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Stochastic reduced basis methods

Stochastic reduced basis methods
Stochastic reduced basis methods
Stochastic reduced basis methods for solving large-scale linear random algebraic systems of equations, such as those obtained by discretizing linear stochastic partial differential equations in space, time, and the random dimension, are introduced. The fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients (or random functions). We present a theoretical justification for employing basis vectors spanning the preconditioned stochastic Krylov subspace to approximate the response process. Subsequently, variants of the Bubnov–Galerkin scheme are employed to compute the undetermined coefficients, which allow explicit expressions for the response quantities to be derived. We also examine some theoretical properties of the projection scheme and procedures for computing the response statistics. Numerical studies are presented for static and dynamic analysis of stochastic structural systems. We demonstrate that significant improvements over the Neumann expansion scheme, as well as other relevant techniques in the literature, can be achieved.
stochastic mechanics, stochastic finite element method, linear random algebraic equations, stochastic krylov subspace, stochastic subspace projection theory
0001-1452
1653-1664
Nair, Prasanth B.
d4d61705-bc97-478e-9e11-bcef6683afe7
Keane, Andrew J.
26d7fa33-5415-4910-89d8-fb3620413def
Nair, Prasanth B.
d4d61705-bc97-478e-9e11-bcef6683afe7
Keane, Andrew J.
26d7fa33-5415-4910-89d8-fb3620413def

Nair, Prasanth B. and Keane, Andrew J. (2002) Stochastic reduced basis methods. AIAA Journal, 40 (8), 1653-1664.

Record type: Article

Abstract

Stochastic reduced basis methods for solving large-scale linear random algebraic systems of equations, such as those obtained by discretizing linear stochastic partial differential equations in space, time, and the random dimension, are introduced. The fundamental idea employed is to represent the system response using a linear combination of stochastic basis vectors with undetermined deterministic coefficients (or random functions). We present a theoretical justification for employing basis vectors spanning the preconditioned stochastic Krylov subspace to approximate the response process. Subsequently, variants of the Bubnov–Galerkin scheme are employed to compute the undetermined coefficients, which allow explicit expressions for the response quantities to be derived. We also examine some theoretical properties of the projection scheme and procedures for computing the response statistics. Numerical studies are presented for static and dynamic analysis of stochastic structural systems. We demonstrate that significant improvements over the Neumann expansion scheme, as well as other relevant techniques in the literature, can be achieved.

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Published date: 2002
Keywords: stochastic mechanics, stochastic finite element method, linear random algebraic equations, stochastic krylov subspace, stochastic subspace projection theory

Identifiers

Local EPrints ID: 22061
URI: http://eprints.soton.ac.uk/id/eprint/22061
ISSN: 0001-1452
PURE UUID: 83d3fd1c-e89d-495d-b49a-96f30816779f

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Date deposited: 20 Mar 2006
Last modified: 01 May 2020 16:31

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