Karhunen-Loève approximation of random fields by generalized fast multipole methods
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.
accelerated matrix-vector multiplication
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