Inverse source problems without (pseudo) convexity assumptions

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.

Inverse Scattering Source Problems

“…As I mentioned before, there are many scientist and researcher have been working on inverse scattering and more specifically on inverse source problems. To expand your knowledge and further mathematical development in this field of research, please see the result authors in [29][30][31][32][33][34][35][36][37][38][39][40][41], which were discussed different aspects of the problems.…”

On increasing stability of the continuation for elliptic equations of second order without (pseudo)convexity assumptions

“…An increasing stability is expected in the inverse source problem, where one looks for f in the Helmholtz equation (∆+k 2 )u = f (not depending on k) in Ω = {x : 1 < |x| < R} from the Cauchy data u, ∂ ν u on Γ = {x : |x| = 1}, k * < k < k * . General uniqueness results and convincing numerical examples of increasing stability are given in [14]. One needs to obtain stability estimates improving with growing k * and to give more of numerical evidence of better resolution for larger k * .…”

A convergent numerical method for a multi-frequency inverse source problem in inhomogenous media

“…To the best of our knowledge, past numerical methods for these problems are based on various methods of the minimization of mismatched least squares functionals. Good quality numerical solutions are obtained in [4,5,22]. However, those minimization procedures do not allow to establish convergence rates of minimizers to the exact solution when the noise in the data tends to zero.…”