A Quasi‐Newton Reformulated Geostatistical Approach on Reduced Dimensions for Large‐Dimensional Inverse Problems

For classic geostatistical approaches relying on Gauss‐Newton or quasilinear methods, iterative evaluations of forward models to compute the Jacobians is a major computational challenge for large‐dimensional, underdetermined nonlinear inverse problems. Recent developments in dimension reduction based on principal component analysis have lower the number of forward model implementations in each iteration to the number of retained principal components given prior covariance or structural parameters. The present research develops a quasi‐Newton approach on reduced dimensions within the reformulated geostatistical approach framework to further lower the number of forward model calls and thus decrease the overall computational cost. The proposed algorithm uses Broyden's method to update the approximate Jacobian of the forward model with respect to the projections on the retained principal axes and the drift coefficients. When the secant condition is satisfied, no additional forward model runs or adjoint solver runs are needed. The efficiency and validity of the developed approach are tested by numerical experiments of large‐dimensional hydraulic tomography with true or biased prior information. Compared with Newton method using the full‐rank update of the Jacobians, the quasi‐Newton reformulated geostatistical approach provides similarly high‐quality inverse estimates with a 60–70% reduction of required forward model simulations. For inverse problems with biased prior information, the iterative correction of the principal axes or the update of the prior covariance may be influenced by the rank‐one update of the Jacobian, yielding slightly lower map accuracies, but still providing satisfactory inverse results with substantially reduced computational costs.