Accelerating the Convergence of Lattice Methods by Importance Sampling-Based Transformations

Abstract

Importance sampling is a powerful technique for improving the stochastic solution of quadrature problems as well as problems associated with the solution of integral equations, and a generalization of importance sampling, called weighted importance sampling, provides even more potential for error reduction. Additionally, lattice methods are particularly effective for integrating sufficiently smooth periodic functions. We will discuss the advantage of combining these ideas to transform non-periodic to periodic integrands over the unit hypercube to improve the convergence rates of lattice-based quadrature formulas. We provide a pair of examples that show that with the proper choice of importance transformation, the order in the rate of convergence of a quadrature formula can be increased significantly. This technique becomes even more effective when implemented using a family of multidimensional dyadic sequences generally called extensible lattices. Based on an extension of an idea of Soboĺ [17] extensible lattices are both infinite and at the same time return to lattice-based methods with the appropriate choice of sample size. The effectiveness of these sequences, both theoretically and with numerical results, is discussed. Also, there is an interesting parallel with low discrepancy sequences generated by the fractional parts of integer multiples of irrationals which may point the way to a useful construction method for extensible lattices.