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Research Papers

Estimating Physics Models and Quantifying Their Uncertainty Using Optimization With a Bayesian Objective Function

[+] Author and Article Information
Stephen A. Andrews

Verification and Analysis (XCP-8),
Los Alamos National Laboratory,
Mail Stop P365,
P.O. Box 1663,
Los Alamos, NM 87545
e-mail: saandrews@lanl.gov

Andrew M. Fraser

Verification and Analysis (XCP-8),
Los Alamos National Laboratory,
Mail Stop F644,
P.O. Box 1663,
Los Alamos, NM 87545
e-mail: afraser@lanl.gov

1Corresponding author.

Manuscript received July 30, 2018; final manuscript received May 3, 2019; published online June 18, 2019. Assoc. Editor: Tao Xing.The United States Government retains, and by accepting the article for publication, the publisher acknowledges that the United States Government retains, a nonexclusive, paid-up, irrevocable, worldwide license to publish or reproduce the published form of this work, or allow others to do so, for United States government purposes.

J. Verif. Valid. Uncert 4(1), 011002 (Jun 18, 2019) (15 pages) Paper No: VVUQ-18-1024; doi: 10.1115/1.4043807 History: Received July 30, 2018; Revised May 03, 2019

This paper reports a verification study for a method that fits functions to sets of data from several experiments simultaneously. The method finds a maximum a posteriori probability estimate of a function subject to constraints (e.g., convexity in the study), uncertainty about the estimate, and a quantitative characterization of how data from each experiment constrains that uncertainty. While this work focuses on a model of the equation of state (EOS) of gasses produced by detonating a high explosive, the method can be applied to a wide range of physics processes with either parametric or semiparametric models. As a verification exercise, a reference EOS is used and artificial experimental data sets are created using numerical integration of ordinary differential equations and pseudo-random noise. The method yields an estimate of the EOS that is close to the reference and identifies how each experiment most constrains the result.

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Figures

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Fig. 1

An example set of twelve basis functions equi-log spaced between 0.1 g cm−3 and 1.0 g cm−3. Knot locations are shown as filled points.

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Fig. 2

Isentrope in pρ coordinates with ±5% uncertainty bounds

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Fig. 3

Transformed isentrope in fg coordinates with ±5% uncertainty bounds

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Fig. 4

Forces acting on a differential section of the cylinder

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Fig. 5

Forces acting on a differential section of the sandwich

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Fig. 6

The pressure on both prior and optimal isentrope functions normalized by the pressure from the true model of the isentrope

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Fig. 7

Comparison of the prior and optimal values for the isentrope and all three experiments: (a) the isentrope, (b) cylinder, (c) sandwich, and (d) stick

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Fig. 8

Analysis of the Fisher information for two different experiments: (a) cylinder eigenvalues, (b) cylinder eigenfunctions, (c) stick eigenvalues, and (d) stick eigenfunction

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Fig. 9

Analysis of the prior and posterior covariance and its effect on the cylinder experiment: (a) 100 prior samples, (b) 100 posterior samples, (c) cylinder simulation using ten prior samples, and (d) cylinder simulation using ten posterior samples

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Fig. 10

Univariate and bivariate marginals of the posterior distribution in the three directions of greatest variance from the unconstrained Laplace approximation of the posterior. Legend: MCMC (red), constrained Laplace approximation (blue), and unconstrained Laplace approximation (black line).

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Fig. 11

The cumulative probability distribution of the normalized residuals from the MCMC chain and samples from the constrained Laplace approximation of the posterior. A zero mean, unit variance normal distribution is shown for comparison.

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