Inverse problems for linear hyperbolic equations using mixed formulations

TL;DR

This paper presents a new numerical method using mixed formulations and finite element discretization to solve inverse problems for linear hyperbolic equations, enabling reconstruction of wave solutions and sources from partial observations.

Contribution

It introduces a direct least-squares approach with mixed formulations for hyperbolic inverse problems, proving well-posedness and convergence of the numerical scheme.

Findings

Proved well-posedness under geometric conditions

Established strong convergence of the finite element approximation

Demonstrated effectiveness through numerical examples in 1D and 2D

Abstract

We introduce in this document a direct method allowing to solve numerically inverse type problems for linear hyperbolic equations. We first consider the reconstruction of the full solution of the wave equation posed in $\Omega\times (0,T)$ - $\Omega$ a bounded subset of $\mathbb{R}^N$ - from a partial distributed observation. We employ a least-squares technique and minimize the $L^2$-norm of the distance from the observation to any solution. Taking the hyperbolic equation as the main constraint of the problem, the optimality conditions are reduced to a mixed formulation involving both the state to reconstruct and a Lagrange multiplier. Under usual geometric optic conditions, we show the well-posedness of this mixed formulation (in particular the inf-sup condition) and then introduce a numerical approximation based on space-time finite elements discretization. We prove the strong convergence of the approximation and then discussed several examples for $N=1$ and $N=2$. The problem of the reconstruction of both the state and the source term is also addressed.