Weighted Tensor Product Algorithms for Linear Multivariate Problems

We study the ε-approximation of linear multivariate problems defined over weighted tensor product Hilbert spaces of functions f of d variables. A class of weighted tensor product (WTP) algorithms is defined which depends on a number of parameters. Two classes of permissible information are studied. Λ^all consists of all linear functionals while Λ^std consists of evaluations of f or its derivatives. We show that these multivariate problems are sometimes tractable even with a worst-case assurance. We study problem tractability by investigating when a WTP algorithm is a polynomial-time algorithm, that is, when the minimal number of information evaluations is a polynomial in 1/ε and d. For Λ^all we construct an optimal WTP algorithm and provide a necessary and sufficient condition for tractability in terms of the sequence of weights and the sequence of singular values for d=1. ForΛ^std we obtain a weaker result by constructing a WTP algorithm which is optimal only for some weight sequences.