Stable reconstruction of simple Riemannian manifolds from unknown interior sources

TL;DR

This paper addresses the inverse problem of reconstructing simple Riemannian manifolds from boundary measurements of waves emitted by unknown interior sources, providing explicit error bounds for finite data.

Contribution

It introduces a method for approximate reconstruction of simple Riemannian manifolds from finite boundary measurements with explicit error estimates.

Findings

Finite data yields approximate manifold reconstructions.

Infinite measurements recover the true manifold.

Convergence of finite-time approximations to the true manifold.

Abstract

Consider the geometric inverse problem: There is a set of delta-sources in spacetime that emit waves travelling at unit speed. If we know all the arrival times at the boundary cylinder of the spacetime, can we reconstruct the space, a Riemannian manifold with boundary? With a finite set of sources we can only hope to get an approximate reconstruction, and we indeed provide a discrete metric approximation to the manifold with explicit data-driven error bounds when the manifold is simple. This is the geometrization of a seismological inverse problem where we measure the arrival times on the surface of waves from an unknown number of unknown interior microseismic events at unknown times. The closeness of two metric spaces with a marked boundary is measured by a labeled Gromov--Hausdorff distance. If measurements are done for infinite time and spatially dense sources, our construction produces the true Riemannian manifold and the finite-time approximations converge to it in the metric sense.