PNNL

Abstract

We present a new approach for constructing a data-driven surrogate model and using it for Bayesian parameter estimation in partial differential equation (PDE) models. We use parameter observations to condition the Karhunen-Lo\'{e}ve expansion of the space-dependent parameter and construct the conditional generalized Polynomial Chaos (gPC) surrogate model of the PDE state. Next, the state observations are used to estimate the parameter by computing coefficients in the KL expansion using Markov chain Monte Carlo (MCMC) method. Our approach addresses two major challenges in the Bayesian parameter estimation. First, it reduces dimensionality of the parameter space and replaces expensive direct solutions of PDEs with the conditional gPC surrogates. Second, in the absence of measurement error, the estimated parameter field exactly matches the parameter measurements. In addition, we show that the conditional gPC surrogate can be used to estimate the variance of state variables, which in turn can be used to guide the data acquisition. We demonstrate that our approach simplifies the parameter estimation problem and improves its accuracy with application to one- and two-dimensional diffusion equations with (unknown) space-dependent diffusion coefficients. We also discuss the effect of measurement locations on the accuracy of parameter estimation.