LAUSR

17 Markov Chain Monte Carlo (MCMC) methods have become standard in Bayesian in18 ference and multi-observable inversions in almost every discipline of the Earth sciences. 19 In the case of geodynamic and/or coupled geophysical-geodynamic inverse problems, 20 however, the computational cost associated with the solution of large-scale 3D Stokes 21 forward problems has rendered probabilistic formulations impractical. 22 Here we present a novel and extremely efficient method to produce ultra-fast 23 solutions of the 3D Stokes problem for MCMC simulations. Our approach combines 24 the individual benefits of Reduced Basis techniques, goal-oriented error formulations 25 and MCMC algorithms to produce an accurate and computationally efficient surrogate 26 for the forward problem. Importantly, the surrogate adapts itself during the MCMC 27 simulation according to the history of the chain and the goals of the inversion. This 28 maximizes the efficiency of the forward problem and removes the need for pre-inversion 29 offline computations to build a surrogate. We demonstrate the benefits and limitations 30 of the method with several numerical examples and show that in all cases the compu31 tational cost is of the order of < 1% compared to a traditional MCMC approach. The 32 method is general enough to be applied to a range of problems, including uncertainty 33 quantification/propagation, adjoint-based geodynamic inversions, sensitivity analyses 34 in mantle convection problems, as well as in the creating surrogate models for complex 35 forward problems (e.g. heat transfer, seismic tomography, Magnetotellurics). 36