Bayesian and variational Bayesian approaches for flows in heterogeneous random media

Yang, Keren; Guha, Nilabja; Efendiev, Yalchin; Mallick, Bani K.

doi:10.1016/j.jcp.2017.04.034

Title: Bayesian and variational Bayesian approaches for flows in heterogeneous random media

Abstract

In this paper, we study porous media flows in heterogeneous stochastic media. We propose an efficient forward simulation technique that is tailored for variational Bayesian inversion. As a starting point, the proposed forward simulation technique decomposes the solution into the sum of separable functions (with respect to randomness and the space), where each term is calculated based on a variational approach. This is similar to Proper Generalized Decomposition (PGD). Next, we apply a multiscale technique to solve for each term (as in [1]) and, further, decompose the random function into 1D fields. As a result, our proposed method provides an approximation hierarchy for the solution as we increase the number of terms in the expansion and, also, increase the spatial resolution of each term. We use the hierarchical solution distributions in a variational Bayesian approximation to perform uncertainty quantification in the inverse problem. Finally, we conduct a detailed numerical study to explore the performance of the proposed uncertainty quantification technique and show the theoretical posterior concentration.

Authors:

Yang, Keren ^[1]; Guha, Nilabja ^[2]; Efendiev, Yalchin ^[3]; Mallick, Bani K. ^[2]

Fudan Univ., Shanghai (China)
Texas A & M Univ., College Station, TX (United States)
Texas A & M Univ., College Station, TX (United States); North-Eastern Federal University, Yakutsk (Russian Federation)

Publication Date:: Fri May 05 00:00:00 EDT 2017

Research Org.:: Texas A & M Univ., College Station, TX (United States)

Sponsoring Org.:: National Science Foundation (NSF); Russian Federation Government; USDOD; USDOE Office of Science (SC), Advanced Scientific Computing Research (ASCR); Qatar National Research Fund

OSTI Identifier:: 1533959

Alternate Identifier(s):: OSTI ID: 1398704

Grant/Contract Number:: SC0010713; 1620318; N 14.Y26.31.0013; FA9550-15-1-0071; FG02-13ER26165; NPRP 7-1482-1278; 14.Y26.31.0013

Resource Type:: Accepted Manuscript

Journal Name:: Journal of Computational Physics

Additional Journal Information:: Journal Volume: 345; Journal Issue: C; Journal ID: ISSN 0021-9991

Publisher:: Elsevier

Country of Publication:: United States

Language:: English

Subject:: 97 MATHEMATICS AND COMPUTING; Computer Science; Physics; Uncertainty quantification; Variational Bayesian method; Proper generalized decomposition

Citation Formats


                    Yang, Keren, Guha, Nilabja, Efendiev, Yalchin, and Mallick, Bani K. Bayesian and variational Bayesian approaches for flows in heterogeneous random media.  United States: N. p., 2017. 
Web.  doi:10.1016/j.jcp.2017.04.034.

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                    Yang, Keren, Guha, Nilabja, Efendiev, Yalchin, & Mallick, Bani K. Bayesian and variational Bayesian approaches for flows in heterogeneous random media.  United States.  https://doi.org/10.1016/j.jcp.2017.04.034

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                    Yang, Keren, Guha, Nilabja, Efendiev, Yalchin, and Mallick, Bani K. Fri .  
"Bayesian and variational Bayesian approaches for flows in heterogeneous random media".  United States.  https://doi.org/10.1016/j.jcp.2017.04.034.  https://www.osti.gov/servlets/purl/1533959.

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@article{osti_1533959,

  title        = {Bayesian and variational Bayesian approaches for flows in heterogeneous random media},

  author       = {Yang, Keren and Guha, Nilabja and Efendiev, Yalchin and Mallick, Bani K.},

  abstractNote = {In this paper, we study porous media flows in heterogeneous stochastic media. We propose an efficient forward simulation technique that is tailored for variational Bayesian inversion. As a starting point, the proposed forward simulation technique decomposes the solution into the sum of separable functions (with respect to randomness and the space), where each term is calculated based on a variational approach. This is similar to Proper Generalized Decomposition (PGD). Next, we apply a multiscale technique to solve for each term (as in [1]) and, further, decompose the random function into 1D fields. As a result, our proposed method provides an approximation hierarchy for the solution as we increase the number of terms in the expansion and, also, increase the spatial resolution of each term. We use the hierarchical solution distributions in a variational Bayesian approximation to perform uncertainty quantification in the inverse problem. Finally, we conduct a detailed numerical study to explore the performance of the proposed uncertainty quantification technique and show the theoretical posterior concentration.},

  doi          = {10.1016/j.jcp.2017.04.034},

  journal      = {Journal of Computational Physics},

  number       = C,

  volume       = 345,

  place        = {United States},

  year         = {Fri May 05 00:00:00 EDT 2017},

  month        = {Fri May 05 00:00:00 EDT 2017}

}

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Title: Bayesian and variational Bayesian approaches for flows in heterogeneous random media

Abstract

Citation Formats

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