Computational aspects of the stochastic finite element method

Eiermann, Michael; Ernst, Oliver G.; Ullmann, Elisabeth

doi:10.1007/s00791-006-0047-4

Cited by 86 publications

(82 citation statements)

References 31 publications

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“…Of interest are applications related to e.g. steady or unsteady simulations of nonlinear equations [7] or stochastic finite element methods [12,33] in three dimensions where variable preconditioning using approximate solvers has to be usually considered. We also note that when all right-hand sides are available simultaneously and when the matrix is fixed, block subspace methods may be also suitable.…”

Section: Discussionmentioning

confidence: 99%

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

Carvalho¹,

Gratton²,

Lago³

et al. 2011

SIAM J. Matrix Anal. & Appl.

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This work is concerned with the development and study of a minimum residual norm subspace method based on the Generalized Conjugate Residual method with inner Orthogonalization (GCRO) method that allows flexible preconditioning and deflated restarting for the solution of non-symmetric or non-Hermitian linear systems. First we recall the main features of Flexible Generalized Minimum Residual with deflated restarting (FGMRES-DR), a recently proposed algorithm of the same family but based on the GMRES method. Next we introduce the new inner-outer subspace method named FGCRO-DR. A theoretical comparison of both algorithms is then made in the case of flexible preconditioning. It is proved that FGCRO-DR and FGMRES-DR are algebraically equivalent if a collinearity condition is satisfied. While being nearly as expensive as FGMRES-DR in terms of computational operations per cycle, FGCRO-DR offers the additional advantage to be suitable for the solution of sequences of slowly changing linear systems (where both the matrix and right-hand side can change) through subspace recycling. Numerical experiments on the solution of multidimensional elliptic partial differential equations show the efficiency of FGCRO-DR when solving sequences of linear systems.

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Section: Discussionmentioning

confidence: 99%

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

Carvalho¹,

Gratton²,

Lago³

et al. 2011

SIAM J. Matrix Anal. & Appl.

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“…The assembly of an H 2 -matrix costs O(n log n) operations. The vector a (0) m solves a discretized version of the KL integral eigenproblem (5) and can be computed with O(n log n) complexity as outlined in [2]. In summary, then, approximations to each component of the gradient ∇a m of KL eigenfunctions can be obtained with O(n log n) complexity; thus computing ∇a (M ) costs O(n log n) operations.…”

Section: Differentiation Of Karhunen-loève Eigenfunctionsmentioning

confidence: 99%

“…We assess the quality of the expansions ∇a (M ) and ∇ (M ) a by computing the quantities ∇a (M ) 2 = M m=1 λ m ∇a m 2 and ∇ (M ) a 2 = M m=1 γ m , and comparing these with the total variance of ∇a, ∇a 2 2 , then ∇a (M ) converges to ∇a in the m.-s. sense uniformly on D for M → ∞. 3 Thus, for sufficiently smooth covariance functions ∇a (M ) converges to ∇a in the same sense as the truncated KL expansion ∇ (M ) a does (see, e.g., [8,Chapter 10]).…”

Section: Numerical Experimentsmentioning

confidence: 99%

“…An effective way to solve this matrix eigenvalue problem is outlined in [2] where hierarchical matrix techniques [3,4] are combined with a thick-restart Lanczos method [5]; this approach costs O(n log n) operations.…”

Section: Introductionmentioning

confidence: 99%

“…In the second approach, the KL expansion of ∇a is computed directly. We build upon the work in [2] and achieve a total cost of O(n log n) operations for both strategies.…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Expansion of random field gradients using hierarchical matrices

Busch

Ernst

Ullmann

2011

Proc Appl Math and Mech

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We present two expansions for the gradient of a random field. In the first approach, we differentiate its truncated KarhunenLoève expansion. In the second approach, the Karhunen-Loève expansion of the random field gradient is computed directly. Both strategies require the solution of dense, symmetric matrix eigenvalue problems which can be handled efficiently by combining hierachical matrix techniques with a thick-restart Lanczos method.

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Optimization with constraints considering polymorphic uncertainties

et al. 2019

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In this contribution, a numerical design strategy for the optimization under polymorphic uncertainty is introduced and applied to a self‐weight minimization of a framework structure. The polymorphic uncertainty, which affects the constraint function of the optimization problem, is thereby modeled in terms of stochastic variables, fuzzy sets, and intervals to account for variability, imprecision and insufficient information. The stochastic quantities are computed using polynomial chaos expansion resulting in a purely fuzzy‐valued formulation of the constraint functions which can be computed using α‐cut optimization. Afterward, the constraint function can be interpreted in a possibilistic manner, resulting in a flexible formulation to include expert knowledge and to achieve a robust design.

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Computational aspects of the stochastic finite element method

Abstract: Abstract. We present an overview of the stochastic finite element method with an emphasis on the computational tasks involved in its implementation.

Cited by 86 publications

References 31 publications

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

A Flexible Generalized Conjugate Residual Method with Inner Orthogonalization and Deflated Restarting

Expansion of random field gradients using hierarchical matrices

Optimization with constraints considering polymorphic uncertainties

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