Uncertainty quantification in geostatistical approach to inverse problems

We consider the computational challenges associated with uncertainty quantification in high-dimensional parameter estimation using geostatis-tical approach inverse problems. Traditional approaches to uncertainty quantification involve sampling from the high-dimensional posterior prob-ability density function. However, due to the high computational costs associated with sampling, we consider an efficient representation of the posterior covariance matrix at the maximum a posteriori (MAP) point as the sum of the prior covariance matrix and a low-rank update that contains information from the dominant generalized eigenmodes of the data misfit part of the Hessian and the inverse covariance matrix. The rank of the low-rank update is typically independent of the dimension of the unknown parameter. We provide an efficient algorithm for computing the dominant eigenmodes of the generalized eigenvalue problem (and as a result, the low-rank decomposition) that avoids forming square-roots of the covariance matrix or its inverse. As a result, we have a method that scales almost linearly with the dimension of unknown parameter space and the data dimension. Further, we show how to efficiently compute some measures of uncertainty that are based on scalar invariants of the poste-rior covariance matrix. The resulting uncertainty measures can be used in the context of optimal experimental design. The performance of our algorithms is demonstrated by application to model problems in synthetic travel-time tomography and steady-state hydraulic tomography.