Hölder Stability from Exact Uniqueness for Finite-Dimensional Analytic Inverse Problems

Technical Summary: Hölder Stability from Exact Uniqueness for Finite-Dimensional Analytic Inverse Problems

Problem Statement
The paper addresses the inverse boundary value problem of recovering interior coefficients from boundary measurements. Specifically, it considers finite-dimensional families of coefficients (e.g., piecewise constant anisotropic conductivities or elasticity tensors) where the forward map $F$ (mapping parameters to boundary operators) is real analytic. The central question is whether an exact uniqueness result (i.e., $F(p) = F(q) \implies R(p) = R(q)$, where $R$ is the quantity to be recovered) implies a quantitative stability estimate.

While inverse problems in infinite-dimensional settings are typically severely ill-posed (exhibiting only logarithmic stability), the paper investigates whether the restriction to finite-dimensional, real-analytic parameter spaces allows for stronger stability, specifically Hölder stability, without requiring problem-specific quantitative PDE estimates (such as singular solutions or propagation of smallness).

Methodology
The author employs a geometric and analytic approach rather than traditional PDE-based quantitative methods. The core methodology involves three main components:

1. Hilbert–Schmidt Scalarization: To handle operator-valued equations ($F(p) = F(q)$), the author introduces Hilbert–Schmidt operators $J_X$ and $J_Y$ with dense ranges. This allows the construction of a scalar function $\Phi(p, q) = \|J_Y^*(F(p) - F(q))J_X\|_{S_2}^2$. The dense range property ensures that $\Phi(p, q) = 0$ if and only if $F(p) = F(q)$. Since $F$ is real analytic, $\Phi$ is a scalar real analytic function on $U \times U$.
2. Łojasiewicz Distance Inequality: The proof utilizes the Łojasiewicz inequality, which states that for a real analytic function $\phi$ vanishing on a set $Z$, the distance from a point to $Z$ is bounded by a power of $|\phi|$. Applied to $\Phi$, this relates the distance between parameters $(p, q)$ and the zero set of the data mismatch to the magnitude of the mismatch itself.
3. Finite Determinacy of Analytic Zero Sets: To reduce the infinite-dimensional operator data to a finite number of scalar measurements, the paper uses the fact that the local ring of real analytic germs is Noetherian. This implies that the common zero set of a countable family of analytic equations is locally defined by a finite subfamily.

Key Contributions and Results

1. Abstract Hölder Stability Theorem (Theorem A):
   The paper proves that if $F: U \to \mathcal{L}(X, Y)$ is real analytic and $R: U \to \mathbb{R}^r$ is $C^1$, and if exact recovery holds ($F(p)=F(q) \implies R(p)=R(q)$), then for any compact subset $K \subset U$, there exist constants $C > 0$ and $\theta \in (0, 1]$ such that:
   $$|R(p) - R(q)| \leq C \|F(p) - F(q)\|^\theta$$
   This result establishes that exact uniqueness in a finite-dimensional analytic setting automatically forces a global Hölder stability estimate on compact sets.

2. Finite Measurement Theorem (Theorem B):
   The author demonstrates that the full operator norm is not necessary for this stability. By fixing countable dense families of inputs and tests, the paper proves that there exists a finite set of scalar matrix elements (measurements) that suffices to recover the coefficients with the same Hölder stability. Specifically, for a compact $K$, there exist finitely many pairs $(x_{i_\ell}, y_{j_\ell})$ such that:
   $$|R(p) - R(q)| \leq C \left( \sum_{\ell=1}^M |m_{i_\ell j_\ell}(p) - m_{i_\ell j_\ell}(q)|^2 \right)^{\theta/2}$$
   where $m_{ij}(p) = \langle F(p)x_i, y_j \rangle$.

3. Applications:
   The abstract theorems are applied to two specific physical problems, leveraging existing uniqueness theorems:

   • Anisotropic Conductivity: For piecewise constant anisotropic conductivities, the paper combines the abstract result with the local Neumann-to-Dirichlet uniqueness theorem of Alessandrini–de Hoop–Gaburro. This yields a Hölder stability estimate for recovering the conductivity matrices on compact ellipticity classes.
   • Anisotropic Elasticity: For piecewise homogeneous anisotropic elasticity tensors, the paper applies the result to the localized Dirichlet-to-Neumann map, utilizing the layer-stripping uniqueness theorem of Cârstea–Honda–Nakamura. This provides stability for recovering tensors in cells reachable from the boundary via admissible chains.

Significance and Claims
The paper positions itself as a "transfer principle" within the tradition of abstract finite-dimensional stability. Its primary significance lies in the division of labor it proposes:

• The inverse-problem content (the specific physics and uniqueness) is supplied by existing exact uniqueness theorems.
• The quantitative stability (the transition from exact recovery to Hölder estimates and finite measurements) is supplied purely by the real analyticity of the forward map and finite-dimensionality.

The author explicitly notes the qualitative nature of the results:

• The stability exponent $\theta$ and the constant $C$ are non-constructive; they depend on Łojasiewicz exponents and local finite generation of analytic ideals, which are not effectively computable from the proof.
• The finite set of measurements is guaranteed to exist but is not explicitly identified or claimed to be optimal or experimentally convenient.
• The stability is Hölder rather than Lipschitz, which is a weaker rate than some problem-specific PDE proofs achieve, but the advantage is that it requires no additional quantitative PDE machinery (like singular solutions) beyond the exact uniqueness theorem and analyticity.

The paper concludes that this approach isolates the analytic mechanism driving stability, showing that for finite-dimensional analytic families, exact uniqueness is sufficient to guarantee Hölder stability and finite measurement recovery, even if the specific constants and measurement sets remain non-constructive.