Project 4

An efficient Bayesian inference approach to inverse problems based on an adaptive sparse grid collocation method

In this project, a new approach for modeling inverse problems using a Bayesian inference method is introduced. The Bayesian approach considers the unknown parameters as random variables and seeks the probabilistic distribution of the unknowns. By introducing the concept of the stochastic prior state space to the Bayesian formulation, we reformulate the deterministic forward problem as a stochastic one. The adaptive hierarchical sparse gridcollocation method (ASGC) developed in Project 2 is used for constructing an interpolant of the solution to this stochastic forward model in this prior space which is large enough to capture all the variability/uncertainty in the posterior distribution of the unknown parameters. This solution can be considered as a stochastic function of the random unknowns in the stochastic space. This function serves as a stochastic surrogate model for the likelihood calculation. Hierarchical Bayesian formulation is used to derive the posterior probability density function (PPDF). The spatial model is represented as a convolution of a smooth kernel and Markov random field to achieve dimension reduction for facilitating the inference algorithm. Then the state space of the PPDF is explored using Markov chain Monte Carlo (MCMC) algorithms in order to obtain statistics of the unknowns. Instead of repeated evaluations of the deterministic forward model, the likelihood calculation is reduced by directly sampling the approximate stochastic solution obtained through the ASGC method. The technique is assessed on two non-linear inverse problems: source inversion and permeability estimation in flow through porous media.

In the first example, we consider a simple heat source inversion problem on unit square with adiabatic boundaries:

The source location m = (m0, m1) is unknown.

Table 1. The posterior results with four source locations given by ASGC

From Table 1, it can be seen that no matter where the source location is, the method can always infer the exact value without performing any additional direct FEM computation as in the traditional MCMC method. In addition, we only need to solve the forward model once and reuse the same surrogate model. The computational time required by the ASGC method is only a small fraction of that by direct FEM analysis. In ASGC, most of the time is spent on constructing the surrogate model. It is noted that this process is embarrassingly parallel which only needs 84.18 second on 20 nodes on our in-house Linux cluster and it took only 26.9 second for a single MCMC chain with a length of 30,000 on a single processor. On the other hand, the FEM takes nearly 34 hours on one processor to infer one source location since we have to solve the deterministic forward problem for each proposal move sequentially.

The following figure showcase the result from the permeability estimation for flow through porous media of a classical quarter-five spot problem.

Fig 1. The true (left) and posterior mean (right) permeability estimation with 5% noisy in the data

Fig 2. The marginal probability density function of the two hyper parameters

The paper and PowerPoint presentation can be downloaded from here.