Application of Quasi-Monte Carlo Methods to Elliptic PDEs with Random Diffusion Coefficients: A Survey of Analysis and Implementation

• Published in: Foundations of computational mathematics Vol. 16; no. 6; pp. 1631 - 1696
• Main Authors: Kuo, Frances Y., Nuyens, Dirk
• Format: Journal Article
• Language: English
• Published: New York Springer US 01-12-2016; Springer; Springer Nature B.V
• Subjects: Algorithms; Analysis; Applications of Mathematics; Computer programs; Computer Science; Differential equations, Partial; Diffusion coefficient; Economics; Elliptic differential equations; Elliptic functions; Error analysis; Linear and Multilinear Algebras; Math Applications in Computer Science; Mathematical analysis; Mathematics; Mathematics and Statistics; Matrix Theory; Monte Carlo method; Monte Carlo methods; Monte Carlo simulation; Numerical Analysis; Partial differential equations; Software; United States; Infinite-dimensional integration; Single-level; Uniform; Multi-level; Deterministic; Quasi-Monte Carlo methods; Partial differential equations with random coefficients; Lognormal; 65D30; Randomized; First order; 65D32; Higher order; 65N30
• ISSN: 1615-3375; 1615-3383
• DOI: 10.1007/s10208-016-9329-5

This article provides a survey of recent research efforts on the application of quasi-Monte Carlo (QMC) methods to elliptic partial differential equations (PDEs) with random diffusion coefficients. It considers and contrasts the uniform case versus the lognormal case, single-level algorithms versus multi-level algorithms, first-order QMC rules versus higher-order QMC rules, and deterministic QMC methods versus randomized QMC methods. It gives a summary of the error analysis and proof techniques in a unified view, and provides a practical guide to the software for constructing and generating QMC points tailored to the PDE problems. The analysis for the uniform case can be generalized to cover a range of affine parametric operator equations.