Adaptive Fourier decomposition‐type sparse representations versus the Karhunen–Loève expansion for decomposing stochastic processes

Abstract

The present study devotes to introducing and analyzing the use of the adaptive Fourier decomposition (AFD)‐type methods in the area of stochastic processes and random fields. We involve two types of algorithms, namely, stochastic AFD (SAFD) and stochastic pre‐orthogonal AFD (SPOAFD), for, respectively, the Hardy space format and non‐Hardy space ones, as may be regarded. We provide both their theoretical results and practical algorithms and compare them with the well adopted and, in fact, dominating Karhunen–Loève (KL)‐type expansions. The AFD methods involve a finite or infinite sequence of optimally chosen parameters; they, in contrast, do not rely on and hence not have to compute the eigenpairs of the second type Fredholm integral equation with the covariance function as the kernel. Apart from such computational conveniences, they with the same convergence rate have flexibility of choosing best suitable dictionaries for doing the specific task in the practice. We include a number of experiments showing that in terms of effectiveness, the AFD methods give better approximations before all the positive eigenvalues running out in the case the integral operator being of a finite rank or before the KL iteration step becomes excessively large: The AFD methods normally give better approximations from the very beginning of the iterations.