On the Construction of Virtual Interior Point Source Travel Time Distances from the Hyperbolic Neumann-to-Dirichlet Map

TL;DR

This paper presents a novel algorithm and its numerical implementation for constructing interior-to-boundary travel time distances on Riemannian manifolds using boundary data, advancing inverse geometric problems.

Contribution

It introduces the first numerical implementation of a Boundary Control method variant for geometric inverse problems using the hyperbolic Neumann-to-Dirichlet map.

Findings

Successfully reconstructs interior travel time distances from boundary data.

Demonstrates stability of the reconstruction method.

Provides a practical algorithm for inverse Riemannian geometry problems.

Abstract

We introduce a new algorithm to construct travel time distances between a point in the interior of a Riemannian manifold and points on the boundary of the manifold, and describe a numerical implementation of the algorithm. It is known that the travel time distances for all interior points determine the Riemannian manifold in a stable manner. We do not assume that there are sources or receivers in the interior, and use the hyperbolic Neumann-to-Dirichlet map, or its restriction, as our data. Our algorithm is a variant of the Boundary Control method, and to our knowledge, this is the first numerical implementation of the method in a geometric setting.