Fast Stokes Flow Simulations for Geophysical‐Geodynamic  Inverse Problems and Sensitivity Analyses Based On Reduced Order Modeling

Abstract

Markov chain Monte Carlo (MCMC) methods have become standard in Bayesian inference and multi‐observable inversions in almost every discipline of the Earth sciences. In the case of geodynamic and/or coupled geophysical‐geodynamic inverse problems, however, the computational cost associated with the solution of large‐scale 3‐D Stokes forward problems has rendered probabilistic formulations impractical. Here we present a novel and extremely efficient method to produce ultrafast solutions of the 3‐D Stokes problem for MCMC simulations. Our approach combines the individual benefits of Reduced Basis techniques, goal‐oriented error formulations, and MCMC algorithms to produce an accurate and computationally efficient surrogate for the forward problem. Importantly, the surrogate adapts itself during the MCMC simulation according to the history of the chain and the goals of the inversion. This maximizes the efficiency of the forward problem and removes the need for preinversion off‐line computations to build a surrogate. We demonstrate the benefits and limitations of the method with several numerical examples and show that in all cases the computational cost is of the order of <1% compared to a traditional MCMC approach. The method is general enough to be applied to a range of problems, including uncertainty quantification/propagation, adjoint‐based geodynamic inversions, sensitivity analyses in mantle convection problems, and in the creating surrogate models for complex forward problems (e.g., heat transfer, seismic tomography, and magnetotellurics).

A reduced order approach for probabilistic inversions of 3-D magnetotelluric data I: general formulation

“…Previous work has shown that RB can speedup the numerical solutions of complex problems by several orders of magnitude without compromising the accuracy of the solutions (e.g. Ortega-Gelabert et al, 2020;Lieberman et al, 2010;Chen et al, 2010;Rozza et al, 2007Rozza et al, , 2013Florentin & Díez, 2012;Cui et al, 2015). In this paper, we will show that staggering gains in computational time can also be achieved for the 3D MT problem.…”

“…Another way of further achieving a small basis size is to seek for accurate solutions only within specific regions of the numerical domain (e.g. Ortega-Gelabert et al, 2020;Alvarez Aramberri, 2015). This stems from the fact that in many practical applications we are not interested in high accuracy at every point inside the numerical box, but rather within a restricted region.…”

“…Our hybrid approach is similar to that put forward by Ortega-Gelabert et al (2020) in that it automatically updates/enriches the RB space V RB during the MCMC inversion by adding high-fidelity solutions (i.e. new bases) as needed by the evolution of the chain.…”

“…While other choices can be implemented for this purpose, they usually require additional high-fidelity computations at the current state of the chain m t−1 (e.g. Yan & Zhou, 2019;Ortega-Gelabert et al, 2020). Also, although the delayed rejection step is not strictly necessary from a MCMC algorithmic point of view, it is rather important in practice.…”

“…At the same time, it is imperative that the number of elements in the basis is kept as small as possible without compromising the quality of their predictions (cf. Ortega-Gelabert et al, 2020). In order to achieve this, we have implemented the use of variable tolerances and Singular Value Decomposition (SVD) of the basis as additional functionalities.…”