Inverse obstacle problem for the non-stationary wave equation with an unknown background

We consider boundary measurements for the wave equation on a bounded domain $M \subset \R^2$ or on a compact Riemannian surface, and introduce a method to locate a discontinuity in the wave speed. Assuming that the wave speed consist of an inclusion in a known smooth background, the method can determine the distance from any boundary point to the inclusion. In the case of a known constant background wave speed, the method reconstructs a set contained in the convex hull of the inclusion and containing the inclusion. Even if the background wave speed is unknown, the method can reconstruct the distance from each boundary point to the inclusion assuming that the Riemannian metric tensor determined by the wave speed gives simple geometry in $M$. The method is based on reconstruction of volumes of domains of influence by solving a sequence of linear equations. For \tau \in C(\p M) the domain of influence $M(\tau)$ is the set of those points on the manifold from which the distance to some boundary point $x$ is less than $\tau(x)$.Comment: 4 figure

Recovery of zeroth order coefficients in non-linear wave equations

This paper is concerned with the resolution of an inverse problem related to the recovery of a scalar (potential) function $V$ from the source to solution map, of the semi-linear equation $(\Box_{g}+V)u+u^3=0$ on a globally hyperbolic Lorentzian manifold $(M,g)$. We first study the simpler model problem where the geometry is the Minkowski space and prove the uniqueness of $V$ through the use of geometric optics and a three-fold wave interaction arising from the cubic non-linearity. Subsequently, the result is generalized to globally hyperbolic Lorentzian manifolds by using Gaussian beams

A stabilized finite element method for inverse problems subject to the convection-diffusion equation. I: diffusion-dominated regime

The numerical approximation of an inverse problem subject to the convection--diffusion equation when diffusion dominates is studied. We derive Carleman estimates that are on a form suitable for use in numerical analysis and with explicit dependence on the P\'eclet number. A stabilized finite element method is then proposed and analysed. An upper bound on the condition number is first derived. Combining the stability estimates on the continuous problem with the numerical stability of the method, we then obtain error estimates in local $H^1$- or $L^2$-norms that are optimal with respect to the approximation order, the problem's stability and perturbations in data. The convergence order is the same for both norms, but the $H^1$-estimate requires an additional divergence assumption for the convective field. The theory is illustrated in some computational examples.Comment: 21 pages, 6 figures; in v2 we added two remarks and an appendix on psiDOs, and made some minor correction

A finite element data assimilation method for the wave equation

We design a primal-dual stabilized finite element method for the numerical approximation of a data assimilation problem subject to the acoustic wave equation. For the forward problem, piecewise affine, continuous, finite element functions are used for the approximation in space and backward differentiation is used in time. Stabilizing terms are added on the discrete level. The design of these terms is driven by numerical stability and the stability of the continuous problem, with the objective of minimizing the computational error. Error estimates are then derived that are optimal with respect to the approximation properties of the numerical scheme and the stability properties of the continuous problem. The effects of discretizing the (smooth) domain boundary and other perturbations in data are included in the analysis.Comment: 23 page

Regularization strategy for inverse problem for 1+1 dimensional wave equation

An inverse boundary value problem for a 1+1 dimensional wave equation with wave speed $c(x)$ is considered. We give a regularisation strategy for inverting the map $\mathcal A:c\mapsto \Lambda,$ where $\Lambda$ is the hyperbolic Neumann-to-Dirichlet map corresponding to the wave speed $c$. More precisely, we consider the case when we are given a perturbation of the Neumann-to-Dirichlet map $\tilde \Lambda=\Lambda +\mathcal E$, where $\mathcal E$ corresponds to the measurement errors, and reconstruct an approximate wave speed $\tilde c$. We emphasize that $\tilde \Lambda$ may not not be in the range of the map $\mathcal A$. We show that the reconstructed wave speed $\tilde c$ satisfies $\| \tilde c-c\|_{L^\infty}<C \|E\|^{1/18}$. Our regularization strategy is based on a new formula to compute $c$ from $\Lambda$