On increasing stability in the two dimensional inverse source scattering problem with many frequencies

Abstract

In this paper, we will study increasing stability in the inverse source problem for the Helmholtz equation in the plane when the source term is assumed to be compactly supported in a bounded domain Ω with sufficiently smooth boundary. Using the Fourier transform in the frequency domain, bounds for the Hankel functions and for scattering solutions in the complex plane, improving bounds for the analytic continuation, and exact observability for wave equation led us to our goals which are a sharp uniqueness and increasing stability estimate with larger wave numbers interval.

Inverse Scattering Source Problems

“…First increasing stability results were obtained in [15] by using the spatial Fourier transform. In [16,17] more general and sharp results were obtained in sub-domain of  3 and  2 in an arbitrary domains with C 2 boundary by the temporal Fourier transform, with a possibility of handling spatially variable coefficients. The recent results showed that the estimate for source functions is a logarithmic type.…”

“…While Bao, Li and Lu used Dirichlet to Neumann map to simplify the boundary conditions for two dimensional and three dimensional domains (disks and balls), Isakov, Lu, Chang and Entekhabi used the Fourier transform and observability bound for corresponding hyperbolic initial value boundary problem (wave equation) for two and three dimensional domain with C 2 -boundary. In papers [19,20], authors considered inverse source scattering problems with damping factor for two and three dimensional domains, that is, they considered the following equation:…”

Direct and inverse problems for the Schrödinger operator in a three-dimensional planar waveguide

“…The first stability result was obtained in [4] for the inverse source problem of the Helmholtz equation by using multi-frequency data. Later on, the increasing stability was studied for the inverse source problems of various wave equations including the acoustic, elastic, and electromagnetic wave equations, as well as the wave equation with the biharmonic Schrödinger operator [5,12,17,18,[22][23][24]. A more recent study on the stability for the inverse medium problem can be found in [6].…”

Runge approximation and stability improvement for a partial data Calderón problem for the acoustic Helmholtz equation

“…In this article we study improvement of stability effects in Runge approximation originating from the interplay of geometry and an increasing frequency parameter for the acoustic Helmholtz equation. These effects had first been observed in [15] and have subsequently been the object of intensive study, both in the context of unique continuation [24,23,21,51,50] and with regards to their effects on inverse problems [10,26,22,12,13,19,20,25,28,29,5,42,6,27]. Due to the notorious instability in many inverse problems, these improved stability estimates are of great significance, both from a theoretical and practical point of view [9].…”

“…In order to put our results into a proper context, we recall some of the earlier literature on improved stability properties. Due to their ability to stabilize notoriously ill-posed inverse problems, the stabilization effects at high frequency which had first been established in [15] in the context of improved (interior) unique continuation properties were subsequently extended to improved unique continuation properties in various other geometric settings and other model equations [24,23,21,51,50,10,26,22,12,13,19,20,25,28,29,5,42,6,27]. The optimality of exponential k-dependences in unique continuation (in the form of three balls inequalities) was further established recently in [8] for the exact Helmholtz equation (which can be studied by investigating explicit behaviour of Bessel functions).…”