Fast computation of uncertainty quantification measures in the geostatistical approach to solve inverse problems

摘要

We consider the computational challenges associated with uncertaintyquantification involved in parameter estimation such as seismic slowness andhydraulic transmissivity fields. The reconstruction of these parameters can bemathematically described as Inverse Problems which we tackle using theGeostatistical approach. The quantification of uncertainty in theGeostatistical approach involves computing the posterior covariance matrixwhich is prohibitively expensive to fully compute and store. We consider anefficient representation of the posterior covariance matrix at the maximum aposteriori (MAP) point as the sum of the prior covariance matrix and a low-rankupdate that contains information from the dominant generalized eigenmodes ofthe data misfit part of the Hessian and the inverse covariance matrix. The rankof the low-rank update is typically independent of the dimension of the unknownparameter. The cost of our method scales as $\bigO(m\log m)$ where $m $dimension of unknown parameter vector space. Furthermore, we show how toefficiently compute measures of uncertainty that are based on scalar functionsof the posterior covariance matrix. The performance of our algorithms isdemonstrated by application to model problems in synthetic travel-timetomography and steady-state hydraulic tomography. We explore the accuracy ofthe posterior covariance on different experimental parameters and show that thecost of approximating the posterior covariance matrix depends on the problemsize and is not sensitive to other experimental parameters.