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Front. Earth Sci. | doi: 10.3389/feart.2018.00203

Parallelized Adaptive Importance Sampling for Solving Inverse Problems

  • 1Laboratoire d'Hydrogéologie Stochastique et Groupe de Géostatistique, Université de Neuchâtel, Switzerland

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 $2 \cdot 10^4$ parameters corresponding to $4$ geological facies values. It is used to show empirically the convergence of the PoPEx method.

Keywords: Adaptive importance sampling, machine learning, uncertainty quantification, Bayesian inversion, monte carlo, Muliple-Point Statistics, parallelization

Received: 23 Jul 2018; Accepted: 26 Oct 2018.

Edited by:

Yuri Fialko, University of California, San Diego, United States

Reviewed by:

Shuai Zhang, Lawrence Livermore National Laboratory, United States Department of Energy (DOE), United States
Thomas Romary, ParisTech École Nationale Supérieure des Mines de Paris, Université de Sciences Lettres de Paris, France
Sarah Minson, Earthquake Science Center, United States Geological Survey, United States  

Copyright: © 2018 Jäggli, Straubhaar and Renard. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Mr. Christoph Jäggli, Laboratoire d'Hydrogéologie Stochastique et Groupe de Géostatistique, Université de Neuchâtel, Neuchâtel, Neuchâtel, Switzerland, christoph.jaeggli@unine.ch