Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic nonlocal equation
Our paper considers uncertainty quantification for an elliptic nonlocal equation. In particular, it is assumed that the parameters which define the kernel in the nonlocal operator are uncertain and a priori distributed according to a probability measure. It is shown that the induced probability measure on some quantities of interest arising from functionals of the solution to the equation with random inputs is welldefined,s as is the posterior distribution on parameters given observations. As the elliptic nonlocal equation cannot be solved approximate posteriors are constructed. The multilevel Monte Carlo (MLMC) and multilevel sequential Monte Carlo (MLSMC) sampling algorithms are used for a priori and a posteriori estimation, respectively, of quantities of interest. Furthermore, these algorithms reduce the amount of work to estimate posterior expectations, for a given level of error, relative to Monte Carlo and i.i.d. sampling from the posterior at a given level of approximation of the solution of the elliptic nonlocal equation.
 Authors:

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 National Univ. of Singapore (Singapore). Dept. of Statistics and Applied Probability
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States). Computer Science and Mathematics Division
 Publication Date:
 Grant/Contract Number:
 AC0500OR22725
 Type:
 Accepted Manuscript
 Journal Name:
 International Journal for Uncertainty Quantification
 Additional Journal Information:
 Journal Volume: 6; Journal Issue: 6; Journal ID: ISSN 21525080
 Publisher:
 Begell House
 Research Org:
 Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)
 Sponsoring Org:
 USDOE
 Country of Publication:
 United States
 Language:
 English
 Subject:
 97 MATHEMATICS AND COMPUTING
 OSTI Identifier:
 1342665
Jasra, Ajay, Law, Kody J. H., and Zhou, Yan. Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic nonlocal equation. United States: N. p.,
Web. doi:10.1615/Int.J.UncertaintyQuantification.2016018661.
Jasra, Ajay, Law, Kody J. H., & Zhou, Yan. Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic nonlocal equation. United States. doi:10.1615/Int.J.UncertaintyQuantification.2016018661.
Jasra, Ajay, Law, Kody J. H., and Zhou, Yan. 2016.
"Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic nonlocal equation". United States.
doi:10.1615/Int.J.UncertaintyQuantification.2016018661. https://www.osti.gov/servlets/purl/1342665.
@article{osti_1342665,
title = {Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic nonlocal equation},
author = {Jasra, Ajay and Law, Kody J. H. and Zhou, Yan},
abstractNote = {Our paper considers uncertainty quantification for an elliptic nonlocal equation. In particular, it is assumed that the parameters which define the kernel in the nonlocal operator are uncertain and a priori distributed according to a probability measure. It is shown that the induced probability measure on some quantities of interest arising from functionals of the solution to the equation with random inputs is welldefined,s as is the posterior distribution on parameters given observations. As the elliptic nonlocal equation cannot be solved approximate posteriors are constructed. The multilevel Monte Carlo (MLMC) and multilevel sequential Monte Carlo (MLSMC) sampling algorithms are used for a priori and a posteriori estimation, respectively, of quantities of interest. Furthermore, these algorithms reduce the amount of work to estimate posterior expectations, for a given level of error, relative to Monte Carlo and i.i.d. sampling from the posterior at a given level of approximation of the solution of the elliptic nonlocal equation.},
doi = {10.1615/Int.J.UncertaintyQuantification.2016018661},
journal = {International Journal for Uncertainty Quantification},
number = 6,
volume = 6,
place = {United States},
year = {2016},
month = {1}
}