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Hybrid Error Estimation Methods and Dimension Reduction for Large-scale Bayesian Inverse Problems
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Abstract
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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. For large-scale problems, the maximum likelihood of the posterior

distribution can be estimated using techniques such as least-squares minimization (e.g., 4D-Var), or ensemble approaches (e.g., EnKF). Both

methods rely, either explicitly (ensemble) or implicitly (Hessian estimate in minimization subspace) on low-rank approximation of the

second-order moment statistics. In this presentation, deterministic and stochastic approaches to estimate the posterior error covariance

matrix for large-scale inverse problems will be presented and discussed. First, in the context of 4D-Var, the standard Monte-Carlo

method will be compared with Hessian-based estimates from a randomization approach and from the deterministic BFGS algorithm. New

preconditioning techniques for the BFGS method that significantly improve the posterior error estimates will be presented and

illustrated with numerical examples. Secondly, posterior error approximations will be discussed in the broader context of dimension

reduction for large-scale inverse problems. The formalism of dimension reduction for Bayesian problems will be introduced and different

optimality criteria presented. An optimal method that maximizes the degree of freedom for signal of the inversion will be proposed. Its

numerical efficiency will be demonstrated with an Observation Simulation System Experiment consisting of a methane source inversion

over North America using space-based observations from GOSAT and the GEOS-Chem chemistry-transport model. To conclude, interesting

theoretical links between this optimal posterior covariance low-rank approximation and existing variational and ensemble data assimilation

system will be discussed.