Imaging acoustic obstacles by singular and hypersingular point sources
Jingzhi Li Hongyu Liu Hongpeng Sun Jun Zou
Inverse Problems & Imaging 2013, 7(2): 545-563 doi: 10.3934/ipi.2013.7.545
We investigate a qualitative method for imaging acoustic obstacles in two and three dimensions by boundary measurements corresponding to hypersingular point sources. Rigorous mathematical justification of the imaging method is established, and numerical experiments are presented to illustrate the effectiveness of the proposed imaging scheme.
keywords: hypersingular point sources. Dirichelet-to-Neumann map sampling method indicator function Inverse acoustic scattering boundary measurment
A direct sampling method for inverse scattering using far-field data
Jingzhi Li Jun Zou
Inverse Problems & Imaging 2013, 7(3): 757-775 doi: 10.3934/ipi.2013.7.757
This work is concerned with a direct sampling method (DSM) for inverse acoustic scattering problems using far-field data. Using one or few incident waves, the DSM provides quite reasonable profiles of scatterers in time-harmonic inverse acoustic scattering without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer. We shall first present a novel derivation of the DSM using far-field data, then carry out a systematic evaluation of the performances and distinctions of the DSM using both near-field and far-field data. A new interpretation from the physical perspective is provided based on some numerical observations. It is shown from a variety of numerical experiments that the method has several interesting and promising potentials: a) ability to identify not only medium scatterers, but also obstacles, and even cracks, using measurement data from one or few incident directions, b) robustness with respect to large noise, and c) computational efficiency with only inner products involved.
keywords: far-field data. direct sampling method indicator function Inverse acoustic scattering
Fast imaging of electromagnetic scatterers by a two-stage multilevel sampling method
Jingzhi Li Hongyu Liu Qi Wang
Discrete & Continuous Dynamical Systems - S 2015, 8(3): 547-561 doi: 10.3934/dcdss.2015.8.547
Some effective imaging schemes for inverse scattering problems were recently proposed in [13,14] for locating multiple multiscale electromagnetic (EM) scatterers, namely a combination of components of possible small size and regular size compared to the detecting EM wavelength. In this paper, instead of using a single far-field measurement, we relax the assumption of one fixed frequency to multiple ones, and develop efficient numerical techniques to speed up those imaging schemes by adopting multi-frequency and Multilevel ideas in a two-stage manner. Numerical tests are presented to demonstrate the efficiency and the salient features of the proposed fast imaging scheme.
keywords: single-shot multilevel. sampling method Inverse scattering two-stage
On a gesture-computing technique using electromagnetic waves
Jingzhi Li Hongyu Liu Hongpeng Sun
Inverse Problems & Imaging 2018, 12(3): 677-696 doi: 10.3934/ipi.2018029

This paper is concerned with a conceptual gesture-based instruction/input technique using electromagnetic wave detection. The gestures are modeled as the shapes of some impenetrable or penetrable scatterers from a certain admissible class, called a dictionary. The gesture-computing device generates time-harmonic electromagnetic point signals for the gesture recognition and detection. It then collects the scattered wave in a relatively small backscattering aperture on a bounded surface containing the point sources. The recognition algorithm consists of two stages and requires only two incident waves of different wavenumbers. The location of the scatterer is first determined approximately by using the measured data at a small wavenumber and the shape of the scatterer is then identified using the computed location of the scatterer and the measured data at a regular wavenumber. We provide the corresponding mathematical principle with rigorous analysis. Numerical experiments show that the proposed device works effectively and efficiently.

keywords: Gesture recognition instruction/input device electromagnetic wave propagation inverse scattering
Conditional Stability and Numerical Reconstruction of Initial Temperature
Jingzhi Li Masahiro Yamamoto Jun Zou
Communications on Pure & Applied Analysis 2009, 8(1): 361-382 doi: 10.3934/cpaa.2009.8.361
In this paper, we address an inverse problem of reconstruction of the initial temperature in a heat conductive system when some measurement data of temperature are available, which may be observed in a subregion inside or on the boundary of the physical domain, along a time period which may start at some point, possibly far away from the initial time. A conditional stability estimate is first achieved by the Carleman estimate for such reconstruction. Numerical reconstruction algorithm is proposed based on the output least-squares formulation with the Tikhonov regularization using the finite element discretization, and the existence and convergence of the finite element solution are presented. Numerical experiments are carried out to demonstrate the applicability and effectiveness of the proposed method.
keywords: conditional stability Tikhonov regularization Carleman estimate finite element method backward heat problem output least-squares

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