Karhunen-Loève approximation of random fields by generalized fast multipole methods
type: Item
instance of: scholarly article
title: Karhunen-Loève approximation of random fields by generalized fast multipole methods
zbMATH Open document ID: 1104.65008
author: Christoph Schwab
author: Radu Todor
published in: Journal of Computational Physics
publication date: 2006-10-10T00:00:00Z
review text: Due to the high cost of the stochastic Galerkin finite element method (FEM) for large the \(M\) in \(M\)-term approximate of the Karhunen-Loève (KL) expansion for a possibly instationary random field \textbf{a}, a considerable computational work has been spent on finding `optimal' (separated) approximations \(\mathbf a_M\) of \textbf{a}. The paper addresses the problem of computing a KL-approximation of the instantionary random field \textbf{a} by truncating the KL expansion of \(\mathbf a\) after \(M\) terms and by Galerkin approximation of the first \(M\) KL-eigenpairs, the mean field and its covariant (or the two point correlation) being given. The authors' approach proposed to efficiently compute the approximate of KL-expansions is based on a Ritz-Galerkin approximation of the KL-eigenvalue problem. Section 2 presents formal definitions and properties of the KL-expansion of random fields. Estimates on the decay rate of the KL-eigenvalues are provided. These estimates are essential for determining the approximation rate of the \(M\)-term truncated KL-expansion. Section 3 investigates the Galerkin approximation of truncated \(M\)-term KL-expansions. Section 4 addresses the generalized fast multipole method using accelerated matrix-vector multiplication, the covariance function being given. Algorithm details on kernel interpolation and realization of shift operations are specified, and exponential convergence with respect to the expansion order of the multipole error is given.
Mathematics Subject Classification ID: 65C50
Mathematics Subject Classification ID: 65M60
Mathematics Subject Classification ID: 60G60
Mathematics Subject Classification ID: 65C30
Mathematics Subject Classification ID: 60H15
Mathematics Subject Classification ID: 65N25
zbMATH DE Number: 5061725
zbMATH Keywords: Karhunen-Loève expansion
zbMATH Keywords: non-stationary random field
zbMATH Keywords: \(M\)-term approximate
zbMATH Keywords: Ritz-Galerkin approximation
zbMATH Keywords: Karhunen-Loève eigenvalue problem
zbMATH Keywords: generalized fast multipole method
zbMATH Keywords: accelerated matrix-vector multiplication
zbMATH Keywords: exponential convergence
MaRDI profile type: Publication
full work available at URL: j.jcp.2006.01.048
OpenAlex ID: W2062993069
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DOI: 10.1016/J.JCP.2006.01.048
description: scientific article