Shifted rank-1 lattice rules, a special class of quasi-Monte Carlo methods, have recently been proposed by the present authors for the integration of functions belonging to certain “weighted” Sobolev… (More)

An algorithm to generate Sobol' sequences to approximate integrals in up to 40 dimensions has been previously given by Bratley and Fox in Algorithm 659. Here, we provide more primitive polynomials… (More)

Direction numbers for generating Sobol′ sequences that satisfy the so-called Property A in up to 1111 dimensions have previously been given in Joe and Kuo [ACM Trans. Math. Software, 29 (2003), pp.… (More)

We devise and implement quasi-Monte Carlo methods for computing the expectations of nonlinear functionals of solutions of a class of elliptic partial differential equations with random coefficients.… (More)

The construction of randomly shifted rank-1 lattice rules, where the number of points n is a prime number, has recently been developed by Sloan, Kuo and Joe for integration of functions in weighted… (More)

Many recent papers considered the problem of multivariate integration, and studied the tractability of the problem in the worst case setting as the dimensionality d increases. The typical question… (More)

We present formulas that allow us to decompose a function f of d variables into a sum of 2d terms fu indexed by subsets u of {1, . . . , d}, where each term fu depends only on the variables with… (More)

We develop and justify an algorithm for the construction of quasi– Monte Carlo (QMC) rules for integration in weighted Sobolev spaces; the rules so constructed are shifted rank-1 lattice rules. The… (More)