Approximate Bayesian Model Inversion for PDEs with Heterogeneous and StateDependent Coefficients
Abstract
We present two approximate Bayesian inference methods for parameter estimation in partial differential equation (PDE) models with spacedependent and statedependent parameters. We demonstrate that these methods provide accurate and costeffective alternatives to Markov Chain Monte Carlo simulation. We assume a parameterized Gaussian prior on the unknown functions, and approximate the posterior density by a parameterized multivariate Gaussian density. The parameters of the prior and posterior are estimated from sparse observations of the PDE model's states and the unknown functions themselves by maximizing the evidence lower bound (ELBO), a lower bound on the log marginal likelihood of the observations. The first method, LaplaceEM, employs the expectation maximization algorithm to maximize the ELBO, with a Laplace approximation of the posterior on the Estep, and minimization of a KullbackLeibler divergence on the Mstep. The second method, DSVIEB, employs the doubly stochastic variational inference (DSVI) algorithm, in which the ELBO is maximized via gradientbased stochastic optimization, with nosiy gradients computed via simple Monte Carlo sampling and Gaussian backpropagation. We apply these methods to identifying diffusion coefficients in linear and nonlinear diffusion equations, and we find that both methods provide accurate estimates of posterior densities and the hyperparameters of Gaussian priors. While the LaplaceEM methodmore »
 Authors:

 BATTELLE (PACIFIC NW LAB)
 Publication Date:
 Research Org.:
 Pacific Northwest National Lab. (PNNL), Richland, WA (United States)
 Sponsoring Org.:
 USDOE
 OSTI Identifier:
 1572471
 Report Number(s):
 PNNLSA141387
 DOE Contract Number:
 AC0576RL01830
 Resource Type:
 Journal Article
 Journal Name:
 Journal of Computational Physics
 Additional Journal Information:
 Journal Volume: 395
 Country of Publication:
 United States
 Language:
 English
 Subject:
 approximate Bayesian inference, model inversion, variational inference, empirical Bayes
Citation Formats
BarajasSolano, David A., and Tartakovsky, Alexandre M. Approximate Bayesian Model Inversion for PDEs with Heterogeneous and StateDependent Coefficients. United States: N. p., 2019.
Web. doi:10.1016/j.jcp.2019.06.010.
BarajasSolano, David A., & Tartakovsky, Alexandre M. Approximate Bayesian Model Inversion for PDEs with Heterogeneous and StateDependent Coefficients. United States. doi:10.1016/j.jcp.2019.06.010.
BarajasSolano, David A., and Tartakovsky, Alexandre M. Tue .
"Approximate Bayesian Model Inversion for PDEs with Heterogeneous and StateDependent Coefficients". United States. doi:10.1016/j.jcp.2019.06.010.
@article{osti_1572471,
title = {Approximate Bayesian Model Inversion for PDEs with Heterogeneous and StateDependent Coefficients},
author = {BarajasSolano, David A. and Tartakovsky, Alexandre M.},
abstractNote = {We present two approximate Bayesian inference methods for parameter estimation in partial differential equation (PDE) models with spacedependent and statedependent parameters. We demonstrate that these methods provide accurate and costeffective alternatives to Markov Chain Monte Carlo simulation. We assume a parameterized Gaussian prior on the unknown functions, and approximate the posterior density by a parameterized multivariate Gaussian density. The parameters of the prior and posterior are estimated from sparse observations of the PDE model's states and the unknown functions themselves by maximizing the evidence lower bound (ELBO), a lower bound on the log marginal likelihood of the observations. The first method, LaplaceEM, employs the expectation maximization algorithm to maximize the ELBO, with a Laplace approximation of the posterior on the Estep, and minimization of a KullbackLeibler divergence on the Mstep. The second method, DSVIEB, employs the doubly stochastic variational inference (DSVI) algorithm, in which the ELBO is maximized via gradientbased stochastic optimization, with nosiy gradients computed via simple Monte Carlo sampling and Gaussian backpropagation. We apply these methods to identifying diffusion coefficients in linear and nonlinear diffusion equations, and we find that both methods provide accurate estimates of posterior densities and the hyperparameters of Gaussian priors. While the LaplaceEM method is more accurate, it requires computing Hessians of the physics model. The DSVIEB method is found to be less accurate but only requires gradients of the physics model.},
doi = {10.1016/j.jcp.2019.06.010},
journal = {Journal of Computational Physics},
number = ,
volume = 395,
place = {United States},
year = {2019},
month = {10}
}