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We consider an inverse problem for the Helmholtz equation of reconstructing a solution from measurements taken on a segment inside a semi-infinite strip. Homogeneous Neumann conditions are prescribed on both side boundaries of the strip and an unknown Dirichlet condition on the remaining part of the boundary. Additional complexity is that the radiation condition at infinity is unknown. Our aim is to find the unknown function in the Dirichlet boundary condition and the radiation condition. Such problems appear in acoustics to determine acoustical sources and surface vibrations from acoustic field measurements. The problem is split into two sub-problems, a well-posed and an ill-posed problem. We analyse the theoretical properties of both problems; in particular, we show that the radiation condition is described by a stable non-linear problem. The second problem is ill-posed, and we use the Landweber iteration method together with the discrepancy principle to regularize it. Numerical tests show that the approach works well. [ABSTRACT FROM AUTHOR]

We consider the Cauchy problem for the Helmholtz equation with a domain in ℝ d , d ≥ 2 with N cylindrical outlets to infinity with bounded inclusions in ℝ d - 1. Cauchy data are prescribed on the boundary of the bounded domains and the aim is to find solution on the unbounded part of the boundary. In 1989, Kozlov and Maz'ya [14] proposed an alternating iterative method for solving Cauchy problems associated with elliptic, self-adjoint and positive-definite operators in bounded domains. Different variants of this method for solving Cauchy problems associated with Helmholtz-type operators exists. We consider the variant proposed by Berntsson, Kozlov, Mpinganzima and Turesson (2018) [4] for bounded domains and derive the necessary conditions for the convergence of the procedure in unbounded domains. For the numerical implementation, a finite difference method is used to solve the problem in a simple rectangular domain in ℝ 2 that represent a truncated infinite strip. The numerical results shows that by appropriate truncation of the domain and with appropriate choice of the Robin parameters μ 0 and μ 1 , the Robin–Dirichlet alternating iterative procedure is convergent. [ABSTRACT FROM AUTHOR]

The Cauchy problem for the Helmholtz equation is considered. It was demonstrated in a previous paper by the authors that the alternating algorithm suggested by V.A. Kozlov and V.G. Maz’ya does not converge for large wavenumbers k in the Helmholtz equation. Here, we present some simple modifications of the algorithm which may restore the convergence. They consist of the replacement of the Neumann-Dirichlet iterations by the Robin-Dirichlet ones which repairs the convergence for less than the first Dirichlet-Laplacian eigenvalue. In order to treat large wavenumbers, we present an algorithm based on iterative solution of Robin-Dirichlet boundary value problems in a sufficiently narrow border strip. Numerical implementations obtained using the finite difference method are presented. The numerical results illustrate that the algorithms suggested in this paper, produce convergent iterative sequences. [ABSTRACT FROM AUTHOR]