Increasing stability for the inverse source scattering problem with multi-frequencies
Peijun Li Ganghua Yuan
Inverse Problems & Imaging 2017, 11(4): 745-759 doi: 10.3934/ipi.2017035

Consider the scattering of the two-or three-dimensional Helmholtz equation where the source of the electric current density is assumed to be compactly supported in a ball. This paper concerns the stability analysis of the inverse source scattering problem which is to reconstruct the source function. Our results show that increasing stability can be obtained for the inverse problem by using only the Dirichlet boundary data with multi-frequencies.

keywords: Stability multi-frequencies inverse source problem Helmholtz equation partial differential equation
Near-field imaging of obstacles
Peijun Li Yuliang Wang
Inverse Problems & Imaging 2015, 9(1): 189-210 doi: 10.3934/ipi.2015.9.189
A novel method is developed for solving the inverse obstacle scattering problem in near-field imaging. The obstacle surface is assumed to be a small and smooth deformation of a circle. Using the transformed field expansion, the direct obstacle scattering problem is reduced to a successive sequence of two-point boundary value problems. Analytical solutions of these problems are derived by a Green's function method. The nonlinear inverse problem is linearized by dropping the higher order terms in the power series expansion. Based on the linear model and analytical solutions, an explicit reconstruction formula is obtained. In addition, a nonlinear correction scheme is devised to improve the results dramatically when the deformation is large. The method requires only a single incident wave at a fixed frequency. Numerical tests show that the method is stable and effective for near-field imaging of obstacles with subwavelength resolution.
keywords: inverse obstacle scattering Near-field imaging transformed field expansion.
A fast direct imaging method for the inverse obstacle scattering problem with nonlinear point scatterers
Jun Lai Ming Li Peijun Li Wei Li
Inverse Problems & Imaging 2018, 12(3): 635-665 doi: 10.3934/ipi.2018027

Consider the scattering of a time-harmonic plane wave by heterogeneous media consisting of linear or nonlinear point scatterers and extended obstacles. A generalized Foldy–Lax formulation is developed to take fully into account of the multiple scattering by the complex media. A new imaging function is proposed and an FFT-based direct imaging method is developed for the inverse obstacle scattering problem, which is to reconstruct the shape of the extended obstacles. The novel idea is to utilize the nonlinear point scatterers to excite high harmonic generation so that enhanced imaging resolution can be achieved. Numerical experiments are presented to demonstrate the effectiveness of the proposed method.

keywords: Foldy–Lax formulation point scatterers inverse obstacle scattering problem the Helmholtz equation boundary integral equation nonlinear optics
Inverse source problems in electrodynamics
Guanghui Hu Peijun Li Xiaodong Liu Yue Zhao
Inverse Problems & Imaging 2018, 12(6): 1411-1428 doi: 10.3934/ipi.2018059

This paper concerns inverse source problems for the time-dependent Maxwell equations. The electric current density is assumed to be the product of a spatial function and a temporal function. We prove uniqueness and stability in determining the spatial or temporal function from the electric field, which is measured on a sphere or at a point over a finite time interval.

keywords: Stability uniqueness inverse source problem Maxwell's equations

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