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.

翻译：本文研究了与Helmholtz方程相关的数据完备化问题。目标是通过利用从可访问区域采集的测量数据，识别无法直接访问的边界部分上的未知边界条件。此类反问题在Hadamard意义下是病态的，这使得寻找稳定可靠的解变得尤为困难。为应对这些挑战，我们提出一种仿生方法，将粒子群优化与Tikhonov正则化相结合。数值实验结果表明，该方法能够获得精确、稳定且收敛可靠的解。总体而言，该方法为处理此类反问题固有的不稳定性与敏感性提供了一种有前景的途径。