79 research outputs found
Compressive Inverse Scattering II. SISO Measurements with Born scatterers
Inverse scattering methods capable of compressive imaging are proposed and
analyzed. The methods employ randomly and repeatedly (multiple-shot) the
single-input-single-output (SISO) measurements in which the probe frequencies,
the incident and the sampling directions are related in a precise way and are
capable of recovering exactly scatterers of sufficiently low sparsity.
For point targets, various sampling techniques are proposed to transform the
scattering matrix into the random Fourier matrix. The results for point targets
are then extended to the case of localized extended targets by interpolating
from grid points. In particular, an explicit error bound is derived for the
piece-wise constant interpolation which is shown to be a practical way of
discretizing localized extended targets and enabling the compressed sensing
techniques.
For distributed extended targets, the Littlewood-Paley basis is used in
analysis. A specially designed sampling scheme then transforms the scattering
matrix into a block-diagonal matrix with each block being the random Fourier
matrix corresponding to one of the multiple dyadic scales of the extended
target. In other words by the Littlewood-Paley basis and the proposed sampling
scheme the different dyadic scales of the target are decoupled and therefore
can be reconstructed scale-by-scale by the proposed method. Moreover, with
probes of any single frequency \om the coefficients in the Littlewood-Paley
expansion for scales up to \om/(2\pi) can be exactly recovered.Comment: Add a new section (Section 3) on localized extended target
Self-Averaging Scaling Limits of Two-Frequency Wigner Distribution for Random Paraxial Waves
Two-frequency Wigner distribution is introduced to capture the asymptotic
behavior of the space-frequency correlation of paraxial waves in the radiative
transfer limits. The scaling limits give rises to deterministic transport-like
equations. Depending on the ratio of the wavelength to the correlation length
the limiting equation is either a Boltzmann-like integral equation or a
Fokker-Planck-like differential equation in the phase space. The solutions to
these equations have a probabilistic representation which can be simulated by
Monte Carlo method. When the medium fluctuates more rapidly in the longitudinal
direction, the corresponding Fokker-Planck-like equation can be solved exactly.Comment: typos correcte
Mutual Coherence of Polarized Light in Disordered Media: Two-Frequency Method Extended
The paper addresses the two-point correlations of electromagnetic waves in
general random, bi-anisotropic media whose constitutive tensors are complex
Hermitian, positive- or negative-definite matrices. A simplified version of the
two-frequency Wigner distribution (2f-WD) for polarized waves is introduced and
the closed form Wigner-Moyal equation is derived from the Maxwell equations. In
the weak-disorder regime with an arbitrarily varying background the
two-frequency radiative transfer (2f-RT) equations for the associated coherence matrices are derived from the Wigner-Moyal equation by using the
multiple scale expansion. In birefringent media, the coherence matrix becomes a
scalar and the 2f-RT equations take the scalar form due to the absence of
depolarization. A paraxial approximation is developed for spatialy anisotropic
media. Examples of isotropic, chiral, uniaxial and gyrotropic media are
discussed
Space-frequency correlation of classical waves in disordered media: high-frequency and small scale asymptotics
Two-frequency radiative transfer (2f-RT) theory is developed for geometrical
optics in random media. The space-frequency correlation is described by the
two-frequency Wigner distribution (2f-WD) which satisfies a closed form
equation, the two-frequency Wigner-Moyal equation. In the RT regime it is
proved rigorously that 2f-WD satisfies a Fokker-Planck-like equation with
complex-valued coefficients. By dimensional analysis 2f-RT equation yields the
scaling behavior of three physical parameters: the spatial spread, the
coherence length and the coherence bandwidth. The sub-transport-mean-free-path
behavior is obtained in a closed form by analytically solving a paraxial 2f-RT
equation
On time reversal mirrors
The concept of time reversal (TR) of scalar wave is reexamined from basic
principles. Five different time reversal mirrors (TRM) are introduced and their
relations are analyzed. For the boundary behavior, it is shown that for
