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

TL;DR

This paper introduces a fast computational method combining Laplace transforms and Krylov solvers to efficiently solve large-scale parabolic PDE inverse problems, significantly reducing inversion time.

Contribution

The paper presents a novel solver that accelerates inverse problem computations by efficiently solving shifted systems using Laplace transforms and flexible Krylov methods.

Findings

Achieved significant speedup in forward and adjoint problem solutions.

Demonstrated computational efficiency on a synthetic Transient Hydraulic Tomography example.

Reduced overall inversion time for large-scale parabolic PDE inverse problems.

Abstract

We consider the problem of estimating parameters in large-scale weakly nonlinear inverse problems for which the underlying governing equations is a linear, time-dependent, parabolic partial differential equation. A major challenge in solving these inverse problems using Newton-type methods is the computational cost associated with solving the forward problem and with repeated construction of the Jacobian, which represents the sensitivity of the measurements to the unknown parameters. Forming the Jacobian can be prohibitively expensive because it requires repeated solutions of the forward and adjoint time-dependent parabolic partial differential equations corresponding to multiple sources and receivers. We propose an efficient method based on a Laplace transform-based exponential time integrator combined with a flexible Krylov subspace approach to solve the resulting shifted systems of equations efficiently. Our proposed solver speeds up the computation of the forward and adjoint problems, thus yielding significant speedup in total inversion time. We consider an application from Transient Hydraulic Tomography (THT), which is an imaging technique to estimate hydraulic parameters related to the subsurface from pressure measurements obtained by a series of pumping tests. The algorithms discussed are applied to a synthetic example taken from THT to demonstrate the resulting computational gains of this proposed method.