A Flexible Krylov Solver for Shifted Systems with Application to Oscillatory Hydraulic Tomography

We discuss efficient solutions to systems of shifted linear systems arising in computations for oscillatory hydraulic tomography (OHT). The reconstruction of hydrogeological parameters such as hydraulic conductivity and specific storage using limited discrete measurements of pressure (head) obtained from sequential oscillatory pumping tests, leads to a nonlinear inverse problem. We tackle this using the quasi-linear geostatistical approach \cite{kitanidis1995quasi}. This method requires repeated solution of the forward (and adjoint) problem for multiple frequencies, for which we use flexible preconditioned Krylov subspace solvers specifically designed for shifted systems based on ideas in \cite{gu2007flexible}. The solvers allow the preconditioner to change at each iteration. We analyze the convergence of the solver and perform an error analysis when an iterative solver is used for inverting the preconditioner matrices. Finally, we apply our algorithm to a challenging application taken from oscillatory hydraulic tomography to demonstrate the computational gains by using the resulting method.