Fourier mode analysis of multigrid methods for partial differential equations with random coefficients

Seynaeve, Bert; Rosseel, Eveline; Nicolaı̈, Bart; Vandewalle, Stefan

doi:10.1016/j.jcp.2006.12.011

Cited by 23 publications

(28 citation statements)

References 18 publications

(42 reference statements)

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“…In this section we discuss the AMG convergence behavior with respect to the stochastic discretization. The conclusions agree with the properties of the geometric multigrid variant, as observed in [15] and theoretically analyzed in [16,27]. The AMG convergence behavior with respect to the IRK discretization, i.e.…”

Section: A Discussion Of the Theoretical Convergence Propertiessupporting

confidence: 90%

“…The univariate polynomials are chosen from the Wiener-Askey scheme according to the probability distributions of the random variables i . The second criterion can be used to create an alternative set of basis functions { q } [4,13,16], which possess a double orthogonality property:…”

Section: Generalized Polynomial Chaos Consider a Hilbert Spacementioning

confidence: 99%

“…A local Fourier analysis of geometric multigrid for stochastic, stationary PDEs can be found in [16]. This analysis cannot be directly applied to AMG methods.…”

Section: Convergence Analysismentioning

confidence: 99%

“…,C L * can be seen as a perturbation of the system matrix. A more thorough (geometric) multigrid analysis for the general stationary case can be found in [16].…”

Section: E Rosseel T Boonen and S Vandewallementioning

confidence: 99%

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Algebraic multigrid for stationary and time‐dependent partial differential equations with stochastic coefficients

Rosseel

Boonen

Vandewalle

2007

Numerical Linear Algebra App

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We consider the numerical solution of time-dependent partial differential equations (PDEs) with random coefficients. A spectral approach, called stochastic finite element method, is used to compute the statistical characteristics of the solution. This method transforms a stochastic PDE into a coupled system of deterministic equations by means of a Galerkin projection onto a generalized polynomial chaos. An algebraic multigrid (AMG) method is presented to solve the algebraic systems that result after discretization of this coupled system. High-order time integration schemes of an implicit Runge-Kutta type and spatial discretization on unstructured finite element meshes are considered. The convergence properties of the AMG method are demonstrated by a convergence analysis and by numerical tests.

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Section: A Discussion Of the Theoretical Convergence Propertiessupporting

confidence: 90%

Section: Generalized Polynomial Chaos Consider a Hilbert Spacementioning

confidence: 99%

“…A local Fourier analysis of geometric multigrid for stochastic, stationary PDEs can be found in [16]. This analysis cannot be directly applied to AMG methods.…”

Section: Convergence Analysismentioning

confidence: 99%

“…,C L * can be seen as a perturbation of the system matrix. A more thorough (geometric) multigrid analysis for the general stationary case can be found in [16].…”

Section: E Rosseel T Boonen and S Vandewallementioning

confidence: 99%

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Algebraic multigrid for stationary and time‐dependent partial differential equations with stochastic coefficients

Rosseel

Boonen

Vandewalle

2007

Numerical Linear Algebra App

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“…During the last few years there is a great interest in numerical methods for solving stochastic PDEs and ODEs [2,3,10,25,35,[37][38][39]. Examples are stochastic NavierStokes equations, stochastic plasticity equations and stochastic aerodynamic equations.…”

Section: Introductionmentioning

confidence: 99%

Application of hierarchical matrices for computing the Karhunen–Loève expansion

Khoromskij

Litvinenko²,

Matthies³

2008

Computing

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Realistic mathematical models of physical processes contain uncertainties. These models are often described by stochastic differential equations (SDEs) or stochastic partial differential equations (SPDEs) with multiplicative noise. The uncertainties in the right-hand side or the coefficients are represented as random fields. To solve a given SPDE numerically one has to discretise the deterministic operator as well as the stochastic fields. The total dimension of the SPDE is the product of the dimensions of the deterministic part and the stochastic part. To approximate random fields with as few random variables as possible, but still retaining the essential information, the Karhunen-Loève expansion (KLE) becomes important. The KLE of a random field requires the solution of a large eigenvalue problem. Usually it is solved by a Krylov subspace method with a sparse matrix approximation. We demonstrate the use of sparse hierarchical matrix techniques for this. A log-linear computational cost of the matrix-vector product and a log-linear storage requirement yield an efficient and fast discretisation of the random fields presented.

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A multigrid method with reduced phase error for 2D damped Helmholtz equations

Ahmed

Zhang

2021

Math Methods in App Sciences

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This paper deals with a new multigrid method with reduced phase error for solving 2D damped Helmholtz equations. The method is obtained by taking the high‐effective, reduced phase error 5‐point finite difference (FD) scheme as a coarse grid operator and the regular 5‐point FD scheme as a fine grid operator. It is found that the proposed method gives a faster convergent rate than the regular multigrid method. A local mode Fourier analysis confirms the validity of our proposed method. Finally, some numerical results demonstrate the efficiency of the method.

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Fourier mode analysis of multigrid methods for partial differential equations with random coefficients

Cited by 23 publications

References 18 publications

Algebraic multigrid for stationary and time‐dependent partial differential equations with stochastic coefficients

Algebraic multigrid for stationary and time‐dependent partial differential equations with stochastic coefficients

Application of hierarchical matrices for computing the Karhunen–Loève expansion

A multigrid method with reduced phase error for 2D damped Helmholtz equations

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