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DCDS
We consider the theory of correctors to homogenization in stationary
transport equations with rapidly oscillating, random coefficients.
Let ε << 1 be the ratio of the correlation length in the random
medium to the overall distance of propagation. As ε $ \downarrow
0$, we show that the heterogeneous transport solution is
well-approximated by a homogeneous transport solution. We then show
that the rescaled corrector converges in (probability) distribution
and weakly in the space and velocity variables, to a Gaussian
process as an application of a central limit result. The latter
result requires strong assumptions on the statistical structure of
randomness and is proved for random processes constructed by
means of a Poisson point process.
IPI
This paper concerns the reconstruction of a diffusion coefficient in an elliptic equation from knowledge of several power densities. The power density is the product of the diffusion coefficient with the square of the modulus of the gradient of the elliptic solution. The derivation of such internal functionals comes from perturbing the medium of interest by acoustic (plane) waves, which results in small changes in the diffusion coefficient. After appropriate asymptotic expansions and (Fourier) transformation, this allow us to construct the power density of the equation point-wise inside the domain. Such a setting finds applications in ultrasound modulated electrical impedance tomography and ultrasound modulated optical tomography.
We show that the diffusion coefficient can be uniquely and stably reconstructed from knowledge of a sufficient large number of power densities. Explicit expressions for the reconstruction of the diffusion coefficient are also provided. Such results hold for a large class of boundary conditions for the elliptic equation in the two-dimensional setting. In three dimensions, the results are proved for a more restrictive class of boundary conditions constructed by means of complex geometrical optics solutions.
We show that the diffusion coefficient can be uniquely and stably reconstructed from knowledge of a sufficient large number of power densities. Explicit expressions for the reconstruction of the diffusion coefficient are also provided. Such results hold for a large class of boundary conditions for the elliptic equation in the two-dimensional setting. In three dimensions, the results are proved for a more restrictive class of boundary conditions constructed by means of complex geometrical optics solutions.
IPI
We study the stability of the reconstruction of the scattering and
absorption coefficients in a stationary linear transport equation
from knowledge of the full albedo operator in dimension $n\geq3$.
The albedo operator is defined as the mapping from the incoming
boundary conditions to the outgoing transport solution at the
boundary of a compact and convex domain. The uniqueness of the
reconstruction was proved in [2, 3]
and partial stability estimates were obtained in [12]
for spatially independent scattering coefficients. We generalize
these results and prove an $L^1$-stability estimate for spatially
dependent scattering coefficients.
IPI
This paper concerns the reconstruction of a scalar diffusion coefficient $\sigma(x)$ from redundant functionals of the form $H_i(x)=\sigma^{2\alpha}(x)|\nabla u_i|^2(x)$ where $\alpha\in\mathbb{R}$ and $u_i$ is a solution of the elliptic problem $\nabla\cdot \sigma \nabla u_i=0$ for $1\leq i\leq I$. The case $\alpha=\frac12$ is used to model measurements obtained from modulating a domain of interest by ultrasound and finds applications in ultrasound modulated electrical impedance tomography (UMEIT), ultrasound modulated optical tomography (UMOT) as well as impedance acoustic computerized tomography (ImpACT). The case $\alpha=1$ finds applications in Magnetic Resonance Electrical Impedance Tomography (MREIT).
We present two explicit reconstruction procedures of $\sigma$ for appropriate choices of $I$ and of traces of $u_i$ at the boundary of a domain of interest. The first procedure involves the solution of an over-determined system of ordinary differential equations and generalizes to the multi-dimensional case and to (almost) arbitrary values of $\alpha$ the results obtained in two and three dimensions in [10] and [5], respectively, in the case $\alpha=\frac12$. The second procedure consists of solving a system of linear elliptic equations, which we can prove admits a unique solution in specific situations.
We present two explicit reconstruction procedures of $\sigma$ for appropriate choices of $I$ and of traces of $u_i$ at the boundary of a domain of interest. The first procedure involves the solution of an over-determined system of ordinary differential equations and generalizes to the multi-dimensional case and to (almost) arbitrary values of $\alpha$ the results obtained in two and three dimensions in [10] and [5], respectively, in the case $\alpha=\frac12$. The second procedure consists of solving a system of linear elliptic equations, which we can prove admits a unique solution in specific situations.
