Efficient methods for large‐scale linear inversion using a geostatistical approach

Abstract

In geophysical inverse problems, such as estimating the unknown parameter field from noisy observations of dependent quantities, e.g., hydraulic conductivity from head observations, stochastic Bayesian and geostatistical approaches are frequently used. To obtain best estimates and conditional realizations it is required to perform several matrix‐matrix computations involving the covariance matrix of the discretized field of the parameters. In realistic three‐dimensional fields that are finely discretized, these operations as performed in conventional algorithms become extremely expensive and even prohibitive in terms of memory and computational requirements. Using Hierarchical Matrices, we show how to reduce the complexity of forming approximate matrix‐vector products involving the Covariance matrices in log linear complexity for an arbitrary distribution of points and a wide variety of generalized covariance functions. The resulting system of equations is solved iteratively using a matrix‐free Krylov subspace approach. Furthermore, we show how to generate unconditional realizations using an approximation to the square root of the covariance matrix using Chebyshev matrix polynomials and use the above to generate conditional realizations. We demonstrate the efficiency of our method on a few standard test problems, such as interpolation from noisy observations and contaminant source identification.