Reconstruction of Boundary Data in the Helmholtz Equation Using Particle Swarm Optimization

This paper tackles the data completion problem related to the Helmholtz equation. The goal is to identify unknown boundary conditions on parts of the boundary that cannot be accessed directly, by making use of measurements collected from accessible regions. Such inverse problems are known to be ill-posed in the Hadamard sense, which makes finding stable and dependable solutions particularly difficult. To address these challenges, we propose a bio-inspired method that combines Particle Swarm Optimization with Tikhonov regularization. The results of our numerical experiments suggest that this approach can yield solutions that are both accurate and stable, converging reliably. Overall, this method provides a promising way to handle the inherent instability and sensitivity of these types of inverse problems.