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Two major bottlenecks to the solution of large-scale Bayesian inverseproblems are the scaling of posterior sampling algorithms to high-dimensionalparameter spaces and the computational cost of forward model evaluations. Yetincomplete or noisy data, the state variation and parameter dependence of theforward model, and correlations in the prior collectively provide usefulstructure that can be exploited for dimension reduction in this setting--bothin the parameter space of the inverse problem and in the state space of theforward model. To this end, we show how to jointly construct low-dimensionalsubspaces of the parameter space and the state space in order to accelerate theBayesian solution of the inverse problem. As a byproduct of state dimensionreduction, we also show how to identify low-dimensional subspaces of the datain problems with high-dimensional observations. These subspaces enableapproximation of the posterior as a product of two factors: (i) a projection ofthe posterior onto a low-dimensional parameter subspace, wherein the originallikelihood is replaced by an approximation involving a reduced model; and (ii)the marginal prior distribution on the high-dimensional complement of theparameter subspace. We present and compare several strategies for constructingthese subspaces using only a limited number of forward and adjoint modelsimulations. The resulting posterior approximations can rapidly becharacterized using standard sampling techniques, e.g., Markov chain MonteCarlo. Two numerical examples demonstrate the accuracy and efficiency of ourapproach: inversion of an integral equation in atmospheric remote sensing,where the data dimension is very high; and the inference of a heterogeneoustransmissivity field in a groundwater system, which involves a partialdifferential equation forward model with high dimensional state and parameters.

Scalable posterior approximations for large-scale Bayesian inverse problems via likelihood-informed parameter and state reduction