A note on several inverse problems with generally random coefficients

Technical Summary: A Note on Several Inverse Problems with Generally Random Coefficients

Problem Statement
This paper investigates inverse problems for elliptic equations with random coefficients, specifically focusing on the Schrödinger equation $(-\Delta + q)u = 0$ on a bounded domain $\Omega \subset \mathbb{R}^n$. The central inquiry is what information about the random potential $q(\omega, x)$ can be recovered from various statistical summaries of the boundary or interior data. The randomness is assumed to be "generally random," meaning no special probabilistic structure (such as Gaussianity, perturbative smallness, microlocal isotropy, or support on inclusions) is imposed.

The study compares the recoverability of the potential's law from four distinct types of data:

1. The full probability law of the Dirichlet-to-Neumann (DN) map, $\text{Law}(\Lambda_q)$.
2. The expectation of the DN map, $\mathbb{E}\Lambda_q$.
3. Finitely many joint moments of the boundary bilinear form $\langle \Lambda_q f, g \rangle$.
4. The averaged interior Green's operator, $\mathbb{E}G_q$, where $G_q = (-\Delta_D + q)^{-1}$.

Methodology
The paper employs a combination of explicit radial constructions, symbolic calculus for pseudodifferential operators, and measure-theoretic arguments.

• Radial Counterexamples: To demonstrate limitations of boundary data, the author constructs explicit smooth, nonnegative, radial potentials in the unit ball of $\mathbb{R}^3$. By analyzing the eigenvalues of the DN map in the spherical harmonic basis, the paper identifies an affine dependence of these eigenvalues on the square of a parameter $N$ (where the potential is defined via $Q_N$). This allows the construction of distinct potentials $q_1, q_2$ and a deterministic potential $q$ such that $\Lambda_q = \frac{1}{2}\Lambda_{q_1} + \frac{1}{2}\Lambda_{q_2}$.
• Moment Hierarchies: Using Vandermonde matrix properties, the paper constructs random potentials taking finitely many values such that their joint moments of the DN map up to any fixed order $M$ coincide, yet their mean potentials differ.
• Symbolic Analysis: For interior data, the paper utilizes the full interior symbol of the Green's operator. By averaging the parametrix expansion of the resolvent $(-\Delta + q)^{-1}$, the author isolates terms in the symbol expansion that correspond to pointwise moments of the potential and its derivatives.
• Deterministic Uniqueness Lifts: The paper applies the Lusin–Souslin theorem to show that if a deterministic inverse problem is injective on a Borel class, then the joint law of the DN map entries (with respect to a dense sequence of boundary functions) uniquely determines the law of the random coefficient.

Key Contributions and Results

1. Obstruction for Averaged Boundary Maps:
   The primary negative result is that the expected Dirichlet-to-Neumann map, $\mathbb{E}\Lambda_q$, does not determine the mean potential $\mathbb{E}q$. The paper constructs a two-valued random potential $Q$ and a deterministic potential $q^*$ such that $\mathbb{E}\Lambda_Q = \Lambda_{q^*}$, but $q^* \neq \mathbb{E}Q$. This obstruction extends to any finite hierarchy of joint moments; knowing all moments of $\langle \Lambda_q f, g \rangle$ up to order $M$ is insufficient to determine $\mathbb{E}q$. This indicates that averaging and finite moment truncation cause a loss of information that is not merely a failure of deterministic uniqueness.

2. Recovery from Full Law:
   In contrast to the moment-based results, the paper establishes that if the full joint law of the DN map is known (specifically, the joint law of countably many matrix entries $\langle \Lambda_q f_i, f_j \rangle$), then the law of the random potential is uniquely determined, provided the underlying deterministic inverse problem is injective. This result lifts classical deterministic uniqueness theorems (e.g., Sylvester–Uhlmann for $n \ge 3$) directly to the probabilistic setting without requiring specific statistical structures on the noise.

3. Recovery from Averaged Green's Operators:
   The paper demonstrates that the averaged interior Green's operator, $\mathbb{E}G_q$, contains significantly more local information than the averaged boundary map.

   • Pointwise Moments: The full interior symbol of $\mathbb{E}G_q$ determines the pointwise mean $\mathbb{E}q$ and variance $\text{Var}(q)$ of the potential.
   • Two-Atom Rigidity: For a two-valued random potential, the averaged Green's operator determines all pointwise moments of the two-point law. This implies that the map from the potential to the averaged Green's operator is injective for two-atom distributions, unlike the boundary map.
   • Finite-Dimensional Models: If the random potential is of the form $q(x) = q_0(x) + \sum X_j V_j(x)$ with known profiles $V_j$ and a compactly supported random vector $X$, the averaged Green's operator determines all mixed moments of $X$, and thus the full law of $X$, under a non-degeneracy condition on the profiles.

4. Conductivity Analogues:
   The appendices extend these findings to the conductivity equation $\nabla \cdot (\gamma \nabla u) = 0$. Similar counterexamples exist for the expected Neumann-to-Dirichlet map. However, for the averaged conductivity Green's operator, the principal symbol determines $\mathbb{E}(\gamma^{-1})$, and the next order determines a weighted covariance matrix involving logarithmic gradients of $\gamma^{-1}$.

Significance and Claims
The paper claims to clarify the fundamental differences between boundary and interior inverse problems in the presence of general randomness.

• Boundary vs. Interior: The results highlight a sharp distinction: boundary data (even with full statistical knowledge) suffers from non-uniqueness when averaged or truncated, whereas interior source-to-solution data (the Green's operator) retains local moment information that allows for the recovery of the mean and variance, and in specific cases, the full law.
• Range Characterization: The counterexamples for the DN map are interpreted not just as probabilistic obstructions but as geometric statements about the range of the nonlinear map $q \mapsto \Lambda_q$. Specifically, the range contains nontrivial affine combinations of deterministic maps that do not correspond to the affine combination of the potentials.
• Contrast with Existing Literature: The work distinguishes itself from inverse scattering literature (e.g., Lassas, Pääivärinta, Saksman; Caro, Helin, Lassas) which typically assumes Gaussian microlocally isotropic fields and recovers the principal symbol of the covariance operator. This paper addresses the recovery of the pointwise mean and full law without assuming such specific structures, showing that such recovery is impossible from boundary moments but possible from interior averages.

The paper concludes that while the full law of the boundary map is theoretically sufficient for recovery (given deterministic uniqueness), practical or partial statistical data (expectations, finite moments) is insufficient for boundary problems but sufficient for interior problems under the conditions described.