Increasing stability in the two dimensional inverse source scattering problem with attenuation and many frequencies

Abstract

In this paper, we investigate the interior inverse source problem for the Helmholtz equation with attenuation in the plane from boundary Cauchy data of multiple frequencies when the source term is assumed to be compactly supported in an arbitrary domain Ω with sufficiently smooth boundary. The main goal of this paper is to understand the dependence of increasing stability on the attenuation factor or constant damping. Using Fourier transform with respect to the wave numbers, explicit bounds for the analytic continuation and Hankel function and exact observability and Carleman estimates for the wave equation led us to our goal which is an increasing stability estimates with larger wave numbers interval.

Increasing stability of the inverse source problem for one dimensional domain

“…using the Schwartz inequality and integrating with respect to s, using the bound for |k| in S, we complete the proof of (8). Using the same technique, we can prove the (9).…”

“…Because of the wide applications, these problems have being attracted considerable attention. These kinds of problems have been extensively investigated by many researchers (see, for example, [1,5,6,7,8,9,10,14,15,16,18,19,20]). We also note that these type of problem and technique can apply to systems.…”

Stability Analysis for the Helmholtz Equation with Many Frequencies

“…Due to the significant applications, these problems have been attracted considerable attention. These kinds of problems have been extensively investigated by many researchers such as In the paper [2,4,7,8,10,11,12,15,16,17,19,20] and [21]. It is worth mentioning that this type of problems has also application in important system such classical elasticity system and Maxwell system.…”

Increasing stability for the inverse source scattering problem with multi-frequencies

“…Motivated by these significant applications, the inverse source problems, as an important research subject in inverse scattering theory, have continuously attracted much attention by many researchers [3,4,5,9,11,13,14,17,21,22]. Consequently, a great deal of mathematical and numerical results are available, especially for the acoustic waves or the Helmholtz equations.…”

“…Interested readers can also see [19] that claims a uniqueness result and a numerical algorithm for recovering the location and the shape of a supported acoustic source function from boundary measurements at many frequencies. See also [3,17,18,13,14,21] and the references therein. As for increasing stability results proved for coefficients inverse problems, we can refer for example to [10] and [15], in which inverse problems of recovering an electric potential appearing in a Schrödinger equation have been studied (see also the references therein).…”