Fast QMC matrix-vector multiplication

TL;DR

This paper introduces a fast method for matrix-vector multiplication in Quasi-Monte Carlo (QMC) rules, enabling efficient high-dimensional integral approximations with significant computational speed-ups.

Contribution

The paper develops a novel approach to perform QMC matrix-vector products using FFT, applicable to various QMC rules, reducing computational complexity from quadratic to near-linear.

Findings

Achieves $oldsymbol{O}(N \, ext{log} N)$ complexity for QMC matrix-vector multiplication.

Demonstrates significant speed-up in high-dimensional integral computations.

Validates the method through three numerical experiments on different PDE and statistical problems.

Abstract

Quasi-Monte Carlo (QMC) rules $1/N \sum_{n=0}^{N-1} f(\boldsymbol{y}_n A)$ can be used to approximate integrals of the form $\int_{[0,1]^s} f(\boldsymbol{y} A) \,\mathrm{d} \boldsymbol{y}$, where $A$ is a matrix and $\boldsymbol{y}$ is row vector. This type of integral arises for example from the simulation of a normal distribution with a general covariance matrix, from the approximation of the expectation value of solutions of PDEs with random coefficients, or from applications from statistics. In this paper we design QMC quadrature points $\boldsymbol{y}_0, ..., \boldsymbol{y}_{N-1} \in [0,1]^s$ such that for the matrix $Y = (\boldsymbol{y}_{0}^\top, ..., \boldsymbol{y}_{N-1}^\top)^\top$ whose rows are the quadrature points, one can use the fast Fourier transform to compute the matrix-vector product $Y \boldsymbol{a}^\top$, $\boldsymbol{a} \in \mathbb{R}^s$, in $\mathcal{O}(N \log N)$ operations and at most $s-1$ extra additions. The proposed method can be applied to lattice rules, polynomial lattice rules and a certain type of Korobov $p$-set. The approach is illustrated computationally by three numerical experiments. The first test considers the generation of points with normal distribution and general covariance matrix, the second test applies QMC to high-dimensional, affine-parametric, elliptic partial differential equations with uniformly distributed random coefficients, and the third test addresses Finite-Element discretizations of elliptic partial differential equations with high-dimensional, log-normal random input data. All numerical tests show a significant speed-up of the computation times of the fast QMC matrix method compared to a conventional implementation as the dimension becomes large.