Good Lattice Rules in Weighted Korobov Spaces with General Weights |
| |
Authors: | Josef Dick Ian H Sloan Xiaoqun Wang Henryk Woźniakowski |
| |
Institution: | (1) School of Mathematics, University of New South Wales, Sydney, 2052, Australia;(2) Department of Mathematical Sciences, Tsinghua University, Beijing, 100084, China;(3) Department of Computer Science, Columbia University, New York, NY 10027, USA;(4) Institute of Applied Mathematics and Mechanics, University of Warsaw, Poland |
| |
Abstract: | We study the problem of multivariate integration and the construction of good lattice rules in weighted Korobov spaces with
general weights. These spaces are not necessarily tensor products of spaces of univariate functions. Sufficient conditions
for tractability and strong tractability of multivariate integration in such weighted function spaces are found. These conditions
are also necessary if the weights are such that the reproducing kernel of the weighted Korobov space is pointwise non-negative.
The existence of a lattice rule which achieves the nearly optimal convergence order is proven. A component-by-component (CBC)
algorithm that constructs good lattice rules is presented. The resulting lattice rules achieve tractability or strong tractability
error bounds and achieve nearly optimal convergence order for suitably decaying weights. We also study special weights such
as finite-order and order-dependent weights. For these special weights, the cost of the CBC algorithm is polynomial. Numerical
computations show that the lattice rules constructed by the CBC algorithm give much smaller worst-case errors than the mean
worst-case errors over all quasi-Monte Carlo rules or over all lattice rules, and generally smaller worst-case errors than
the best Korobov lattice rules in dimensions up to hundreds. Numerical results are provided to illustrate the efficiency of
CBC lattice rules and Korobov lattice rules (with suitably chosen weights), in particular for high-dimensional finance problems. |
| |
Keywords: | Quasi-Monte Carlo methods lattice rules multivariate integration |
本文献已被 SpringerLink 等数据库收录! |
|