Symplectic Reconstruction of Data for Heat and Wave Equations

TL;DR

This paper introduces a novel method for solving inverse problems related to heat and wave equations by using Hamilton-Jacobi theory to regularize and solve the resulting nonlinear PDE system.

Contribution

It develops a general approach combining Hamilton-Jacobi regularization with Newton's method for efficiently solving inverse PDE problems.

Findings

Effective regularization of ill-posed inverse problems

Successful application to heat and wave equations

Improved computational stability and accuracy

Abstract

This report concerns the inverse problem of estimating a spacially dependent coefficient of a partial differential equation from observations of the solution at the boundary. Such a problem can be formulated as an optimal control problem with the coefficient as the control variable and the solution as state variable. The heat or the wave equation is here considered as state equation. It is well known that such inverse problems are ill-posed and need to be regularized. The powerful Hamilton-Jacobi theory is used to construct a simple and general method where the first step is to analytically regularize the Hamiltonian; next its Hamiltonian system, a system of nonlinear partial differential equations, is solved with the Newton method and a sparse Jacobian.