Abstract
We study the inverse source problem for the Helmholtz equation from boundary Cauchy data with multiple wave numbers. The main goal of this paper is to study the uniqueness and increasing stability when the (pseudo)convexity or non-trapping conditions for the related hyperbolic problem are not satisfied. We consider general elliptic equations of the second order and arbitrary observation sites. To show the uniqueness we use the analytic continuation, the Fourier transform with respect to the wave numbers and uniqueness in the lateral Cauchy problem for hyperbolic equations. Numerical examples in 2 spatial dimension support the analysis and indicate the increasing stability for large intervals of the wave numbers, while analytic proofs of the increasing stability are not available.
Highlights
Introduction and main resultsThis paper studies the inverse source problem for the Helmholtz equation from boundary Cauchy data with multiple frequencies
In this paper we rigorously demonstrate uniqueness in the inverse source problems without any convexity/non-trapping condition and give a strong numerical evidence of increasing stability for source identification
We shall emphasize that because the wave number interval [1, 200] might not be sufficient in this inverse source problem and the error between both sources is still comparably large
Summary
∂j (ajm∂m) + i bj ∂j + c, j,m=1 j=1 ajm, bj ∈ C1(Rn), c ∈ L∞(Rn), and A := ∆ in Rn \ Ω. A solution (f0, f1, g1) to the inverse source problems (3)-(5) with the additional Cauchy data (7) is unique in each of the cases a): f0 = 0, b0 = 0; b): g1 = 0, Rn \ Ω is connected. This result will be proven by using analyticity of u(·, k) with respect to k and an auxiliary hyperbolic initial boundary value problem obtained after the Fourier-Laplace transform of the scattering problem with respect to k. We shall emphasize that because the wave number interval [1, 200] might not be sufficient in this inverse source problem and the error between both sources is still comparably large
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