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Improved Efficiency of a Multi-Index FEM for Computational Uncertainty Quantification

by J. Dick and M. Feischl and Ch. Schwab

(Report number 2018-22)

Abstract
We propose a multi-index algorithm for the Monte Carlo discretization of a linear, elliptic PDE with affine-parametric input. We prove an error vs. work analysis which allows a multi-level finite-element approximation in the physical domain, and apply the multi-index analysis with isotropic, unstructured mesh refinement in the physical domain for the solution of the forward problem, for the approximation of the random field, and for the Monte-Carlo quadrature error. Our approach allows general spatial domains and unstructured mesh hierarchies. The improvement in complexity is obtained from combining spacial discretization, dimension truncation and MC sampling in a multi-index fashion. Our analysis improves cost estimates compared to multi-level algorithms for similar problems and mathematically underpins the outstanding practical performance of multi-index algorithms for partial differential equations with random coefficients.

Keywords: multi-index, Monte Carlo, finite element method, uncertainty quantification

@Techreport{DFS18_776,
  author = {J. Dick and M. Feischl and Ch. Schwab},
  title = {Improved Efficiency of a Multi-Index FEM for Computational Uncertainty Quantification},
  institution = {Seminar for Applied Mathematics, ETH Z{\"u}rich},
  number = {2018-22},
  address = {Switzerland},
  url = {https://www.sam.math.ethz.ch/sam_reports/reports_final/reports2018/2018-22.pdf },
  year = {2018}
}

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