Uncertainty quantification and stochastic polynomial chaos expansion for recovering random data in Darcy and Diffusion equations

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Abstract

A probabilistic collocation based polynomial chaos expansion method is developed to solve stochastic boundary value problems with random coefficients and randomly distributed initial data. In this paper we deal with two different boundary value problems with random data: the Darcy equation with random lognormally distributed hydraulic conductivity, and a diffusion equation with absorption, with random distribution of the initial concentration under periodic boundary conditions. Special attention is paid to the extension of the probabilistic collocation method to input data with arbitrary correlation functions defined both analytically and through measurements. We construct the relevant Karhunen-Loève expansion from a special randomized singular value decomposition of the correlation matrix, which makes possible to treat problems of high dimension. We show that the unknown statistical characteristics of the random input data can be recovered from the correlation analysis of the solution field.

Original languageEnglish
Pages (from-to)733-745
Number of pages13
JournalJournal of Inverse and Ill-Posed Problems
Volume25
Issue number6
DOIs
Publication statusPublished - 1 Dec 2017

Keywords

  • Darcy equation
  • Monte Carlo direct simulation
  • polynomial chaos
  • probabilistic collocation
  • Uncertainty quantification
  • TRANSPORT
  • APPROXIMATIONS
  • KINETICS
  • MEDIA
  • ALGORITHMS
  • SIMULATION
  • COLLOCATION

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