QUASI-MONTE CARLO INTEGRATION FOR AFFINE-PARAMETRIC, ELLIPTIC PDEs: LOCAL SUPPORTS AND PRODUCT WEIGHTS

摘要

We analyze convergence rates of first-order quasi-Monte Carlo (QMC) integration with randomly shifted lattice rules and for higher-order, interlaced polynomial lattice rules for a class of countably parametric integrands that result from linear functionals of solutions of linear, elliptic diffusion equations with affine-parametric, uncertain coefficient function a(x,y) = (a) over bar (x) + Sigma(j = 1)y(j) psi(j)(x) in a bounded domain D subset of R-d. Extending the result in [F. Y. Kuo, C. Schwab, and I. H. Sloan, SIAM J. Numer. Anal., 50 (2012), pp. 3351-3374], where psi(j) was assumed to have global support in the domain D, we assume in the present paper that supp(psi(j)) is localized in D and that we have control on the overlaps of these supports. Under these conditions we prove dimension-independent convergence rates in [1/2,1) of randomly shifted lattice rules with product weights and corresponding higher-order convergence rates by higher-order, interlaced polynomial lattice rules with product weights. The product structure of the QMC weights facilitates work bounds for the fast, component-by-component constructions of [D. Nuyens and R. Cools, Math. Comp., 75 (2006), pp. 903-920] which scale linearly with respect to the parameter dimension s. The dimension-independent convergence rates are only limited by the degree of digit interlacing used in the construction of the higher-order QMC quadrature rule and, for locally supported coefficient functions, by the summability of the locally supported coefficient sequence in the affine-parametric coefficient.