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

Journal Article · · SIAM Journal on Scientific Computing
DOI:https://doi.org/10.1137/18M1181675· OSTI ID:1479491
 [1];  [2];  [2]
  1. Univ. of Colorado, Boulder, CO (United States)
  2. Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
+ Show Author Affiliations

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.

Research Organization:
Sandia National Lab. (SNL-NM), Albuquerque, NM (United States)
Sponsoring Organization:
USDOE National Nuclear Security Administration (NNSA); National Science Foundation (NSF)
Grant/Contract Number:
AC04-94AL85000; DMS-1818941
OSTI ID:
1479491
Alternate ID(s):
OSTI ID: 1595429
Report Number(s):
SAND-2018-6887J; SAND-2020-0021J; 664970
Journal Information:
SIAM Journal on Scientific Computing, Vol. 40, Issue 5; ISSN 1064-8275
Publisher:
SIAMCopyright Statement
Country of Publication:
United States
Language:
English
Citation Metrics:
Cited by: 11 works
Citation information provided by
Web of Science

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Cited By (6)

Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis
  • Jakeman, John D.; Eldred, Michael S.; Geraci, Gianluca
  • International Journal for Numerical Methods in Engineering, Vol. 121, Issue 6 https://doi.org/10.1002/nme.6268
journal November 2019
Adaptive multi‐index collocation for uncertainty quantification and sensitivity analysis
  • Jakeman, John D.; Eldred, Michael S.; Geraci, Gianluca
  • International Journal for Numerical Methods in Engineering, Vol. 121, Issue 19 https://doi.org/10.1002/nme.6450
journal August 2020
Adaptive multi-index collocation for uncertainty quantification and sensitivity analysis report November 2019
Solving Stochastic Inverse Problems for Property–Structure Linkages Using Data-Consistent Inversion and Machine Learning journal November 2020
Adaptive sampling-based quadrature rules for efficient Bayesian prediction text January 2019
Convergence of Probability Densities using Approximate Models for Forward and Inverse Problems in Uncertainty Quantification: Extensions to $L^p$ preprint January 2020

Figures / Tables (12)

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

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Related Subjects

97 MATHEMATICS AND COMPUTING
inverse problems
uncertainty quantification
density estimation
surrogate modeling
response surface approximations
discretization errors
LP convergence
approximate modeling