Abstract
In this paper, the Cauchy problem for the Helmholtz equation is investigated. The objective is to recover the missing data on some part of the boundary of a bounded domain from overspecified data on the remaining part of the boundary. We propose a preconditioned Krylov algorithm to solve this ill-posed problem, based on the representation of the solution with surface integral equations and the Steklov–Poincaré operator. We give a theoretical and numerical validation of the proposed method conducted in the 3D setting. We show the fast convergence of the proposed algorithm tested on various synthetic examples. The numerical results show a high precision of the reconstruction obtained for different levels of noisy data.
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