Model adaptivity for goal-oriented inference using adjoints

Abstract

An inverse problem seeks to infer unknown model parameters using observed data. We consider a goaloriented inverse problem, where the goal of inferring parameters is to use them in predicting a quantity of interest (QoI). Recognizing that multiple models of varying fidelity and computational cost may be available to describe the physical system, we formulate a goal-oriented model adaptivity approach that leverages multiple models while controlling the error in the QoI prediction. In particular, we adaptively form a mixedfidelity model by using models of different levels of fidelity in different subregions of the domain. Taking the solution of the inverse problem with the highest-fidelity model as our reference QoI prediction, we derive an adjoint-based third-order estimate for the QoI error from using a lower-fidelity model. Localization of this error then guides the formation of mixed-fidelity models. We demonstrate the method for example problems described by convection-diffusion-reaction models. For these examples, our mixed-fidelity models use the high-fidelity model over only a small portion of the domain, but result in QoI estimates with small relative errors. We also demonstrate that the mixed-fidelity inverse problems can be cheaper to solve and less sensitive to the initial guess than the high-fidelity inverse problems.

Data-driven forward discretizations for Bayesian inversion

“…For instance, our forward model discretizations may be used within multilevel MCMC methods [18] or within two-stage sampling methods [20,41,6,10,13], and thus help to reduce the sampling cost. Also, forward model discretizations may be combined with parameter reduction and model adaptation techniques, as in [28,26]. It is important, however, to distinguish between the parameter and model reduction problems.…”

Data-Driven Forward Discretizations for Bayesian Inversion

“…For instance, our forward model discretizations may be used within multilevel MCMC methods [18] or within two-stage sampling methods [20,40,6,10,13], and thus help to reduce the sampling cost. Also, forward model discretizations may be combined with parameter reduction and model adaptation techniques, as in [28,26]. It is important, however, to distinguish between the parameter and model reduction problems.…”

Goal-oriented adaptive modeling of random heterogeneous media and model-based multilevel Monte Carlo methods

“…This is not the case for a sequence of surrogate models, where the unavailability of convergence rates (at least a priori) constitutes the main challenge in the construction of an efficient multilevel estimator. The use of sets of low-fidelity models correlated with a high-fidelity base model as a tool for deriving MC solvers for large-scale simulations has led to the development of Multi-Fidelity Monte Carlo Methods (MFMC) [29,30,18]. In these methods, the selection of surrogate models and the optimal distribution of samples among the levels are tied to the correlation between the approximate QoIs of each surrogate and the QoI of the high-fidelity model, and by the cost of evaluating the QoI for each surrogate model.…”