EXPLICIT COST BOUNDS OF ALGORITHMS FOR MULTIVARIATE TENSOR PRODUCT PROBLEMS

We study multivariate tenser product problems in the worst case and average case settings. They are defined on functions of d variables. For arbitrary d, we provide explicit upper bounds on the costs of algorithms which compute an epsilon-approximation to the solution. The cost bounds are of the form (c(d) + 2)beta(1)(beta(2) + beta(3)(ln 1/epsilon)/(d - 1))(beta 4(d-1))(1/epsilon)(beta 5). Here c(d) is the cost of one function evaluation (or one linear functional evaluation), and beta(i)'s do not depend on d; they are determined by the properties of the problem for d = 1. For certain tenser product problems, these cost bounds do not exceed c(d)K epsilon(-p) for some numbers K and p, both independent of d. However, the exponents p which we obtain are too large. We apply these general estimates to certain integration and approximation problems in the worst and average case settings. We also obtain an upper bound, which is independent of d, for the number, n(epsilon, d), of points for which discrepancy (with unequal weights) is at most epsilon, n(epsilon, d) less than or equal to 7.26 epsilon(-2.454) , For All d, epsilon less than or equal to 1. (C) 1995 Academic Press, Inc.