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

<title>Abstract</title> <p>This paper presents a Nash game formulation for solving the ill-posed Cauchy problem associated with Laplace’s equation. The data completion task is recast as a two-player static game, where each player controls a virtual Dirichlet condition on the inaccessible boundary. The first player minimizes a Neumann gap functional, while the second minimizes a Dirichlet gap functional, with both objectives coupled through a multiplier-based coupling term. A rigorous functional framework is developed in energy trace spaces. We prove the existence and uniqueness of the Nash equilibrium and show that, for compatible exact data, this equilibrium coincides with the missing trace of the exact solution of the Cauchy problem. In the presence of noisy measurements, Tikhonov regularization is incorporated into both players’ cost functionals. The main contribution of the paper is the derivation of explicit stability and error bound estimates for the regularized Nash equilibrium in a practical finite-noise regime. These estimates quantify the dependence of the reconstruction error on the noise level, the regularization parameter, and the coupling strength. In contrast with classical asymptotic convergence estimates , the proposed bounds are established uniformly for regularization parameters belonging to a fixed admissible interval [ε, M ], with ε &gt; 0, and do not require additional spectral source-type assumptions on the exact missing data. A conforming finite element discretization and an alternating minimization algorithm are then proposed. Numerical experiments on annular and half-annular domains confirm the robustness and accuracy of the method for nonzero noise levels and reduced-regularity configurations. Mathematics Subject Classification 65F22, 65N12, 65N15, 65N21, 65N22</p>