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.
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
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.
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.
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.
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.