Reconstruction of the Radiation Condition and Solution for the Helmholtz Equation in a Semi-infinite Strip from Cauchy Data on an Interior Segment

• Published in: Journal of computational methods in applied mathematics Vol. 24; no. 4; pp. 813 - 828
• Main Authors: Achieng, Pauline, Berntsson, Fredrik, Kozlov, Vladimir
• Format: Journal Article
• Language: English
• Published: De Gruyter 25.07.2023
• Subjects: 35J05; 65N21; Cauchy Problem; Helmholtz Equation; Ill-Posed Problem; Inverse Problem; Landweber Method; Well-Posed Problem; Cauchy Problem; Ill-Posed Problem; Landweber Method; Inverse Problem; Helmholtz Equation; Well-Posed Problem
• ISSN: 1609-4840; 1609-9389; 1609-9389
• DOI: 10.1515/cmam-2022-0244

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.