TY - JOUR

T1 - Approximate Bayesian model inversion for PDEs with heterogeneous and state-dependent coefficients

AU - Barajas-Solano, D. A.

AU - Tartakovsky, A. M.

N1 - Funding Information:
This work was supported by the Applied Mathematics Program within the U.S. Department of Energy Office of Advanced Scientific Computing Research under Contract DE-AC02-06CH11347. Pacific Northwest National Laboratory is operated by Battelle for the DOE under Contract DE-AC05-76RL01830.
Funding Information:
This work was supported by the Applied Mathematics Program within the U.S. Department of Energy Office of Advanced Scientific Computing Research under Contract DE-AC02-06CH11347 . Pacific Northwest National Laboratory is operated by Battelle for the DOE under Contract DE-AC05-76RL01830.
Publisher Copyright:
© 2019 Elsevier Inc.

PY - 2019/10/15

Y1 - 2019/10/15

N2 - We present two approximate Bayesian inference methods for parameter estimation in partial differential equation (PDE) models with space-dependent and state-dependent parameters. We demonstrate that these methods provide accurate and cost-effective 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, Laplace-EM, employs the expectation maximization algorithm to maximize the ELBO, with a Laplace approximation of the posterior on the E-step, and minimization of a Kullback-Leibler divergence on the M-step. The second method, DSVI-EB, employs the doubly stochastic variational inference (DSVI) algorithm, in which the ELBO is maximized via gradient-based stochastic optimization, with noisy 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 Laplace-EM method is more accurate, it requires computing Hessians of the physics model. The DSVI-EB method is found to be less accurate but only requires gradients of the physics model.

AB - We present two approximate Bayesian inference methods for parameter estimation in partial differential equation (PDE) models with space-dependent and state-dependent parameters. We demonstrate that these methods provide accurate and cost-effective 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, Laplace-EM, employs the expectation maximization algorithm to maximize the ELBO, with a Laplace approximation of the posterior on the E-step, and minimization of a Kullback-Leibler divergence on the M-step. The second method, DSVI-EB, employs the doubly stochastic variational inference (DSVI) algorithm, in which the ELBO is maximized via gradient-based stochastic optimization, with noisy 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 Laplace-EM method is more accurate, it requires computing Hessians of the physics model. The DSVI-EB method is found to be less accurate but only requires gradients of the physics model.

KW - Approximate Bayesian inference

KW - Empirical Bayes

KW - Model inversion

KW - Variational inference

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U2 - 10.1016/j.jcp.2019.06.010

DO - 10.1016/j.jcp.2019.06.010

M3 - Article

AN - SCOPUS:85067473108

VL - 395

SP - 247

EP - 262

JO - Journal of Computational Physics

JF - Journal of Computational Physics

SN - 0021-9991

ER -