On the theoretical foundations of stochastic reduced basis methods
On the theoretical foundations of stochastic reduced basis methods
Stochastic reduced basis methods (SRBMs) are a class of numerical techniques for approximately computing the response of stochastic systems. The basic idea is to approximate the response using a linear combination of stochastic basis vectors with undetermined coefficients. In this paper, we examine the theoretical foundations of SRBMs by exploring their relationship with Krylov subspace methods for deterministic systems. The mathematical justification for employing the terms of the stochastic Krylov subspace as basis vectors is presented. It is shown that SRBMs are a stochastic generalization of preconditioned Krylov subspace methods. Subsequently, some approaches for stochastic generalization of the Bubnov-Galerkin scheme are analyzed. We also address the issue of computing a posteriori error estimates of SRBMs. Some preliminary numerical studies are presented for examining the accuracy of the error estimates. The paper concludes with a discussion of ongoing work on algebraic random eigenvalue problems.
1677
American Institute of Aeronautics and Astronautics
Nair, Prasanth B.
ecde4c5a-d628-4128-93b9-ce3e4561f0e8
2001
Nair, Prasanth B.
ecde4c5a-d628-4128-93b9-ce3e4561f0e8
Nair, Prasanth B.
(2001)
On the theoretical foundations of stochastic reduced basis methods.
In Proceedings of the 42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference and Exhibition.
American Institute of Aeronautics and Astronautics.
.
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Abstract
Stochastic reduced basis methods (SRBMs) are a class of numerical techniques for approximately computing the response of stochastic systems. The basic idea is to approximate the response using a linear combination of stochastic basis vectors with undetermined coefficients. In this paper, we examine the theoretical foundations of SRBMs by exploring their relationship with Krylov subspace methods for deterministic systems. The mathematical justification for employing the terms of the stochastic Krylov subspace as basis vectors is presented. It is shown that SRBMs are a stochastic generalization of preconditioned Krylov subspace methods. Subsequently, some approaches for stochastic generalization of the Bubnov-Galerkin scheme are analyzed. We also address the issue of computing a posteriori error estimates of SRBMs. Some preliminary numerical studies are presented for examining the accuracy of the error estimates. The paper concludes with a discussion of ongoing work on algebraic random eigenvalue problems.
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Published date: 2001
Venue - Dates:
42nd AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics and Materials Conference and Exhibition, Seattle, USA, 2001-04-16 - 2001-04-19
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Local EPrints ID: 21847
URI: http://eprints.soton.ac.uk/id/eprint/21847
PURE UUID: ff63801b-b2f0-40cc-b836-e4bded8b84f9
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Date deposited: 01 Mar 2007
Last modified: 15 Mar 2024 06:33
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Author:
Prasanth B. Nair
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