Quantitative stability of Gel'fand's inverse boundary problem

TL;DR

This work analyzes the Gel'fand inverse boundary problem on compact manifolds with boundary under bounded-geometry assumptions and proves a quantitative stability result: from a $\delta$-approximation of the Neumann boundary spectral data, one can construct a finite metric space $X$ that approximates the manifold $M$ in the Gromov-Hausdorff sense with a explicit log-log stability rate. The core method is a uniform quantitative unique continuation for the wave operator on manifolds with boundary, developed via a distance-based propagation framework within Alexandrov-CAT geometry to avoid boundary-induced degeneracies. The authors also provide an algorithmic reconstruction pipeline that (i) propagates stability to approximate domain volumes and boundary-distance functions, (ii) reconstructs interior geometry from these data, and (iii) proves stability of the resulting manifold approximation. The results connect boundary spectral data to interior geometry through a controlled sequence of extensions, domain constructions, and spectral-projection techniques, with potential implications for medical imaging and inverse problems in anisotropic media.

Abstract

In Gel'fand's inverse problem, one aims to determine the topology, differential structure and Riemannian metric of a compact manifold $M$ with boundary from the knowledge of the boundary $\partial M,$ the Neumann eigenvalues $λ_j$ and the boundary values of the eigenfunctions $\varphi_j|_{\partial M}$. We show that this problem has a stable solution with quantitative stability estimates in a class of manifolds with bounded geometry. More precisely, we show that finitely many eigenvalues and the boundary values of corresponding eigenfunctions, known up to small errors, determine a metric space that is close to the manifold in the Gromov-Hausdorff sense. We provide an algorithm to construct this metric space. This result is based on an explicit estimate on the stability of the unique continuation for the wave operator.

Figures (6)

• Figure 1: Domains of unique continuation. The blue vertical line is $\Gamma\times [-T,T]$. The domain enclosed by the blue lines is the optimal domain $K(\Gamma,T)$. The domain enclosed by the red lines is $\Omega (h)$ defined in \ref{['Omegaht']}, obtained by propagating local unique continuation. The distance between the blue and red lines is $\sqrt{h}$.
• Figure 2: Domains for the initial step. Enclosed by the red solid line is the domain we work in, and it is close to $\Gamma$.
• Figure 3: Domains for Case 1 or the first step in Case 2. Enclosed by the red solid lines is the domain we work in, and its boundary consists of two disjoint parts. This domain never reaches outside distance $\rho_0$, which is marked by the upper red dotted line. The blue dashed line $\Gamma_2$ is the reference set for the second step in Case 2.
• Figure 4: Domains for the second step in Case 2. Enclosed by the red solid lines is the domain we work in. The blue dashed line $\Gamma_3$ is the reference set for the third step. From here, the procedure is entirely done in $M$.
• Figure 5: The procedure of a three-step propagation besides the initial step. The red solid lines enclose the whole region $\Omega=\cup_{i,j}\Omega_{i,j}$ propagated by the unique continuation. The black dotted line represents the optimal region, while the blue dotted line represents the actual region we can estimate.

Theorems & Definitions (74)

• Definition 1.1
• Theorem 1
• Theorem 2
• Remark 1
• Theorem 3
• Definition 2.1
• Theorem 2.2
• Theorem 3.1
• Remark 2
• Proposition 3.2