Stable recovery of a metric tensor from the partial hyperbolic Dirichlet to Neumann map

TL;DR

This paper proves that the metric tensor of a compact Riemannian manifold can be uniquely and stably recovered from boundary measurements of wave equations, advancing inverse boundary value problem theory.

Contribution

It establishes unique determination and logarithmic stability of the metric tensor from the Dirichlet-to-Neumann map in dimensions two and higher.

Findings

Unique recovery of the metric tensor from boundary data

Logarithm-type stability estimate

Extension to dimensions n ≥ 2

Abstract

In this paper we consider the inverse problem of determining on a compact Riemannian manifold the metric tensor in the wave equation with Dirichlet data from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the wave equation. We prove in dimension $n\geq 2$ that the knowledge of the Dirichlet-to-Neumann map for the wave equation uniquely determines the metric tensor and we establish logarithm-type stability.