198 research outputs found
Generalized solutions and distributional shadows for Dirac equations
We discuss the application of recent results on generalized solutions to the
Cauchy problem for hyperbolic systems to Dirac equations with external fields.
In further analysis we focus on the question of existence of associated
distributional limits and derive their explicit form in case of free Dirac
fields with regularizations of initial values corresponding to point-like
probability densities
Generalized Qualification and Qualification Levels for Spectral Regularization Methods
The concept of qualification for spectral regularization methods for inverse
ill-posed problems is strongly associated to the optimal order of convergence
of the regularization error. In this article, the definition of qualification
is extended and three different levels are introduced: weak, strong and
optimal. It is shown that the weak qualification extends the definition
introduced by Mathe and Pereverzev in 2003, mainly in the sense that the
functions associated to orders of convergence and source sets need not be the
same. It is shown that certain methods possessing infinite classical
qualification, e.g. truncated singular value decomposition (TSVD), Landweber's
method and Showalter's method, also have generalized qualification leading to
an optimal order of convergence of the regularization error. Sufficient
conditions for a SRM to have weak qualification are provided and necessary and
sufficient conditions for a given order of convergence to be strong or optimal
qualification are found. Examples of all three qualification levels are
provided and the relationships between them as well as with the classical
concept of qualification and the qualification introduced by Mathe and
Perevezev are shown. In particular, spectral regularization methods having
extended qualification in each one of the three levels and having zero or
infinite classical qualification are presented. Finally several implications of
this theory in the context of orders of convergence, converse results and
maximal source sets for inverse ill-posed problems, are shown.Comment: 20 pages, 1 figur
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
The weakly coupled fractional one-dimensional Schr\"{o}dinger operator with index
We study fundamental properties of the fractional, one-dimensional Weyl
operator densely defined on the Hilbert space
and determine the asymptotic behaviour of
both the free Green's function and its variation with respect to energy for
bound states. In the sequel we specify the Birman-Schwinger representation for
the Schr\"{o}dinger operator
and extract the finite-rank portion which is essential for the asymptotic
expansion of the ground state. Finally, we determine necessary and sufficient
conditions for there to be a bound state for small coupling constant .Comment: 16 pages, 1 figur
Generalized Smoluchowski equation with correlation between clusters
In this paper we compute new reaction rates of the Smoluchowski equation
which takes into account correlations. The new rate K = KMF + KC is the sum of
two terms. The first term is the known Smoluchowski rate with the mean-field
approximation. The second takes into account a correlation between clusters.
For this purpose we introduce the average path of a cluster. We relate the
length of this path to the reaction rate of the Smoluchowski equation. We solve
the implicit dependence between the average path and the density of clusters.
We show that this correlation length is the same for all clusters. Our result
depends strongly on the spatial dimension d. The mean-field term KMFi,j = (Di +
Dj)(rj + ri)d-2, which vanishes for d = 1 and is valid up to logarithmic
correction for d = 2, is the usual rate found with the Smoluchowski model
without correlation (where ri is the radius and Di is the diffusion constant of
the cluster). We compute a new rate: the correlation rate K_{i,j}^{C}
(D_i+D_j)(r_j+r_i)^{d-1}M{\big(\frac{d-1}{d_f}}\big) is valid for d \leq
1(where M(\alpha) = \sum+\infty i=1i\alphaNi is the moment of the density of
clusters and df is the fractal dimension of the cluster). The result is valid
for a large class of diffusion processes and mass radius relations. This
approach confirms some analytical solutions in d 1 found with other methods. We
also show Monte Carlo simulations which illustrate some exact new solvable
models
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
A variational approach to strongly damped wave equations
We discuss a Hilbert space method that allows to prove analytical
well-posedness of a class of linear strongly damped wave equations. The main
technical tool is a perturbation lemma for sesquilinear forms, which seems to
be new. In most common linear cases we can furthermore apply a recent result
due to Crouzeix--Haase, thus extending several known results and obtaining
optimal analyticity angle.Comment: This is an extended version of an article appeared in
\emph{Functional Analysis and Evolution Equations -- The G\"unter Lumer
Volume}, edited by H. Amann et al., Birkh\"auser, Basel, 2008. In the latest
submission to arXiv only some typos have been fixe
Reconstructing initial data using observers: error analysis of the semi-discrete and fully discrete approximations
A new iterative algorithm for solving initial data inverse problems from partial observations has been recently proposed in Ramdani et al. (Automatica 46(10), 1616-1625, 2010 ). Based on the concept of observers (also called Luenberger observers), this algorithm covers a large class of abstract evolution PDE's. In this paper, we are concerned with the convergence analysis of this algorithm. More precisely, we provide a complete numerical analysis for semi-discrete (in space) and fully discrete approximations derived using finite elements in space and an implicit Euler method in time. The analysis is carried out for abstract Schrödinger and wave conservative systems with bounded observation (locally distributed)
Quantum Kinetic Evolution of Marginal Observables
We develop a rigorous formalism for the description of the evolution of
observables of quantum systems of particles in the mean-field scaling limit.
The corresponding asymptotics of a solution of the initial-value problem of the
dual quantum BBGKY hierarchy is constructed. Moreover, links of the evolution
of marginal observables and the evolution of quantum states described in terms
of a one-particle marginal density operator are established. Such approach
gives the alternative description of the kinetic evolution of quantum
many-particle systems to generally accepted approach on basis of kinetic
equations.Comment: 18 page
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