Adaptive Multidimensional Integration Based on Rank-1 Lattices

Quasi-Monte Carlo methods are used for numerically integrating multivariate functions. However, the error bounds for these methods typically rely on a priori knowledge of some semi-norm of the integrand, not on the sampled function values. In this article, we propose an error bound based on the discrete Fourier coefficients of the integrand. If these Fourier coefficients decay more quickly, the integrand has less fine scale structure, and the accuracy is higher. We focus on rank-1 lattices because they are a commonly used quasi-Monte Carlo design and because their algebraic structure facilitates an error analysis based on a Fourier decomposition of the integrand. This leads to a guaranteed adaptive cubature algorithm with computational cost $O(mb^m)$, where $b$ is some fixed prime number and $b^m$ is the number of data points

First-Year Student Success Initiative: Financial Working Group Video

Jessica Hickernell, Associate Director of Student Financial Aid discusses the goals of the financial working group that is part of the First-Year Student Success Initiative at UMaine

A Multivariate Fast Discrete Walsh Transform with an Application to Function Interpolation

For high dimensional problems, such as approximation and integration, one cannot afford to sample on a grid because of the curse of dimensionality. An attractive alternative is to sample on a low discrepancy set, such as an integration lattice or a digital net. This article introduces a multivariate fast discrete Walsh transform for data sampled on a digital net that requires only $O(N \log N)$ operations, where $N$ is the number of data points. This algorithm and its inverse are digital analogs of multivariate fast Fourier transforms. This fast discrete Walsh transform and its inverse may be used to approximate the Walsh coefficients of a function and then construct a spline interpolant of the function. This interpolant may then be used to estimate the function's effective dimension, an important concept in the theory of numerical multivariate integration. Numerical results for various functions are presented

The Discrepancy and Gain Coefficients of Scrambled Digital Nets

AbstractDigital sequences and nets are among the most popular kinds of low discrepancy sequences and sets and are often used for quasi-Monte Carlo quadrature rules. Several years ago Owen proposed a method of scrambling digital sequences and recently Faure and Tezuka have proposed another method. This article considers the discrepancy of digital nets under these scramblings. The first main result of this article is a formula for the discrepancy of a scrambled digital (λ, t, m, s)-net in base b with n=λbm points that requires only O(n) operations to evaluate. The second main result is exact formulas for the gain coefficients of a digital (t, m, s)-net in terms of its generator matrices. The gain coefficients, as defined by Owen, determine both the worst-case and random-case analyses of quadrature error