Conditioning by Projection for the Sampling from Prior Gaussian Distributions

TL;DR

This paper introduces a new Gaussian field conditioning method using projection for faster convergence in Bayesian inverse problems, demonstrated through porous media flow modeling.

Contribution

A novel projection-based conditioning technique for Gaussian fields that accelerates MCMC convergence in Bayesian inverse problems.

Findings

The method improves MCMC convergence speed.

Conditioning reduces uncertainty effectively.

Numerical experiments confirm faster convergence.

Abstract

In this work we are interested in the (ill-posed) inverse problem for absolute permeability characterization that arises in predictive modeling of porous media flows. We consider a Bayesian statistical framework with a preconditioned Markov Chain Monte Carlo (MCMC) algorithm for the solution of the inverse problem. Reduction of uncertainty can be accomplished by incorporating measurements at sparse locations (static data) in the prior distribution. We present a new method to condition Gaussian fields (the log of permeability fields) to available sparse measurements. A truncated Karhunen-Lo\`eve expansion (KLE) is used for dimension reduction. In the proposed method the imposition of static data is made through the projection of a sample (expressed as a vector of independent, identically distributed normal random variables) onto the nullspace of a data matrix, that is defined in terms of the KLE. The numerical implementation of the proposed method is straightforward. Through numerical experiments for a model of second-order elliptic equation, we show that the proposed method in multi-chain studies converges much faster than the MCMC method without conditioning. These studies indicate the importance of conditioning in accelerating the MCMC convergence.