Fast estimation of continuous Karhunen-Loeve eigenfunctions using wavelets

TLDR: A new wavelet method for the fast estimation of continuous Karhunen-Loeve eigenfunctions is developed by projecting the ensemble functions onto orthogonal or biorthogonal interpolating function spaces and the covariance matrix may be sparsified by a multiresolution decomposition.

Abstract: This paper develops a new wavelet method for the fast estimation of continuous Karhunen-Loeve eigenfunctions. The method of snapshots is modified by projecting the ensemble functions onto orthogonal or biorthogonal interpolating function spaces. Under well-behaved piecewise smooth polynomial ensemble functions, the size of the covariance matrix produced is greatly reduced, without sacrificing much accuracy. Moreover, the covariance matrix C/spl tilde/ may be easily decomposed such that C/spl tilde/ = A/sup T/ A, and thus, the more stable singular value decomposition (SVD) algorithm may be applied. An interpolating scheme that reduces the computation of projecting the ensemble functions onto the biorthogonal subspace to a single sample is also developed. Furthermore, by projecting the ensemble functions onto wavelet spaces, the covariance matrix may be sparsified by a multiresolution decomposition. Error bounds for the eigenvalues between the sparsified and nonsparsified covariance matrix are also derived.