Weighted tensor product algorithms for linear multivariate problems

We study the epsilon-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..Lambda(all) consists of all linear functionals while...Lambda(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/epsilon and d. For Lambda(all) we construct an optimal WTP algorithm and provide a necessary and sufficient condition for tractability in terms of the sequence of wrights and the sequence of singular values for d = 1. For Lambda(std) we obtain a weaker result by constructing a WTP algorithm which is optimal only for some weight sequences. (C) 1999 Academic Press.