paraxial wave only the monopole TR scheme satisfies the exact boundary
condition while for spherical wave only one of the mixed mode TR scheme, after
multiplication by two, satisfies the exact boundary condition. The asymptotic
analysis of the near-field focusing property is presented. It is shown that to
have a subwavelength focal spot the TRM should involve dipole fields. The
monopole TR is extremely ineffective to focus below wavelength as the focal
spot size decreases logarithmically with the distance between the source and
TRM. Contrary to the matched field processing and the phase processor, both of
which resemble TR, TR in a weak- or non-scattering medium is usually biased in
the longitudinal direction, especially when TR is carried out on a {\em single}
plane with a {finite} aperture. This is true for all five TR schemes. On the
other hand, the TR focal spot has been shown repeatedly in the literature, both
theoretically and experimentally, to be centered at the source point when the
medium is multiply scattering. A reconciliation of the two seemingly
conflicting results is found in the random fluctuations in the intensity of the
Green function for a multiply scattering medium and the notion of
scattering-enlarged effective aperture
Quenching and Propagation of Combustion Without Ignition Temperature Cutoff
We study a reaction-diffusion equation in the cylinder , with combustion-type reaction term without
ignition temperature cutoff, and in the presence of a periodic flow. We show
that if the reaction function decays as a power of larger than three as
and the initial datum is small, then the flame is extinguished -- the
solution quenches. If, on the other hand, the power of decay is smaller than
three or initial datum is large, then quenching does not happen, and the
burning region spreads linearly in time. This extends results of
Aronson-Weinberger for the no-flow case. We also consider shear flows with
large amplitude and show that if the reaction power-law decay is larger than
three and the flow has only small plateaux (connected domains where it is
constant), then any compactly supported initial datum is quenched when the flow
amplitude is large enough (which is not true if the power is smaller than three
or in the presence of a large plateau). This extends results of
Constantin-Kiselev-Ryzhik for combustion with ignition temperature cutoff. Our
work carries over to the case , when
the critical power is , as well as to certain non-periodic flows
Dissipation time and decay of correlations
We consider the effect of noise on the dynamics generated by
volume-preserving maps on a d-dimensional torus. The quantity we use to measure
the irreversibility of the dynamics is the dissipation time. We focus on the
asymptotic behaviour of this time in the limit of small noise. We derive
universal lower and upper bounds for the dissipation time in terms of various
properties of the map and its associated propagators: spectral properties,
local expansivity, and global mixing properties. We show that the dissipation
is slow for a general class of non-weakly-mixing maps; on the opposite, it is
fast for a large class of exponentially mixing systems which include uniformly
expanding maps and Anosov diffeomorphisms.Comment: 26 Pages, LaTex. Submitted to Nonlinearit
Gradient Elasticity Theory for Mode III Fracture in Functionally Graded Materials—Part I: Crack Perpendicular to the Material Gradation
Compressive Inverse Scattering I. High Frequency SIMO Measurements
Inverse scattering from discrete targets with the
single-input-multiple-output (SIMO), multiple-input-single-output (MISO) or
multiple-input-multiple-output (MIMO) measurements is analyzed by compressed
sensing theory with and without the Born approximation. High frequency analysis
of (probabilistic) recoverability by the -based
minimization/regularization principles is presented. In the absence of noise,
it is shown that the -based solution can recover exactly the target of
sparsity up to the dimension of the data either with the MIMO measurement for
the Born scattering or with the SIMO/MISO measurement for the exact scattering.
The stability with respect to noisy data is proved for weak or widely separated
scatterers. Reciprocity between the SIMO and MISO measurements is analyzed.
Finally a coherence bound (and the resulting recoverability) is proved for
diffraction tomography with high-frequency, few-view and limited-angle
SIMO/MISO measurements.Comment: A new section on diffraction tomography added; typos fixed; new
figures adde
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