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Research Papers

Optimal Experimental Design Using a Consistent Bayesian Approach

[+] Author and Article Information
Scott N. Walsh

Department of Mathematical
and Statistical Sciences,
University of Colorado Denver,
Denver, CO 80204
e-mail: scott.walsh@ucdenver.edu

Tim M. Wildey

Sandia National Laboratories,
Center for Computing Research,
Albuquerque, NM 87123
e-mail: tmwilde@sandia.gov

John D. Jakeman

Sandia National Laboratories,
Center for Computing Research,
Albuquerque, NM 87123
e-mail: jdjakem@sandia.gov

Manuscript received October 4, 2016; final manuscript received June 26, 2017; published online September 7, 2017. Assoc. Editor: Yan Wang.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.

ASME J. Risk Uncertainty Part B 4(1), 011005 (Sep 07, 2017) (19 pages) Paper No: RISK-16-1132; doi: 10.1115/1.4037457 History: Received October 04, 2016; Revised June 26, 2017

We consider the utilization of a computational model to guide the optimal acquisition of experimental data to inform the stochastic description of model input parameters. Our formulation is based on the recently developed consistent Bayesian approach for solving stochastic inverse problems, which seeks a posterior probability density that is consistent with the model and the data in the sense that the push-forward of the posterior (through the computational model) matches the observed density on the observations almost everywhere. Given a set of potential observations, our optimal experimental design (OED) seeks the observation, or set of observations, that maximizes the expected information gain from the prior probability density on the model parameters. We discuss the characterization of the space of observed densities and a computationally efficient approach for rescaling observed densities to satisfy the fundamental assumptions of the consistent Bayesian approach. Numerical results are presented to compare our approach with existing OED methodologies using the classical/statistical Bayesian approach and to demonstrate our OED on a set of representative partial differential equations (PDE)-based models.

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Figures

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Fig. 1

Approximation of the posterior density obtained using the data Q1 (left), which gives IQ1(πDobs)≈2.015, a set of samples from the posterior (middle), and a comparison of the observed density on Q1 with the push-forward densities of the prior and the posterior (right)

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Fig. 2

Approximation of the posterior density obtained using Q2 (left), which gives IQ2(πDobs)≈0.466, a set of samples from the posterior (middle), and a comparison of the observed density on Q2 with the push-forward densities of the prior and the posterior (right)

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Fig. 3

The approximation of the push-forward of the prior (left), the exact observed density on (Q1, Q2) (middle), the approximation of the posterior density using both Q1 and Q2 (right) which gives IQ(πDobs)≈2.98

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Fig. 4

The push-forward of the prior for the map Q: Λ → (Q1, Q2) introduced in Sec. 3.2 (left), the observed density using the product of the one-dimensional Gaussian (middle), which extends beyond the range of the map, and the normalized observed density that does not extend beyond the range of the map (right)

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Fig. 5

The push-forward and observed densities on Q1 (left) and the push-forward and observed densities on Q2 (right). Notice the support of both of the observed densities is contained within the range of the model, i.e., the observed densities are absolutely continuous with respect to their corresponding push-forward densities.

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Fig. 6

The expected information gained over the design space (which is Ω in this example) approximated using 50, 200, 1000, and 5000 samples from the prior. Notice the higher values in the center of the domain and toward the top right (in the direction of the convection vector from the location of the source); this is consistent with our intuition. Moreover, notice the small changes in the design space as we increase the number of samples from 50 to 5000. This suggests that we compute accurate approximations to the design space using as few as 50 model evaluations.

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Fig. 7

The expected information gain over the design space (which is Ω in this example) approximated using 50, 200, 1000, and 5000 samples from the prior. Notice, the higher values in the corners and the general trend are consistent with Ref. [24].

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Fig. 8

Posteriors, approximated using 5000 samples, using the OED for three realizations of the location of the source. Notice the information gain changes substantially for each posterior, however, this experimental design, placement of the sensor in the bottom left corner, produces the maximum average information gain, E(IQ), over all possible locations of the source.

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Fig. 9

The computational domain and Poisson ratio showing the inclusion for a particular realization of the ellipsoid parameters

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Fig. 10

The set of samples from the posterior and the corresponding kernel density estimate of the posterior for the first sensor location, (3.5294, 1.3049)

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Fig. 11

The set of samples from the posterior and the corresponding kernel density estimate of the posterior for the second sensor location, (1.3902, 1.2100)

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Fig. 12

The expected information gain over the design space (which is Ω in this example) approximated using 10, 50, 100, and 1000 samples from the prior. Notice the higher values near the location of the inclusion.

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Fig. 13

The expected information gain over the design space (which is Ω in this example) approximated using 50, 100, 1000, and 10,000 samples from the prior. Notice the small changes in the design space as we increase the number of samples from 1000 to 10,000. This suggests that we compute accurate approximations to the design space using as few as 1000 model evaluations.

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Fig. 14

The expected information gain over the design space as a function of previously chosen sensor locations. Note that the range of the expected information gained changes in the progression of the figures. In the bottom right, we see the greedy optimal location of eight sensors within the physical domain.

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Fig. 15

In black, we show the possible locations of the sensors and in white, we show the optimal location(s) for 1, 2, 3, 4, and 5 sensors. In the bottom, we see the optimal location of five sensors within the physical domain. Note that the range of the data in each plot is indicative of the expected information gain for the first sensor location, not for the expected information gain for multiple sensors.

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