A Nash-Game Approach to Elliptic Cauchy Data Completion with Robust Error Bounds in Practical Noisy Regimes

Technical Summary: A Nash-Game Approach to Elliptic Cauchy Data Completion with Robust Error Bounds in Practical Noisy Regimes

Problem Statement
The paper addresses the ill-posed Cauchy problem associated with Laplace's equation in a bounded, connected Lipschitz domain $\Omega \subset \mathbb{R}^2$. The boundary $\partial\Omega$ is partitioned into an accessible part $\Gamma_c$ where Cauchy data (Dirichlet $T$ and Neumann $\phi$) are available, and an inaccessible part $\Gamma_i$ where the solution traces are unknown. The goal is to recover the missing Dirichlet and Neumann data on $\Gamma_i$ to reconstruct the solution $u$ throughout $\Omega$. This problem is inherently ill-posed, lacking continuous dependence on the boundary data, which leads to instability and unbounded errors in the presence of noise.

Methodology
The authors propose a gamified approach reformulating the data completion task as a static, non-cooperative two-player Nash game.

1. Virtual Control Formulation: The method introduces two independent virtual Dirichlet controls, $\eta_D$ and $\eta_N$, defined on the inaccessible boundary $\Gamma_i$. These controls govern two complementary auxiliary boundary value problems:

   • Player 1 (Dirichlet): Solves a problem with Dirichlet data $\eta_D$ on $\Gamma_i$ and the known data $T$ on $\Gamma_c$.
   • Player 2 (Neumann): Solves a problem with Dirichlet data $\eta_N$ on $\Gamma_i$ and the known Neumann data $\phi$ on $\Gamma_c$.

2. Cost Functionals: Each player minimizes a specific cost functional:

   • Player 1 minimizes the Neumann gap on the accessible boundary $\Gamma_c$ (the difference between the computed Neumann trace and the measured $\phi$).
   • Player 2 minimizes the Dirichlet gap on $\Gamma_c$ (the difference between the computed Dirichlet trace and the measured $T$).
   • Both functionals include a coupling term penalizing the discrepancy between the Neumann traces of the two players on the inaccessible boundary $\Gamma_i$. This term enforces consistency between the two virtual systems.

3. Regularization: To handle noisy measurements, Tikhonov regularization is incorporated into both players' cost functionals. The regularization parameter $\alpha$ is added to the controls' norms.

4. Numerical Implementation: The continuous problem is discretized using a conforming finite element method. The Nash equilibrium is computed via an alternating minimization algorithm, where each player updates their control while holding the other's fixed. Gradients for the optimization are computed efficiently using an adjoint state method.

Key Contributions

• Existence and Uniqueness: The authors establish a rigorous functional framework in energy trace spaces ($H^{1/2}$ and $H^{-1/2}$). They prove that for compatible exact data, a unique Nash equilibrium exists and coincides with the missing trace of the exact harmonic solution.
• Robust Error Bounds in Finite-Noise Regimes: A primary theoretical contribution is the derivation of explicit stability and error bound estimates for the regularized Nash equilibrium. Unlike classical asymptotic analysis where noise and regularization parameters tend to zero, this work provides bounds valid for a fixed, nonzero noise level $\delta$ and a regularization parameter $\alpha$ belonging to a fixed admissible interval $[\epsilon, M]$ (with $\epsilon > 0$).
• Absence of Source-Type Assumptions: The derived error bounds do not rely on classical spectral source-type assumptions or additional regularity conditions on the exact missing data beyond belonging to the energy space $H^{1/2}(\Gamma_i)$. This contrasts with standard Tikhonov theory, which typically requires such assumptions to derive comparable convergence rates.
• Optimal Parameter Scaling: The analysis identifies optimal scaling relationships between the noise level $\delta$, the regularization parameter $\alpha$, and the coupling parameter $\gamma$. Specifically, the optimal choice is shown to be $\alpha_{opt} \sim \sqrt{\delta}$ and $\gamma_{opt} \sim \sqrt[4]{\delta}$, balancing noise amplification, approximation error, and coupling effects to achieve an error rate of $O(\sqrt{\delta})$.

Results
Numerical experiments were conducted on two geometries: a full annulus and an upper half-annulus (introducing Dirichlet-Neumann transition points).

• Accuracy and Stability: The method demonstrated high accuracy in reconstructing both the interior solution and the missing boundary data for noise-free cases, with relative errors in the $H^1(\Omega)$ norm reaching approximately $10^{-5}$.
• Robustness to Noise: In the presence of multiplicative white noise, the method maintained accurate reconstructions when the regularization parameter was scaled appropriately with the noise level ($\alpha \sim \delta$). The error bounds predicted by the theory were observed to hold.
• Convergence Rates: Numerical convergence rates for finite element discretizations (using polynomial degrees $k=2, 3, 4$) were found to be super-optimal, exceeding the theoretical $O(h^{k/3})$ prediction in many cases.
• Reduced Regularity: The method remained stable and accurate even in the half-annulus configuration, where Dirichlet-Neumann transition points typically induce reduced regularity (singularities) in the solution. The results suggest the Nash-game formulation is robust against these geometric singularities.

Significance and Claims
The paper claims that its primary significance lies in bridging the gap between theoretical asymptotic analysis and practical computational requirements. By providing error bounds that are uniform for positive regularization parameters in a fixed-noise regime, the work offers a theoretical justification for the method's robustness in practical applications where noise levels are fixed and cannot be reduced to zero. The approach complements classical Tikhonov theory by removing the need for restrictive source-type assumptions, thereby making the error analysis applicable to a broader class of practical inverse problems where the exact solution's spectral properties are unknown or difficult to verify. The authors position this as a step toward more reliable data completion methods for non-destructive testing and electro-cardiology, where data is inherently noisy and regularity is often limited.