Recovery of a Smooth Metric via Wave Field and Coordinate Transformation Reconstruction

Abstract

In this paper, we study the inverse boundary value problem for the wave equation with a view towards an explicit reconstruction procedure. We consider both the anisotropic problem where the unknown is a general Riemannian metric smoothly varying in a domain, and the isotropic problem where the metric is conformal to the Euclidean metric. Our objective in both cases is to construct the metric, using either the Neumann-to-Dirichlet (N-to-D) map or Dirichlet-to-Neumann (D-to-N) map as the data. In the anisotropic case we construct the metric in the boundary normal (or semi-geodesic) coordinates via reconstruction of the wave field in the interior of the domain. In the isotropic case we can go further and construct the wave speed in the Euclidean coordinates via reconstruction of the coordinate transformation from the boundary normal coordinates to the Euclidean coordinates. Both cases utilize a variant of the Boundary Control method, and work by probing the interior using special boundary sources. We provide a computational experiment to demonstrate our procedure in the isotropic case with N-to-D data.

Convexification of a 3-D coefficient inverse scattering problem

“…In other words, the number m of free variables in the data equals the number n of free variables in the unknown coefficient, m = n. As to the globally convergent numerical methods for CIPs with redundant data with m > n, see, e.g. [12,14,15].…”

Inverse Problems for Semilinear Wave Equations on Lorentzian Manifolds

“…Their approach is called the boundary control method and it combines both the wave propagation and controllability results [30]. The boundary control method allows also an effective numerical algorithm [15]. Recently, there has been several results on determining a Riemannian manifold from partial data boundary measurements for the linear wave equation and related equations such as the ones in [6,22,28,34,35,40,43,48].…”

“…where u is the unique solution to (14). To have the energy estimate (15) it is sufficient to show that A is continuous. By the closed graph theorem, this is in turn equivalent to showing that if…”

Discrete regularization and convergence of the inverse problem for 1+1 dimensional wave equation

“…For systems it has been considered in [43,44,45]. Stability results for the method have been considered in [1] and [35], and computational implementations in [5,8,21,27,54]. An application of the method to blockage detection in water pipes is in preparation [64].…”