Sparse Evaluation of Compositions of Functions Using Multiscale Expansions

This paper is concerned with the estimation and evaluation of wavelet coefficients of the composition F ◦u of two functions F and u from the wavelet coefficients of u. Our main objective is to show that certain sequence spaces that can be used to measure the sparsity of the arrays of wavelet coefficients are stable under a class of nonlinear mappings F that occur naturally, e.g., in nonlinear PDEs. We indicate how these results can be used to facilitate the sparse evaluation of arrays of wavelet coefficients of compositions at asymptotically optimal computational cost. Furthermore, the basic requirements are verified for several concrete choices of nonlinear mappings. These results are generalized to compositions by a multivariate map F of several functions u1, . . . , un and their derivatives, i.e., F(Dα1u1, . . . , Dnun).