Application of quasi-Monte Carlo methods to elliptic PDEs with random\n diffusion coefficients - a survey of analysis and implementation

2016/06/21 by Frances Y. Kuo, Dirk Nuyens, Kuo, Frances Y. +1 · 10 citations
Decision Sciences · Mathematics · #Affine transformation #Applied mathematics #Computer science #Cover (algebra) #Hybrid Monte Carlo #Markov chain Monte Carlo #Mathematical Approximation and Integration #Mathematical analysis #Mathematical optimization #Mathematics #Monte Carlo method #Parametric statistics #Partial differential equation #Probabilistic and Robust Engineering Design #Pure mathematics #Quasi-Monte Carlo method #Statistics

paper · pdf · doi:10.48550/arxiv.1606.06613

published in arXiv (Cornell University) (Cornell University)

openalex publication_date 2016/06/21 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

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

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