Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic non-local equation

Jasra, Ajay; Law, Kody J.  H.; Zhou, Yan

doi:10.1615/Int.J.UncertaintyQuantification.2016018661

Title: Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic non-local equation

Abstract

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 well-defined,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:

Jasra, Ajay ^[1]; Law, Kody J. H. ^[2]; Zhou, Yan ^[1]

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:: Fri Jan 01 00:00:00 EST 2016

Research Org.:: Oak Ridge National Lab. (ORNL), Oak Ridge, TN (United States)

Sponsoring Org.:: USDOE

OSTI Identifier:: 1342665

Grant/Contract Number:: AC05-00OR22725

Resource Type:: Accepted Manuscript

Journal Name:: International Journal for Uncertainty Quantification

Additional Journal Information:: Journal Volume: 6; Journal Issue: 6; Journal ID: ISSN 2152-5080

Publisher:: Begell House

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING

Citation Formats


                    Jasra, Ajay, Law, Kody J.  H., and Zhou, Yan. Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic non-local equation.  United States: N. p., 2016. 
Web.  doi:10.1615/Int.J.UncertaintyQuantification.2016018661.

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                    Jasra, Ajay, Law, Kody J.  H., & Zhou, Yan. Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic non-local equation.  United States.  https://doi.org/10.1615/Int.J.UncertaintyQuantification.2016018661

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                    Jasra, Ajay, Law, Kody J.  H., and Zhou, Yan. Fri .  
"Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic non-local equation".  United States.  https://doi.org/10.1615/Int.J.UncertaintyQuantification.2016018661.  https://www.osti.gov/servlets/purl/1342665.

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@article{osti_1342665,

  title        = {Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic non-local 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 well-defined,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         = {Fri Jan 01 00:00:00 EST 2016},

  month        = {Fri Jan 01 00:00:00 EST 2016}

}

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Works referencing / citing this record:

Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions
journal, December 2016

Del Moral, Pierre; Jasra, Ajay; Law, Kody J. H.
Stochastic Analysis and Applications, Vol. 35, Issue 3
DOI: 10.1080/07362994.2016.1272421

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Title: Forward and inverse uncertainty quantification using multilevel Monte Carlo algorithms for an elliptic non-local equation

Abstract

Citation Formats

Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions journal, December 2016

Multilevel sequential Monte Carlo: Mean square error bounds under verifiable conditions
journal, December 2016