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An Inverse Problem for symmetric hyperbolic Partial Differential Operators on Complete Riemannian Manifolds

Teemu Saksala, Andrew Shedlock

TL;DR

The paper tackles the inverse problem for symmetric hyperbolic operators on complete Riemannian manifolds, showing that both the geometry and time-invariant lower-order terms can be uniquely reconstructed from a local source-to-solution map. The authors adapt the Boundary Control method to interior data, reducing to travel-time data to recover the manifold’s topology, smooth structure, and Riemannian metric, and then determine the lower-order terms up to a multiplicative gauge using higher-order controllability and the Blagovestchenskii identity. The key contributions are global uniqueness for the geometry and a gauge-robust recovery of the lower-order perturbations in the ultra-static setting, with no a priori compactness or boundedness assumptions for the coefficients. This has significant implications for imaging and probing in physics, geophysics, and related fields, enabling stable, data-driven recovery of internal medium properties from internal measurements. The results extend the BC-method to include unbounded lower-order terms and provide a rigorous pathway from interior measurements to complete geometric and electromagnetic-type data.

Abstract

We show that a complete Riemannian manifold, as well as time independent smooth lower order terms appearing in a first order symmetric perturbation of a Riemannian wave operator can be uniquely recovered, up to the natural obstructions, from a local source to solution map of the respective hyperbolic initial value problem. Our proofs are based on an adaptation of the classical Boundary Control method (BC-method) originally developed by Belishev and Kurylev. The BC-method reduces the PDE-based problem to a purely geometric problem involving the so-called travel time data. For each point in the manifold the travel time data contains the distance function from this point to any point in a fixed \textit{a priori} known compact observation set. It is well known that this geometric problem is solvable. The main novelty of this paper lies in our strategy to recover the lower order terms via a further adaptation of the BC-method.

An Inverse Problem for symmetric hyperbolic Partial Differential Operators on Complete Riemannian Manifolds

TL;DR

The paper tackles the inverse problem for symmetric hyperbolic operators on complete Riemannian manifolds, showing that both the geometry and time-invariant lower-order terms can be uniquely reconstructed from a local source-to-solution map. The authors adapt the Boundary Control method to interior data, reducing to travel-time data to recover the manifold’s topology, smooth structure, and Riemannian metric, and then determine the lower-order terms up to a multiplicative gauge using higher-order controllability and the Blagovestchenskii identity. The key contributions are global uniqueness for the geometry and a gauge-robust recovery of the lower-order perturbations in the ultra-static setting, with no a priori compactness or boundedness assumptions for the coefficients. This has significant implications for imaging and probing in physics, geophysics, and related fields, enabling stable, data-driven recovery of internal medium properties from internal measurements. The results extend the BC-method to include unbounded lower-order terms and provide a rigorous pathway from interior measurements to complete geometric and electromagnetic-type data.

Abstract

We show that a complete Riemannian manifold, as well as time independent smooth lower order terms appearing in a first order symmetric perturbation of a Riemannian wave operator can be uniquely recovered, up to the natural obstructions, from a local source to solution map of the respective hyperbolic initial value problem. Our proofs are based on an adaptation of the classical Boundary Control method (BC-method) originally developed by Belishev and Kurylev. The BC-method reduces the PDE-based problem to a purely geometric problem involving the so-called travel time data. For each point in the manifold the travel time data contains the distance function from this point to any point in a fixed \textit{a priori} known compact observation set. It is well known that this geometric problem is solvable. The main novelty of this paper lies in our strategy to recover the lower order terms via a further adaptation of the BC-method.

Paper Structure

This paper contains 21 sections, 49 theorems, 207 equations, 2 figures.

Table of Contents

  1. Introduction
  2. Problem Setting and the Main result
  3. Previous Literature
  4. Outline of the proof of Theorem \ref{['thm:main_thm']}
  5. Preliminary Material
  6. Change of Coordinates
  7. Definitions for the Cauchy Problem
  8. Closed Manifold Case
  9. Complete Manifold Case
  10. Necessary Tools for the Inverse Problem
  11. Multiplicative Gauge
  12. Recovery of the Geometry
  13. Recovery of Travel Time Data
  14. Recovery of Topology
  15. Recovery of Smooth Structure
  16. ...and 6 more sections

Key Result

Theorem 1.1

Let $(N_i,g_i)$ for $i \in \{1,2\}$ be a complete and connected Riemannian manifold with an open observation set $\mathcal{X}_i \subset N_i$. Let $A_i$ and $V_i$ be a smooth real co-vector field and a smooth real-valued function on $N_i$ respectively. If the Hypothesis hyp:same_data is true, then th

Figures (2)

  • Figure 1: The shaded in region and its boundary is the set $Z_\delta$, the left circle represents $M(\mathcal{Y}_\varepsilon,s)$ and the right circle is the set $M(\mathcal{Y}_\delta,s+\delta)$
  • Figure 2: The dark gray set is $Z_\delta$ and the light gray set is $Z_{\delta'}$ where $0 < \delta < \delta'$.

Theorems & Definitions (108)

  • Theorem 1.1
  • Remark 1.2
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Definition 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • ...and 98 more