Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics

Application of Hierarchical Matrices to Linear Inverse Problems in Geostatistics - Characterizing the uncertainty in the subsurface is an important step for exploration and extraction of natural resources, the storage of nuclear material and gasses such as natural gas or CO2. Imaging the subsurface can be posed as an inverse problem and can be solved using the geostatistical approach [Kitanidis P.K (2007) Geophys. Monogr: Ser. 171, 19-30, doi:10.1029/171GM04; Kitanidis (2011) doi: 10.1002/9780470685853. ch4, pp. 71-85] which is one of the many prevalent approaches. We briefly describe the geostatistical approach in the context of linear inverse problems and discuss some of the challenges in the large-scale implementation of this approach. Using the hierarchical matrix approach, we show how to reduce matrix vector products involving the dense covariance matrix from O(m(2)) to O(m log m), where m is the number of unknowns. Combined with a matrix-free Krylov sub-space solver, this results in a much faster algorithm for solving the system of equations that arise from the geostatistical approach. We illustrate the performance of our algorithm on an application, for monitoring CO2 concentrations using crosswell seismic tomography.