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

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.