A tutorial introduction to the Hamiltonian Monte Carlo solution of weakly nonlinear inverse problems

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Authors

Andreas Fichtner , Andrea Zunino, Lars Gebraad 

Abstract

We present an introduction to Hamiltonian Monte Carlo (HMC) sampling of high-dimensional model spaces with focus on linear and weakly nonlinear inverse problems. This includes the theoretical foundations, intuitive examples, as well as applications to linear and nonlinear, adjoint-based traveltime tomography.

HMC rests on the construction of an artificial Hamiltonian system where a model is treated as a high-dimensional particle moving along a trajectory in an extended model space. Using derivatives of the forward equations, HMC is able to make long-distance moves from the current towards a new model, thereby promoting sample independence while maintaining high acceptance rates. Though HMC is well-known and appreciated in many branches of the physical and computational sciences, it is only starting to be explored for the solution of geophysical inverse problems.

The first part of this article is therefore dedicated to an introduction to HMC using common geophysical terminology. Based on a brief review of equilibrated random walks, we provide a self-contained proof of the HMC algorithm. This leads us to consider simple examples with linear forward problems and Gaussian priors. Having analytical solutions, these problems are not the primary target of Monte Carlo methods. However, they allow us to gain intuition and to study critical tuning issues, the independence of successive samples, the tractable number of dimensions, and limitations of the algorithm.

The second part covers applications to linear traveltime tomography, reflector imaging, and nonlinear traveltime tomography. The latter rests on a fast-sweeping method combined with adjoint techniques for the computation of derivatives. Using HMC allows us to solve the tomographic problem fully probabilistically for several thousand unknowns, and without the requirement of supercomputing resources.

DOI

https://doi.org/10.31223/osf.io/3k9vu

Subjects

Earth Sciences, Geophysics and Seismology, Physical Sciences and Mathematics

Keywords

Seismology, Geophysics, Inverse theory, Bayesian inference, Tomography, Monte Carlo

Dates

Published: 2018-06-11 11:32

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License

Academic Free License (AFL) 3.0

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