Toward Optimal and Scalable Dimension Reduction Methods for large-scale Bayesian Inversions

Abstract:

Many inverse problems in geophysics are solved within the Bayesian framework, in which a prior probability density function of a quantity of interest is optimally updated using newly available observations. A maximum likelihood of the posterior probability density function is estimated using a model of the physics that relates the variables to be optimized to the observations. However, in many practical situations the number of observations is much smaller than the number of variables estimated, which leads to an ill-posed problem. In practice, this means that the data are informative only in a subspace of the initial space. It is both of theoretical and practical interest to characterize this "data-informed" subspace, since it allows a simple interpretation of the inverse solution and its uncertainty, but can also dramatically reduce the computational cost of the optimization by reducing the size of the problem. In this presentation the formalism of dimension reduction in Bayesian methods will be introduced, and different optimality criteria will be discussed (e.g., minimum error variances, maximum degree of freedom for signal). For each criterion, an optimal design for the reduced Bayesian problem will be proposed and compared with other suboptimal approaches. A significant advantage of our method is its high scalability owing to an efficient parallel implementation, making it very attractive for large-scale inverse problems. Numerical results from an Observation Simulation System Experiment (OSSE) consisting of a high spatial resolution (0.5°x0.7°) source inversion of methane over North America using observations from the Greenhouse gases Observing SATellite (GOSAT) instrument and the GEOS-Chem chemistry-transport model will illustrate the computational efficiency of our approach. Although only linear models are considered in this study, possible extensions to the non-linear case will also be discussed