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Title: Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification

Abstract

Here, we analyze the convergence of probability density functions utilizing approximate models for both forward and inverse problems. We consider the standard forward uncertainty quantification problem where an assumed probability density on parameters is propagated through the approximate model to produce a probability density, often called a push-forward probability density, on a set of quantities of interest (QoI). The inverse problem considered in this paper seeks to update an initial probability density assumed on model input parameters such that the subsequent push-forward of this updated density through the parameter-to-QoI map matches a given probability density on the QoI. We prove that the densities obtained from solving the forward and inverse problems, using approximate models, converge to the true densities as the approximate models converge to the true models. Numerical results are presented to demonstrate convergence rates of densities for sparse grid approximations of parameter-to-QoI maps and standard spatial and temporal discretizations of PDEs and ODEs.

Authors:
 [1];  [2];  [2]
+ Show Author Affiliations
  1. Univ. of Colorado, Boulder, CO (United States)
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Publication Date:
Research Org.:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Org.:
USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
OSTI Identifier:
1479491
Alternate Identifier(s):
OSTI ID: 1595429
Report Number(s):
SAND-2018-6887J; SAND-2020-0021J
Journal ID: ISSN 1064-8275; 664970
Grant/Contract Number:  
AC04-94AL85000; DMS-1818941
Resource Type:
Accepted Manuscript
Journal Name:
SIAM Journal on Scientific Computing
Additional Journal Information:
Journal Volume: 40; Journal Issue: 5; Journal ID: ISSN 1064-8275
Publisher:
SIAM
Country of Publication:
United States
Language:
English
Subject:
97 MATHEMATICS AND COMPUTING; inverse problems; uncertainty quantification; density estimation; surrogate modeling; response surface approximations; discretization errors; LP convergence; approximate modeling

Citation Formats

Butler, Troy, Jakeman, John Davis, and Wildey, Timothy Michael. Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification. United States: N. p., 2018. Web. doi:10.1137/18M1181675.
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Butler, Troy, Jakeman, John Davis, & Wildey, Timothy Michael. Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification. United States. https://doi.org/10.1137/18M1181675
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Butler, Troy, Jakeman, John Davis, and Wildey, Timothy Michael. Thu . "Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification". United States. https://doi.org/10.1137/18M1181675. https://www.osti.gov/servlets/purl/1479491.
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@article{osti_1479491,
title = {Convergence of Probability Densities Using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification},
author = {Butler, Troy and Jakeman, John Davis and Wildey, Timothy Michael},
abstractNote = {Here, we analyze the convergence of probability density functions utilizing approximate models for both forward and inverse problems. We consider the standard forward uncertainty quantification problem where an assumed probability density on parameters is propagated through the approximate model to produce a probability density, often called a push-forward probability density, on a set of quantities of interest (QoI). The inverse problem considered in this paper seeks to update an initial probability density assumed on model input parameters such that the subsequent push-forward of this updated density through the parameter-to-QoI map matches a given probability density on the QoI. We prove that the densities obtained from solving the forward and inverse problems, using approximate models, converge to the true densities as the approximate models converge to the true models. Numerical results are presented to demonstrate convergence rates of densities for sparse grid approximations of parameter-to-QoI maps and standard spatial and temporal discretizations of PDEs and ODEs.},
doi = {10.1137/18M1181675},
journal = {SIAM Journal on Scientific Computing},
number = 5,
volume = 40,
place = {United States},
year = {Thu Oct 18 00:00:00 EDT 2018},
month = {Thu Oct 18 00:00:00 EDT 2018}
}
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Journal Article:
Free Publicly Available Full Text
Publisher's Version of Record
https://doi.org/10.1137/18M1181675
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Citation Metrics:
Cited by: 11 works
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Figures / Tables:

Figure 4.1 Figure 4.1: The level-4 sparse grid approximation of the $Q$o$I$ (left), the level-8 sparse grid approximation of the $Q$o$I$ (middle), and the reference solution (right).
All figures and tables (12 total)
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    Works referencing / citing this record:

    Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis
    journal, November 2019

    • Jakeman, John D.; Eldred, Michael S.; Geraci, Gianluca
    • International Journal for Numerical Methods in Engineering, Vol. 121, Issue 6
    • DOI: 10.1002/nme.6268

    Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis
    journal, August 2020

    • Jakeman, John D.; Eldred, Michael S.; Geraci, Gianluca
    • International Journal for Numerical Methods in Engineering, Vol. 121, Issue 19
    • DOI: 10.1002/nme.6450

    Adaptive multi-index collocation for uncertainty quantification and sensitivity analysis
    report, November 2019

    Adaptive sampling-based quadrature rules for efficient Bayesian prediction
    text, January 2019

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      Figures / Tables found in this record:

      Figure 4.1 (p. 16)figure
      Figure 4.2 (p. 16)figure
      Figure 4.3 (p. 17)figure
      Figure 4.4 (p. 18)figure
      Table 4.1 (p. 18)table
      Figure 4.5 (p. 19)figure
      Figure 4.6 (p. 21)figure
      Table 4.2 (p. 21)table
      Figure 4.7 (p. 23)figure
      Table 4.3 (p. 24)table
      Table 4.4 (p. 24)table
      Figure 4.8 (p. 25)figure
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