2,934 research outputs found
Time-dependent angularly averaged inverse transport
This paper concerns the reconstruction of the absorption and scattering
parameters in a time-dependent linear transport equation from knowledge of
angularly averaged measurements performed at the boundary of a domain of
interest. We show that the absorption coefficient and the spatial component of
the scattering coefficient are uniquely determined by such measurements. We
obtain stability results on the reconstruction of the absorption and scattering
parameters with respect to the measured albedo operator. The stability results
are obtained by a precise decomposition of the measurements into components
with different singular behavior in the time domain
Inverse Transport Theory of Photoacoustics
We consider the reconstruction of optical parameters in a domain of interest
from photoacoustic data. Photoacoustic tomography (PAT) radiates high frequency
electromagnetic waves into the domain and measures acoustic signals emitted by
the resulting thermal expansion. Acoustic signals are then used to construct
the deposited thermal energy map. The latter depends on the constitutive
optical parameters in a nontrivial manner. In this paper, we develop and use an
inverse transport theory with internal measurements to extract information on
the optical coefficients from knowledge of the deposited thermal energy map. We
consider the multi-measurement setting in which many electromagnetic radiation
patterns are used to probe the domain of interest. By developing an expansion
of the measurement operator into singular components, we show that the spatial
variations of the intrinsic attenuation and the scattering coefficients may be
reconstructed. We also reconstruct coefficients describing anisotropic
scattering of photons, such as the anisotropy coefficient in a
Henyey-Greenstein phase function model. Finally, we derive stability estimates
for the reconstructions
Corrector theory for MsFEM and HMM in random media
We analyze the random fluctuations of several multi-scale algorithms such as
the multi-scale finite element method (MsFEM) and the finite element
heterogeneous multiscale method (HMM), that have been developed to solve
partial differential equations with highly heterogeneous coefficients. Such
multi-scale algorithms are often shown to correctly capture the homogenization
limit when the highly oscillatory random medium is stationary and ergodic. This
paper is concerned with the random fluctuations of the solution about the
deterministic homogenization limit. We consider the simplified setting of the
one dimensional elliptic equation, where the theory of random fluctuations is
well understood. We develop a fluctuation theory for the multi-scale algorithms
in the presence of random environments with short-range and long-range
correlations. What we find is that the computationally more expensive method
MsFEM captures the random fluctuations both for short-range and long-range
oscillations in the medium. The less expensive method HMM correctly captures
the fluctuations for long-range oscillations and strongly amplifies their size
in media with short-range oscillations. We present a modified scheme with an
intermediate computational cost that captures the random fluctuations in all
cases.Comment: 41 page
Inverse Diffusion Theory of Photoacoustics
This paper analyzes the reconstruction of diffusion and absorption parameters
in an elliptic equation from knowledge of internal data. In the application of
photo-acoustics, the internal data are the amount of thermal energy deposited
by high frequency radiation propagating inside a domain of interest. These data
are obtained by solving an inverse wave equation, which is well-studied in the
literature. We show that knowledge of two internal data based on well-chosen
boundary conditions uniquely determines two constitutive parameters in
diffusion and Schroedinger equations. Stability of the reconstruction is
guaranteed under additional geometric constraints of strict convexity. No
geometric constraints are necessary when internal data for well-chosen
boundary conditions are available, where is spatial dimension. The set of
well-chosen boundary conditions is characterized in terms of appropriate
complex geometrical optics (CGO) solutions.Comment: 24 page
To Use or Not to Use: Nepal Samvat, the National Era of Nepal
