Stable source reconstruction from a finite number of measurements in the multi-frequency inverse source problem. (English) Zbl 1404.65238
Summary: We consider the multi-frequency inverse source problem for the scalar Helmholtz equation in the plane. The goal is to reconstruct the source term in the equation from measurements of the solution on a surface outside the support of the source. We study the problem in a certain finite dimensional setting: from measurements made at a finite set of frequencies we uniquely determine and reconstruct sources in a subspace spanned by finitely many Fourier-Bessel functions. Further, we obtain a constructive criterion for identifying a minimal set of measurement frequencies sufficient for reconstruction, and under an additional, mild assumption, the reconstruction method is shown to be stable. Our analysis is based on a singular value decomposition of the source-to-measurement forward operators and the distribution of positive zeros of the Bessel functions of the first kind. The reconstruction method is implemented numerically and our theoretical findings are supported by numerical experiments.
MSC:
| 65N21 | Numerical methods for inverse problems for boundary value problems involving PDEs |
| 35J05 | Laplace operator, Helmholtz equation (reduced wave equation), Poisson equation |
| 15A18 | Eigenvalues, singular values, and eigenvectors |
| 33C10 | Bessel and Airy functions, cylinder functions, \({}_0F_1\) |
Keywords:
inverse source problem; Helmholtz equation; photo- and thermoacoustic tomography; multi-frequency inverse source problem; stable reconstruction method; measurements at a finite number of frequencies; singular value decompositionReferences:
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