Fast Solvers for Transient Hydraulic Tomography based on Laplace transform

Transient Hydraulic Tomography (THT) is a method to estimate the parameters hydraulic conductivity and specific storage, from measurements of hydraulic heads or pressure obtained in a series of interference tests in aquifer geologic formation such as an aquifer (i.e., pumping at one location and depth while measuring the response at several others). These measurements can be used to reconstruct the spatial variation of hydraulic parameters by solving a nonlinear inverse problem, which we tackle using the geostatistical approach. A central challenge associated with the application of the geostatistical approach to THT, is the computational cost associated with constructing the Jacobian - which represents the sensitivity of the measurements to the unknown parameters. This essentially requires repeated solutions to the 'forward problem' and the 'adjoint problem' for determination of derivatives, which are both time-dependent parabolic partial differential equations. To solve the 'forward problem', we use a Laplace Transform based exponential time integrator combined with a Krylov subspace based method for solving shifted systems. This approach allows us to independently evaluate the transient problem at different time instants at (almost) the cost of solving one steady-state groundwater equation. A similar approach can be used to accelerate the solution of the 'adjoint problem' as well. As we demonstrate, this approach dramatically lowers the computational cost associated with evaluating the Jacobian and as a result, the reconstruction of the parameters. The performance of our algorithm is demonstrated on some challenging synthetic examples; in particular, we apply it to large-scale inverse problems arising from transient hydraulic tomography.