Parallelized Adaptive Importance Sampling for Solving Inverse Problems

Abstract

In the field of groundwater hydrology and more generally geophysics, solving inverse problems in a complex, geologically realistic, and discrete model space often requires the usage of Monte Carlo methods. In a previous paper we introduced PoPEx, a sampling strategy, able to handle such constraints efficiently. Unfortunately, the predictions suffered from a slight bias. In the present work, we propose a series of major modifications of PoPEx. The computational cost of the algorithm is reduced and the underlying uncertainty quantification is improved. Advanced machine learning techniques are combined with an adaptive importance sampling strategy to define a highly efficient and ergodic method that produces unbiased and rapidly convergent predictions. The proposed algorithm may be used for solving a broad range of inverse problems in many different fields. It only requires to obtain a forward problem solver, an inverse problem description and a conditional simulation tool that samples from the prior distribution. Furthermore, its parallel implementation scales perfectly. This means that the required computational time can be decreased almost arbitrarily, such that it is only limited by the available computing resources. The performance of the method is demonstrated using the inversion of a synthetic tracer test problem in an alluvial aquifer. The prior geological knowledge is modeled using multiple-point statistics. The problem consists of the identification of 2 · 104 parameters corresponding to 4 geological facies values. It is used to show empirically the convergence of the PoPEx method.