title: Principal Component Geostatistical Approach for large-dimensional inverse problems.
abstract: The quasi-linear geostatistical approach is for weakly nonlinear underdetermined inverse problems, such as Hydraulic Tomography and Electrical Resistivity Tomography. It provides best estimates as well as measures for uncertainty quantification. However, for its textbook implementation, the approach involves iterations, to reach an optimum, and requires the determination of the Jacobian matrix, i.e., the derivative of the observation function with respect to the unknown. Although there are elegant methods for the determination of the Jacobian, the cost is high when the number of unknowns,m, and the number of observations,n, is high. It is also wasteful to compute the Jacobian for points away from the optimum. Irrespective of the issue of computing derivatives, the computational cost of implementing the method is generally of the order ofm2n, though there are methods to reduce the computational cost. In this work, we present an implementation that utilizes a matrix free in terms of the Jacobian matrix Gauss-Newton method and improves the scalability of the geostatistical inverse problem. For each iteration, it is required to performKruns of the forward problem, whereKis not just much smaller thanmbut can be smaller thatn. The computational and storage cost of implementation of the inverse procedure scales roughly linearly withminstead ofm2as in the textbook approach. For problems of very largem, this implementation constitutes a dramatic reduction in computational cost compared to the textbook approach. Results illustrate the validity of the approach and provide insight in the conditions under which this method perform best.
authors.affiliation: Civil and Environmental Engineering, Stanford University Stanford, California, USA.
authors.forename: P K
authors.initials: PK
authors.lastname: Kitanidis
authors.affiliation: Civil and Environmental Engineering, Stanford University Stanford, California, USA.
authors.forename: J
authors.initials: J
authors.lastname: Lee
keywords: computational; data analysis; geostatistics; inversion; stochastic