Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

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

Lingyun Qiu, Shenwen Yu

TL;DR

<3-5 sentence high-level summary>We address inverse point-source stability for elliptic and parabolic PDEs with spatially varying coefficients by developing a unified optimal-transport framework. The core idea is to realize Kantorovich dual potentials as boundary functionals of suitably constructed adjoint solutions: CGO-based interpolants in the elliptic setting and time-dependent adjoints with boundary controllability in the parabolic setting. This yields global stability estimates of the form $\mathcal{T}_c(\mu,\nu) \le C\|u_1-u_2\|_{L^2(\partial\Omega)}$ (elliptic) and $\mathcal{T}_c(\mu,\nu) \le C\|u_1-u_2\|_{L^2(\partial\Omega\times[0,T])}$, linking source-geometry (OT) to boundary observations. The results establish OT-based stability theory for inverse sources in PDEs with spatially varying coefficients and illuminate pathways for numerical implementations via OT misfits.

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 \[ \mathcal{T}_c(μ,ν) \le C\,\|u_1 - u_2\|_{L^2(\partialΩ)} \quad \text{and} \quad \mathcal{T}_c(μ,ν) \le C\,\|u_1 - u_2\|_{L^2(\partialΩ\times[0,T])}, \] where $μ$ and $ν$ 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.

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

TL;DR

<3-5 sentence high-level summary>We address inverse point-source stability for elliptic and parabolic PDEs with spatially varying coefficients by developing a unified optimal-transport framework. The core idea is to realize Kantorovich dual potentials as boundary functionals of suitably constructed adjoint solutions: CGO-based interpolants in the elliptic setting and time-dependent adjoints with boundary controllability in the parabolic setting. This yields global stability estimates of the form (elliptic) and , linking source-geometry (OT) to boundary observations. The results establish OT-based stability theory for inverse sources in PDEs with spatially varying coefficients and illuminate pathways for numerical implementations via OT misfits.

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 \[ \mathcal{T}_c(μ,ν) \le C\,\|u_1 - u_2\|_{L^2(\partialΩ)} \quad \text{and} \quad \mathcal{T}_c(μ,ν) \le C\,\|u_1 - u_2\|_{L^2(\partialΩ\times[0,T])}, \] where and 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.
Paper Structure (18 sections, 19 theorems, 171 equations)

This paper contains 18 sections, 19 theorems, 171 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Optimal transport and Kantorovich duality
  4. CGO solutions and test function spaces
  5. Boundary control theory
  6. Case for time-independent equations
  7. Problem formulation
  8. Construction of test functions
  9. Global OT stability for elliptic point sources
  10. Case for parabolic equations
  11. Problem setting
  12. Construction of time-dependent test functions
  13. Global OT stability for parabolic point sources
  14. Concluding remarks
  15. Proof of Lemma \ref{['invertible matrix thm']} and Lemma \ref{['lemma:control estimate']}
  16. ...and 3 more sections

Key Result

Theorem 1

Let $X$ and $Y$ be Polish spaces, let $\mu \in P(X)$ and $\nu \in P(Y)$, and let $c: X \times Y \to \mathbb{R}_{+} \cup \{+\infty\}$ be a lower semicontinuous cost function. For $\pi \in P(X \times Y)$ and $(\varphi, \psi) \in L^1(d\mu) \times L^1(d\nu)$, define Let $\Pi(\mu, \nu)$ denote the set of all Borel probability measures $\pi$ on $X \times Y$ such that for every measurable $A \subset X$

Theorems & Definitions (34)

  • Theorem 1: Optimal transport cost and Kantorovich duality
  • Lemma 2
  • Lemma 3
  • Lemma 4
  • Theorem 5
  • Theorem 6
  • proof
  • Theorem 7
  • Remark 1
  • Lemma 8
  • ...and 24 more