Stable Numerical Solution of an Elliptic PDE Inverse Problem Subject to Incomplete Boundary Conditions

This study addresses the challenging problem of solving inverse elliptic Partial Differential Equations (PDE) with incomplete boundary data, data available only on a part of the domain boundary. The aim is to develop a robust, effective numerical framework that consistently recovers parameters and/or sources from incomplete, ill-posed data. In the case of a variational problem discretized by the Finite Element Method (FEM) and solved by an adjoint-based optimization strategy, the framework uses Tikhonov regularization. Morozov's Discrepancy Principle is used to determine regularization parameters that achieve the best balance between accuracy and stability. Even with 5% noise in the measurement data, the unknown model parameters can be successfully reconstructed with an L2 error of less than 5%, according to numerical results. The method can be used to recover parameters for a variety of domain geometries, including those with complex shapes and re-entrant edges. This research demonstrates that the framework is robust and stable, delivering reliable solutions to this challenging problem class. The solution approach has direct application in medical imaging, geophysics, and engineering diagnostics, where boundary data cannot be obtained.