Optimal-Transport Stability of Inverse Point-Source Problems for Elliptic and Parabolic Equations

TL;DR

This paper develops a new stability framework for inverse point-source problems governed by elliptic and parabolic PDEs, using optimal transport costs linked with boundary measurements and adjoint solutions.

Contribution

It introduces a novel approach connecting inverse source problems with optimal transport geometry, providing quantitative stability estimates in terms of OT costs.

Findings

Establishes OT-based stability estimates for inverse source problems.

Links boundary observations with optimal transport potentials.

Provides a unified analytical framework for PDE-based inverse problems.

Abstract

We establish quantitative global stability estimates, formulated in terms of optimal transport (OT) cost, for inverse point-source problems governed by elliptic and parabolic equations with spatially varying coefficients. The key idea is that the Kantorovich dual potential can be represented as a boundary functional of suitable adjoint solutions, thereby linking OT geometry with boundary observations. In the elliptic case, we construct complex geometric optics solutions that enforce prescribed pointwise constraints, whereas in the parabolic case we employ controllable adjoint solutions that transfer interior information to the boundary. Under mild regularity and separation assumptions, we obtain estimates of the form \[ \mathcal{T}_c(\mu,\nu) \le C\,\|u_1 - u_2\|_{L^2(\partial\Omega)} \quad \text{and} \quad \mathcal{T}_c(\mu,\nu) \le C\,\|u_1 - u_2\|_{L^2(\partial\Omega\times[0,T])}, \] where $\mu$ and $\nu$ are admissible point-source measures. These results provide a unified analytical framework connecting inverse source problems and optimal transport, and establish OT-based stability theory for inverse source problems governed by partial differential equations with spatially varying coefficients.