Karhunen–Loéve Expansion of Stochastic Processes with a Modified Exponential Covariance Kernel
Publication: Journal of Engineering Mechanics
Volume 133, Issue 7
Abstract
The spectral representation of stationary stochastic processes via the Karhunen-Loéve (KL) expansion is examined from a numerical efficiency perspective. Attention is focused on processes which have commonly been characterized by covariance kernels decaying exponentially versus the position/time delay variable. By introducing a slight modification in the mathematical description of this covariance kernel, the nondifferentiability at its peak is eliminated, whereas most of its advantageous properties are retained. It is shown that compared to the common exponential model, the requisite number of terms for representing the process in context with the modified kernel is significantly smaller. The effect is demonstrated by means of a specific numerical example. This is done by first determining the eigenfunctions/eigenvalues associated with the KL expansion for the modified kernel model, and by afterwards estimating the approximation errors corresponding to the two kernels considered for specific numerical values. Clearly, the enhanced computational efficiency of the KL expansion associated with the modified kernel can significantly expedite its incorporation in stochastic finite elements and other areas of stochastic mechanics.
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Acknowledgments
The writers gratefully acknowledge the partial support of this work from the Alexander von Humboldt (AvH) Foundation in Germany, the German Research Foundation (DFG) within the framework of the Collaborative Research Center SFB 528, and from the Sandia National Laboratories, DOE. Sandia is a multiprogram laboratory operated by Sandia Corporation, a Lockheed Martin Company, for the United States Department of Energy’s National Nuclear Security Administration under Contract No. DOEDE-AC04-94AL85000.
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Information & Authors
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Published In
Journal of Engineering Mechanics
Volume 133 • Issue 7 • July 2007
Pages: 773 - 779
Copyright
© 2007 ASCE.
History
Received: Nov 15, 2005
Accepted: May 23, 2006
Published online: Jul 1, 2007
Published in print: Jul 2007
Notes
Note. Associate Editor: Joel P. Conte
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