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

Müller, Christopher; Ullmann, Sebastian; Lang, Jens

doi:10.1137/18m1163920

Cited by 4 publications

(7 citation statements)

References 29 publications

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“…In the following, we discuss two different iterative solvers for our SGFE Stokes problem. Similar investigations were conducted in [21] for Stokes flow with deterministic data and in [19] for Stokes flow with uniform random data.…”

Section: Iterative Solversmentioning

confidence: 81%

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

Müller

Ullmann

Lang

2018

Lecture Notes in Computational Science and Engineering

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We study linear systems of equations arising from a stochastic Galerkin finite element discretization of saddle point problems with random data and its iterative solution. We consider the Stokes flow model with random viscosity described by the exponential of a correlated random process and shortly discuss the discretization framework and the representation of the emerging matrix equation. Due to the high dimensionality and the coupling of the associated symmetric, indefinite, linear system, we resort to iterative solvers and problem-specific preconditioners. As a standard iterative solver for this problem class, we consider the block diagonal preconditioned MINRES method and further introduce the Bramble-Pasciak conjugate gradient method as a promising alternative. This special conjugate gradient method is formulated in a non-standard inner product with a block triangular preconditioner. From a structural point of view, such a block triangular preconditioner enables a better approximation of the original problem than the block diagonal one. We derive eigenvalue estimates to assess the convergence behavior of the two solvers with respect to relevant physical and numerical parameters and verify our findings by the help of a numerical test case. We model Stokes flow in a cavity driven by a moving lid and describe the viscosity by the exponential of a truncated Karhunen-Loève expansion. Regarding iteration counts, the Bramble-Pasciak conjugate gradient method with block triangular preconditioner is superior to the MINRES method with block diagonal preconditioner in the considered example.

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Section: Iterative Solversmentioning

confidence: 81%

“…However, this can be done efficiently using iterative methods and problem-specific preconditioners, see e.g. [11,19,22,24,29]. .…”

Section: Introductionmentioning

confidence: 99%

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

Müller

Ullmann

Lang

2018

Lecture Notes in Computational Science and Engineering

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“…Similar idea was, in a simpler form, used already in [28,29]. In the current paper, it is applied in a more general setting, and we believe that the derived technique may lead to an improvement of some other recently introduced estimates, such as [17,23]. The derived technique is also applicable to systems in the form of multi-term matrix equation (see [25, eq.…”

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confidence: 87%

“…[34, p.120], the eigenvalues of (2s − 2) for uniform distribution; see [13,17,23,24,35]. The main difference between (2.12) and (2.13) is that the former is considered point-wise, while the latter uses the norms of a k over D. The condition (2.12) allows us to obtain not only more accurate two-sided guaranteed bounds to the spectra, but these bounds also apply to parameter distribution and functions a k for which no estimate could be obtained using the standard approach; see subsection 4.4.…”

Section: Positive Definitenessmentioning

confidence: 99%

“…In this paper, we focus on matrices arising in the discretized stochastic Galerkin method and present new guaranteed two-sided bounds to the spectra of the preconditioned matrices for several types of preconditioner. We consider only preconditioning with respect to the stochastic parts of problems, and thus we assume that a suitable preconditioning method or an efficient solver for the underlying deterministic problem is available; see, e.g., [12,23,33]. We formulate an idea of obtaining bounds to the spectra of the preconditioned matrix from the spectrum of small Gram matrices depending solely on the stochastic part of the approximation space.…”

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confidence: 99%

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Block Preconditioning of Stochastic Galerkin Problems: New Two-sided Guaranteed Spectral Bounds

Kubínová¹,

Pultarová

2020

SIAM/ASA J. Uncertainty Quantification

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The paper focuses on numerical solution of parametrized diffusion equations with scalar parameterdependent coefficient function by the stochastic (spectral) Galerkin method. We study preconditioning of the related discretized problems using preconditioners obtained by modifying the stochastic part of the partial differential equation. We present a simple but general approach for obtaining two-sided bounds to the spectrum of the resulting matrices, based on a particular splitting of the discretized operator. Using this tool and considering the stochastic approximation space formed by classical orthogonal polynomials, we obtain new spectral bounds depending solely on the properties of the coefficient function and the type of the approximation polynomials for several classes of block-diagonal preconditioners. These bounds are guaranteed and applicable to various distributions of parameters. Moreover, the conditions on the parameter-dependent coefficient function are only local, and therefore less restrictive than those usually assumed in the literature.

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A class of preconditioners based on symmetric-triangular decomposition and matrix splitting for generalized saddle point problems

Wang

2022

IMA Journal of Numerical Analysis

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In this paper, based on the idea of symmetric-triangular (ST) decomposition and matrix splitting, a product-type preconditioner and its inexact version are developed for generalized saddle point problems. These preconditioners can be combined appropriately with the efficient preconditioned conjugate gradient (PCG) method to solve the generalized saddle point problems, although neither these preconditioners nor the generalized saddle point matrices are symmetric and positive definite, which is the advantage of our proposed preconditioners. The proposed PCG method belongs to the group of nonstandard inner product CG methods, so the convergence theorem of the former is given by the use of that of the latter. The difference between the existing nonstandard inner product CG methods and our proposed method is studied. Theoretical analysis shows that the spectrum of the preconditioned matrix corresponding to the proposed inexact preconditioner is contained in a real, positive interval and the quasi-optimal parameter of the preconditioner is effective and easy to apply in practice. Numerical experiments are given to illustrate the effectiveness and robustness of the proposed inexact preconditioner and show the advantages of the preconditioner over the existing state-of-the-art preconditioners for saddle point problems.

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

Cited by 4 publications

References 29 publications

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

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

Block Preconditioning of Stochastic Galerkin Problems: New Two-sided Guaranteed Spectral Bounds

A class of preconditioners based on symmetric-triangular decomposition and matrix splitting for generalized saddle point problems

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