Reconstruction of the solution and the source of hyperbolic equations from boundary measurements: mixed formulations

TL;DR

This paper presents a numerical method for reconstructing solutions and sources of hyperbolic equations from boundary data, using a least-squares approach and mixed formulations, with proven convergence and applications in 1D and 2D.

Contribution

It introduces a novel mixed formulation and numerical scheme for inverse hyperbolic problems, including source reconstruction, with theoretical convergence guarantees.

Findings

Well-posedness of the mixed formulation under geometric conditions

Strong convergence of the finite element approximation

Successful numerical examples in 1D and 2D cases

Abstract

We introduce 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 boundary 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 discuss several examples for $N=1$ and $N=2$. The problem of the reconstruction of both the state and the source term is also addressed.