A dimension-incremental approximation method for bounded orthonormal product bases

We present a dimension-incremental algorithm for approximating high-dimensional, multivariate functions in an arbitrary bounded orthonormal product basis. The goal is a truncation of the basis expansion of the function, where the corresponding significant index set is not known in advance. Our method is based on point evaluations of the considered function and adaptively builds a reasonable index set, such that the approximately largest basis coefficients are included. We provide an proof idea of a detection guarantee for such an index set under certain assumptions on the sub-methods used within our algorithm, which can be used as a foundation for similar statements in various other situations as well.

The application of a modification of the developed adaptive method to partial differential equations which depend on several random parameters allows for a numerical classification of important and non-important random parameters as well as significantly interacting parameters.