Information-based complexity
Tractability and strong tractability of linear multivariate problems
Journal of Complexity
When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?
Journal of Complexity
Complexity and information
Weighted tensor product algorithms for linear multivariate problems
Journal of Complexity
The effective dimension and quasi-Monte Carlo integration
Journal of Complexity
Worst case complexity of multivariate Feynman-Kac path integration
Journal of Complexity
Finite-order weights imply tractability of multivariate integration
Journal of Complexity
Quasi-Monte Carlo methods can be efficient for integration over products of spheres
Journal of Complexity
Why Are High-Dimensional Finance Problems Often of Low Effective Dimension?
SIAM Journal on Scientific Computing
Good Lattice Rules in Weighted Korobov Spaces with General Weights
Numerische Mathematik
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We study the minimal number n(@?,d) of information evaluations needed to compute a worst case @?-approximation of a linear multivariate problem. This problem is defined over a weighted Hilbert space of functions f of d variables. One information evaluation of f is defined as the evaluation of a linear continuous functional or the value of f at a given point. Tractability means that n(@?,d) is bounded by a polynomial in both @?^-^1 and d. Strong tractability means that n(@?,d) is bounded by a polynomial only in @?^-^1. We consider weighted reproducing kernel Hilbert spaces with finite-order weights. This means that each function of d variables is a sum of functions depending only on q^* variables, where q^* is independent of d. We prove that finite-order weights imply strong tractability or tractability of linear multivariate problems, depending on a certain condition on the reproducing kernel of the space. The proof is not constructive if one uses values of f.