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

Rosseel, Eveline; Boonen, Tim J.; Vandewalle, Stefan

doi:10.1002/nla.568

Cited by 30 publications

(46 citation statements)

References 32 publications

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“…Therefore, for some parameters, for example B Initial guess. Each Newton iteration (25) involves an expensive solve of a large system. As such, the choice of a good initial guess can reduce the computational time significantly.…”

Section: Newton's Methodsmentioning

confidence: 99%

“…The expected values ξ 2 Ψ q Ψ i Ψ j can be calculated analytically by using properties of the orthogonal polynomials Ψ q [10,25]. From this representation, statistics of the torque can be computed, in a similar way to (26).…”

Section: Post-processingmentioning

confidence: 99%

“…These methods only split the stochastic discretization matrices, i.e., the D i and G 2 matrices in (25), so that in every iteration several systems with the size of the number of deterministic unknowns need to be inverted. When these block systems are exactly solved, the cost of one Newton iteration (25) is proportional to the cost of Q deterministic Newton iterations, with Q the number of stochastic unknowns.…”

Section: Solving the Linearized Systemsmentioning

confidence: 99%

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Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed

Rosseel

Gersem

Vandewalle

2010

CiCP

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The stochastic Galerkin and stochastic collocation method are two state-of-the-art methods for solving partial differential equations (PDE) containing random coefficients. While the latter method, which is based on sampling, can straightforwardly be applied to nonlinear stochastic PDEs, this is nontrivial for the stochastic Galerkin method and approximations are required. In this paper, both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder. This model can be used for designing solid-rotor induction machines in various machining tools. A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties. A numerical comparison of the computational cost and accuracy of both methods is performed. The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy, however at a higher computational cost. AbstractThe stochastic Galerkin and stochastic collocation method are two state-of-the-art methods for solving partial differential equations (PDE) containing random coefficients. While the latter method, which is based on sampling, can straightforwardly be applied to nonlinear stochastic PDEs, this is nontrivial for the stochastic Galerkin method and approximations are required. In this paper, both methods are used for constructing high-order solutions of a nonlinear stochastic PDE representing the magnetic vector potential in a ferromagnetic rotating cylinder. This model can be used for designing solid-rotor induction machines in various machining tools. A precise design requires to take ferromagnetic saturation effects into account and uncertainty on the nonlinear magnetic material properties. A numerical comparison of the computational cost and accuracy of both methods is performed. The stochastic Galerkin method requires in general less stochastic unknowns than the stochastic collocation approach to reach a certain level of accuracy, however at a higher computational cost.

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Section: Newton's Methodsmentioning

confidence: 99%

Section: Post-processingmentioning

confidence: 99%

Section: Solving the Linearized Systemsmentioning

confidence: 99%

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Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed

Rosseel

Gersem

Vandewalle

2010

CiCP

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“…by orthogonality of ψ Many authors use the space of complete polynomials V p for the stochastic discretization (see, e.g., [13,19,22,21]) to avoid the so-called curse of dimensionality since the number of degrees of freedom in V p grows exponentially with the number of random variables M in contrast to V C p where this growth is only algebraic (cf. (2.3)).…”

Section: Complete Polynomials Turning Now To the Space Vmentioning

confidence: 99%

Stochastic Galerkin Matrices

Ernst¹,

Ullmann²

2010

SIAM J. Matrix Anal. & Appl.

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Abstract. We investigate the structural, spectral, and sparsity properties of Stochastic Galerkin matrices as they arise in the discretization of linear differential equations with random coefficient functions. These matrices are characterized as the Galerkin representation of polynomial multiplication operators. In particular, it is shown that the global Galerkin matrix associated with complete polynomials cannot be diagonalized in the stochastically linear case.

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“…The process of computing the coefficients involves solving a linear system of size N |I| × N |I|, which can be prohibitively large for even a moderate number of input parameters (6 to 10) and low order polynomials (degree < 5). For this reason, there has been a flurry of recent work on solver strategies for the matrix equations arising from the Galerkin methods [30,34,22,11,10,13,35], including papers on preconditioning [32,12,31,36]. Such work has relied on knowing the matrix-valued coefficients A α of a series expansion of the parameterized matrix A(s),…”

mentioning

confidence: 99%

A Factorization of the Spectral Galerkin System for Parameterized Matrix Equations: Derivation and Applications

Constantine¹,

Gleich²,

Iaccarino³

2011

SIAM J. Sci. Comput.

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Abstract. Recent work has explored solver strategies for the linear system of equations arising from a spectral Galerkin approximation of the solution of PDEs with parameterized (or stochastic) inputs. We consider the related problem of a matrix equation whose matrix and right-hand side depend on a set of parameters (e.g., a PDE with stochastic inputs semidiscretized in space) and examine the linear system arising from a similar Galerkin approximation of the solution. We derive a useful factorization of this system of equations, which yields bounds on the eigenvalues, clues to preconditioning, and a flexible implementation method for a wide array of problems. We complement this analysis with (i) a numerical study of preconditioners on a standard elliptic PDE test problem and (ii) a fluids application using existing CFD codes; the MATLAB codes used in the numerical studies are available online.Key words. parameterized systems, spectral methods, stochastic Galerkin AMS subject classifications. 65C99, 65F10 DOI. 10.1137/1007990461. Introduction. Complex engineering models are often described by systems of equations, where the model outputs depend on a set of input parameters. Given values for the input parameters, the model output may be computed by solving a discretized differential equation, which typically involves the solution (or a series of solutions) of a system of linear equations for the unknowns. Each of these computations may be very expensive depending on factors such as grid resolution or physics components in the model. As the role of simulation gains prominence in areas such as decision support, design optimization, and predictive science, understanding the effects of variability in the input parameters on variability in the model output becomes more important. Exhaustive parameter studies (e.g., uncertainty quantification, sensitivity analysis, model calibration) can be prohibitively expensive-particularly for a large number of input parameters-and therefore accurate interpolation and robust surrogate models are essential.We consider the model problem of a matrix equation whose matrix and right-hand side depend on a set of parameters. Let s ∈ S be a set of input parameters from a d-dimensional tensor product parameter space S = S 1 ⊗ · · · ⊗ S d ; the range of each S i may be bounded or unbounded. We equip the parameter space with a bounded, separable, positive weight function ω : S → R + , where

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

Cited by 30 publications

References 32 publications

Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed

Nonlinear Stochastic Galerkin and Collocation Methods: Application to a Ferromagnetic Cylinder Rotating at High Speed

Stochastic Galerkin Matrices

A Factorization of the Spectral Galerkin System for Parameterized Matrix Equations: Derivation and Applications

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