weighted L 2 sense, then this CBS constant only depends on a pair of finite element spaces H 1 , H 2 associated with the spatial domain and their compatibility with respect to an inner product associated with a parameter-free problem. For fixed choices of H 1 , we investigate non-standard choices of H 2 and the associated CBS constants, with the aim of designing efficient error estimators with effectivity indices close to one. When H 1 and H 2 satisfy certain conditions, we also prove new theoretical estimates for the CBS constant using linear algebra arguments.

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“…While there exists a rather firm theory for the a posteriori analysis of random elliptic and parabolic equations in combination with the SG method (Butler et al (2011); Deb et al (2001); Eigel et al (2014); Mathelin & Le Maître (2007); Pultarová (2015) ), the a posteriori analysis for hyperbolic problems is less developed. Beside the missing a posteriori analysis, most applications of the SG method for hyperbolic equations are rather ad-hoc, in the sense that the resolution in the stochastic space is rather arbitrary and, in particular, not related to resolution in space and time.…”

Section: Introductionmentioning

confidence: 99%

A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method

Rohde

Giesselmann

2019

IMA Journal of Numerical Analysis

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In this article we present an a posteriori error estimator for the spatial-stochastic error of a Galerkin-type discretization of an initial value problem for a random hyperbolic conservation law. For the stochastic discretization we use the Stochastic Galerkin method and for the spatial-temporal discretization of the Stochastic Galerkin system a Runge-Kutta Discontinuous Galerkin method. The estimator is obtained using smooth reconstructions of the discrete solution. Combined with the relative entropy stability framework of Dafermos (2016), this leads to computable error bounds for the space-stochastic discretization error. Moreover, it turns out that the error estimator admits a splitting into one part representing the spatial error, and a remaining term, which can be interpreted as the stochastic error. This decomposition allows us to balance the errors arising from spatial and stochastic discretization. We conclude with some numerical examples confirming the theoretical findings.

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Adaptive algorithm for stochastic Galerkin method

Cited by 8 publications

References 25 publications

A Bramble--Pasciak Conjugate Gradient Method for Discrete Stokes Equations with Random Viscosity

A Bramble--Pasciak Conjugate Gradient Method for Discrete Stokes Equations with Random Viscosity

CBS Constants & Their Role in Error Estimation for Stochastic Galerkin Finite Element Methods

A posteriori error analysis for random scalar conservation laws using the stochastic Galerkin method

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