A Fast Algorithm for Parabolic PDE-based Inverse Problems Based on Laplace Transforms and Flexible Krylov Solvers

摘要

We consider the problem of estimating parameters in large-scale weaklynonlinear inverse problems for which the underlying governing equations is alinear, time-dependent, parabolic partial differential equation. A majorchallenge in solving these inverse problems using Newton-type methods is thecomputational cost associated with solving the forward problem and withrepeated construction of the Jacobian, which represents the sensitivity of themeasurements to the unknown parameters. Forming the Jacobian can beprohibitively expensive because it requires repeated solutions of the forwardand adjoint time-dependent parabolic partial differential equationscorresponding to multiple sources and receivers. We propose an efficient methodbased on a Laplace transform-based exponential time integrator combined with aflexible Krylov subspace approach to solve the resulting shifted systems ofequations efficiently. Our proposed solver speeds up the computation of theforward and adjoint problems, thus yielding significant speedup in totalinversion time. We consider an application from Transient Hydraulic Tomography(THT), which is an imaging technique to estimate hydraulic parameters relatedto the subsurface from pressure measurements obtained by a series of pumpingtests. The algorithms discussed are applied to a synthetic example taken fromTHT to demonstrate the resulting computational gains of this proposed method.