Multilevel Representations of Random Fields and Sparse Approximations of Solutions to Random PDEs

In the analysis of random fields as well as in applications, such as in uncertainty quantification, representations of random fields as function series play an important role. While the most usual such expansions are based on Karhunen- Loève decompositions, in many cases of interest, other expansions with independent coefficients exist and may be more relevant. In particular, for certain classes of Gaussian random fields, one can obtain such expansions in terms of function systems withwavelet-type multilevel structure. In this article,we reviewconstructions of such representations that are suitable for numerical methods, as well as their impact on sparse approximations of solutions to random partial differential equations.