This paper presents the importance of Nepal Samvat in Nepalese cultural life and compares it with Vikram Samvat, the official calendar of Nepal. Presenting a discussion on eras prevalent in Nepal, this paper examines the significance of the Nepal Government’s recent recognition of Nepal Samvat as the national calendar of Nepal. It presents a historical and cultural overview of the different eras and calendars that are in use in Nepal. It attempts to demonstrate a continuous historical legitimacy of Nepal Samvat, in contrast with Vikram Samvat, which is shown to be a fairly recent imposition associated with the Rana period, from 1903 onward. This article argues against any claims that the implementation of Nepal Samvat as an official calendar is impractical. In addition, I address the following issues: (1) how to adapt a lunar calendar to practical use, (2) how to coordinate it with governmental and business interests, and (3) how to coordinate it with external calendars (e.g. the Common Era or ‘Christian’ calendar)
Capillary-gravity wave transport over spatially random drift
We derive transport equations for the propagation of water wave action in the presence of a static, spatially random surface drift. Using the Wigner distribution \W(\x,\k,t) to represent the envelope of the wave amplitude at position \x contained in waves with wavevector \k, we describe surface wave transport over static flows consisting of two length scales; one varying smoothly on the wavelength scale, the other varying on a scale comparable to the wavelength. The spatially rapidly varying but weak surface flows augment the characteristic equations with scattering terms that are explicit functions of the correlations of the random surface currents. These scattering terms depend parametrically on the magnitudes and directions of the smoothly varying drift and are shown to give rise to a Doppler coupled scattering mechanism. The Doppler interaction in the presence of slowly varying drift modifies the scattering processes and provides a mechanism for coupling long wavelengths with short wavelengths. Conservation of wave action (CWA), typically derived for slowly varying drift, is extended to systems with rapidly varying flow. At yet larger propagation distances, we derive from the transport equations, an equation for wave energy diffusion. The associated diffusion constant is also expressed in terms of the surface flow correlations. Our results provide a formal set of equations to analyse transport of surface wave action, intensity, energy, and wave scattering as a function of the slowly varying drifts and the correlation functions of the random, highly oscillatory surface flows
Kinetic Limit for Wave Propagation in a Random Medium
We study crystal dynamics in the harmonic approximation. The atomic masses
are weakly disordered, in the sense that their deviation from uniformity is of
order epsilon^(1/2). The dispersion relation is assumed to be a Morse function
and to suppress crossed recollisions. We then prove that in the limit epsilon
to 0 the disorder averaged Wigner function on the kinetic scale, time and space
of order epsilon^(-1), is governed by a linear Boltzmann equation.Comment: 71 pages, 3 figure
Inverse anisotropic diffusion from power density measurements in two dimensions
This paper concerns the reconstruction of an anisotropic diffusion tensor
from knowledge of internal functionals
of the form with for
solutions of the elliptic equation on a two
dimensional bounded domain with appropriate boundary conditions. We show that
for I=4 and appropriately chosen boundary conditions, may uniquely and
stably be reconstructed from such internal functionals, which appear in
coupled-physics inverse problems involving the ultrasound modulation of
electrical or optical coefficients. Explicit reconstruction procedures for the
diffusion tensor are presented and implemented numerically.Comment: 27 pages, 6 figure
Phase Space Models for Stochastic Nonlinear Parabolic Waves: Wave Spread and Singularity
We derive several kinetic equations to model the large scale, low Fresnel
number behavior of the nonlinear Schrodinger (NLS) equation with a rapidly
fluctuating random potential. There are three types of kinetic equations the
longitudinal, the transverse and the longitudinal with friction. For these
nonlinear kinetic equations we address two problems: the rate of dispersion and
the singularity formation.
For the problem of dispersion, we show that the kinetic equations of the
longitudinal type produce the cubic-in-time law, that the transverse type
produce the quadratic-in-time law and that the one with friction produces the
linear-in-time law for the variance prior to any singularity.
For the problem of singularity, we show that the singularity and blow-up
conditions in the transverse case remain the same as those for the homogeneous
NLS equation with critical or supercritical self-focusing nonlinearity, but
they have changed in the longitudinal case and in the frictional case due to
the evolution of the Hamiltonian
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