DCDS
We analyze the refocusing properties of time reversed waves that
propagate in two different media during the forward and backward
stages of a time-reversal experiment. We consider two regimes of
wave propagation modeled by the paraxial wave equation with a smooth
random refraction coefficient and the Itô-Schrödinger equation,
respectively. In both regimes, we rigorously characterize the
refocused signal in the high frequency limit and show that it is
statistically stable, that is, independent of the realizations of
the two media. The analysis is based on a characterization of the
high frequency limit of the Wigner transform of two fields
propagating in different media.
The refocusing quality of the backpropagated signal is determined by the cross correlation of the two media. When the two media decorrelate, two distinct de-focusing effects are observed. The first one is a purely absorbing effect due to the loss of coherence at a fixed frequency. The second one is a phase modulation effect of the refocused signal at each frequency. This causes de-focusing of the backpropagated signal in the time domain.
The refocusing quality of the backpropagated signal is determined by the cross correlation of the two media. When the two media decorrelate, two distinct de-focusing effects are observed. The first one is a purely absorbing effect due to the loss of coherence at a fixed frequency. The second one is a phase modulation effect of the refocused signal at each frequency. This causes de-focusing of the backpropagated signal in the time domain.
IPI
Inverse transport theory concerns the reconstruction of the absorption and scattering coefficients in a transport equation from knowledge of the albedo operator, which models all possible boundary measurements. Uniqueness and stability results are well known and are typically obtained for errors of the albedo operator measured in the $L^1$ sense. We claim that such error estimates are not always very informative. For instance, arbitrarily small blurring and misalignment of detectors result in $O(1)$ errors of the albedo operator and hence in $O(1)$ error predictions on the reconstruction of the coefficients, which are not useful.
This paper revisit such stability estimates by introducing a more forgiving metric on the measurements errors, namely the $1-$Wasserstein distances, which penalize blurring or misalignment by an amount proportional to the width of the blurring kernel or to the amount of misalignment. We obtain new stability estimates in this setting.
We also consider the effect of errors, still measured in the $1-$ Wasserstein distance, on the generation of the probing source. This models blurring and misalignment in the design of (laser) probes and allows us to consider discretized sources. Under appropriate assumptions on the coefficients, we quantify the effect of such errors on the reconstructions.
keywords:
Linear transport
,
inverse problems
,
stability estimates
,
Wasserstein distance
,
albedo operator
IPI
We introduce a technique for recovering a sufficiently smooth function from its ray transforms over rotationally related curves in the unit disc of 2-dimensional Euclidean space. The method is based on a complexification of the underlying vector fields defining the initial transport and inversion formulae are then given in a unified form. The method is used to analyze the attenuated ray transform in the same setting.
KRM
Kinetic equations are often appropriate to model the energy density
of high frequency waves propagating in highly heterogeneous media.
The limitations of the kinetic model are quantified by the
statistical instability of the wave energy density, i.e., by its
sensitivity to changes in the realization of the underlying
heterogeneous medium modeled as a random medium. In the simplified
Itô-Schrödinger regime of wave propagation, we obtain optimal
estimates for the statistical instability of the wave energy density
for different configurations of the source terms and the domains
over which the energy density is measured. We show that the energy
density is asymptotically statistically stable (self-averaging) in
many configurations. In the case of highly localized source terms,
we obtain an explicit asymptotic expression for the scintillation
function in the high frequency limit.
IPI
This paper concerns the reconstruction of a complex-valued anisotropic tensor $\gamma = \sigma + \iota\omega\varepsilon$ from knowledge of several internal magnetic fields $H$, where $H$ satisfies the anisotropic Maxwell system on a bounded domain with prescribed boundary conditions. We show that $\gamma$ can be uniquely reconstructed with a loss of two derivatives from errors in the acquisition of $H$. A minimum number of $6$ such functionals is sufficient to obtain a local reconstruction of $\gamma$ in dimension three provided that the electric field satisfies appropriate boundary conditions. When $\gamma$ is close to a scalar tensor, such boundary conditions are shown to exist using the notion of complex geometric optics (CGO) solutions. For arbitrary symmetric tensors $\gamma$, a Runge approximation property is used instead to obtain partial results. This problem finds applications in the medical imaging modalities Current Density Imaging and Magnetic Resonance Electrical Impedance Tomography.
DCDS-B
We consider the homogenization of the wave equation with high
frequency initial conditions propagating in a medium with highly
oscillatory random coefficients. By appropriate mixing assumptions
on the random medium, we obtain an error estimate between the exact
wave solution and the homogenized wave solution in the energy norm.
This allows us to consider the limiting behavior of the energy
density of high frequency waves propagating in highly heterogeneous
media when the wavelength is much larger than the correlation length
in the medium.
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