Bayesian and variational Bayesian approaches for flows in heterogeneous random media

Abstract

In this paper, we study porous media flows in heterogeneous stochastic media. We propose an efficient forward simulation technique that is tailored for variational Bayesian inversion. As a starting point, the proposed forward simulation technique decomposes the solution into the sum of separable functions (with respect to randomness and the space), where each term is calculated based on a variational approach. This is similar to Proper Generalized Decomposition (PGD). Next, we apply a multiscale technique to solve for each term (as in [1]) and, further, decompose the random function into 1D fields. As a result, our proposed method provides an approximation hierarchy for the solution as we increase the number of terms in the expansion and, also, increase the spatial resolution of each term. We use the hierarchical solution distributions in a variational Bayesian approximation to perform uncertainty quantification in the inverse problem. We conduct a detailed numerical study to explore the performance of the proposed uncertainty quantification technique and show the theoretical posterior concentration.

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

“…where again y = µ q + R q z. The details of the derivations of (20)- (22) are presented in Appendix A. It can be verified that E N (z|0,…”

“…In this section we present the derivation of the gradients (20)- (22). For the gradient with respect to the variational mean, (20), we have by the chain rule, in index notation, ∂ ∂µ q,i log p(D s | y) = ∂y j ∂µ q,i ∂ ∂y j log p(D s | y) = δ ji ∂ ∂y j log p(D s | y),…”

“…By the chain rule, we have Finally, we have ∇ Rq log det R q = (R −1 q ) , from which we recover (21). The gradients with respect to the prior hyperparameters, (22) can be derived from [9], Eqs. (A.14) and (A.15).…”

“…Furthermore, both methods are applicable to non-factorizing likelihoods, do not require computing moments of the likelihood, and do not require third-or higher order derivatives of the physics model. Finally, variational inference and the Laplace approximation have been employed for model inversion [19,20,21,22,13], but to the best of our knowledge, have not been used in the context of the empirical Bayesian framework to estimate GP prior hyperparameters, with the exception of the work of [13]. That work, based on the Laplace approximation, requires computing third-order derivatives of the physics model, which may be costly to compute, whereas the presented methods do not require third-order derivatives.…”

Bayesian Uncertainty Quantification for Channelized Reservoirs via Reduced Dimensional Parameterization

“…This provides a better understanding of their direct and indirect effects on the response of the physical system. Such Bayesian models have been used for uncertainty quantification in subsurface models [13][14][15][16][17], and also in other areas such as seismic modeling [18,19]. Here we use a Bayesian hierarchical model that preserves the facies architecture, at the same time populating the petrophysical properties within the facies in a geologically consistent manner by incorporating available static and dynamic information.…”