Faster computation of the Karhunen-Loeve expansion using its domain independence property

Pranesh, Srikara and Ghosh, Debraj (2015) Faster computation of the Karhunen-Loeve expansion using its domain independence property. In: COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING, 285 . pp. 125-145.

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Abstract

The goal of this work is to reduce the cost of computing the coefficients in the Karhunen-Loeve (KL) expansion. The KL expansion serves as a useful and efficient tool for discretizing second-order stochastic processes with known covariance function. Its applications in engineering mechanics include discretizing random field models for elastic moduli, fluid properties, and structural response. The main computational cost of finding the coefficients of this expansion arises from numerically solving an integral eigenvalue problem with the covariance function as the integration kernel. Mathematically this is a homogeneous Fredholm equation of second type. One widely used method for solving this integral eigenvalue problem is to use finite element (FE) bases for discretizing the eigenfunctions, followed by a Galerkin projection. This method is computationally expensive. In the current work it is first shown that the shape of the physical domain in a random field does not affect the realizations of the field estimated using KL expansion, although the individual KL terms are affected. Based on this domain independence property, a numerical integration based scheme accompanied by a modification of the domain, is proposed. In addition to presenting mathematical arguments to establish the domain independence, numerical studies are also conducted to demonstrate and test the proposed method. Numerically it is demonstrated that compared to the Galerkin method the computational speed gain in the proposed method is of three to four orders of magnitude for a two dimensional example, and of one to two orders of magnitude for a three dimensional example, while retaining the same level of accuracy. It is also shown that for separable covariance kernels a further cost reduction of three to four orders of magnitude can be achieved. Both normal and lognormal fields are considered in the numerical studies. (c) 2014 Elsevier B.V. All rights reserved.

Item Type:	Journal Article
Publication:	COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
Additional Information:	Copy right for this article belongs to the ELSEVIER SCIENCE SA, PO BOX 564, 1001 LAUSANNE, SWITZERLAND
Department/Centre:	Division of Mechanical Sciences > Civil Engineering
Date Deposited:	23 Mar 2015 09:55
Last Modified:	23 Mar 2015 09:55
URI:	http://eprints.iisc.ac.in/id/eprint/51079

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