On increased stability in the continuation of the Helmholtz equation
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Deepak Aralumallige Subbarayappa and Victor Isakov. 2007. On increased stability in the continuation of the Helmholtz equation. Inverse Problems 23(4): 1689.
In this paper, we give analytical and numerical evidence of increasing stability in the Cauchy problem for the Helmholtz equation in the whole domain when frequency is growing. This effect depends upon the convexity properties of the surface where the Cauchy data are given. Proofs use previously obtained estimates in subdomains and the theory of Sobolev spaces: traces, embedding and interpolation theorems. The theory is illustrated by three-dimensional numerical examples. The results show that even in an acoustical frequency range the increase of resolution with growing frequency is quite dramatic. On the other hand, the resolution of continuation outside a unit sphere is decreasing.
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