Comparison between Karhunen–Loeve and wavelet expansions for simulation of Gaussian processes

The series representation consisting of eigenfunctions as the orthogonal basis is called the Karhunen–Loeve expansion. This paper demonstrates that the determination of eigensolutions using a wavelet-Galerkin scheme for Karhunen–Loeve expansion is computationally equivalent to using wavelet directly for stochastic expansion and simulating the correlated random coefficients using eigen decomposition. An alternate but longer wavelet expansion using Cholesky decomposition is shown to be of comparable accuracy. When simulation time dominates over initial overhead incurred by eigen or Cholesky decomposition, it is potentially more efficient to use a shorter truncated K–L expansion that only retains the most significant eigenmodes.