15,276 research outputs found
Fast and accurate algorithms for the computation of spherically symmetric nonlocal diffusion operators on lattices
We present a unified treatment of the Fourier spectra of spherically
symmetric nonlocal diffusion operators. We develop numerical and analytical
results for the class of kernels with weak algebraic singularity as the
distance between source and target tends to . Rapid algorithms are derived
for their Fourier spectra with the computation of each eigenvalue independent
of all others. The algorithms are trivially parallelizable, capable of
leveraging more powerful compute environments, and the accuracy of the
eigenvalues is individually controllable. The algorithms include a Maclaurin
series and a full divergent asymptotic series valid for any spatial
dimensions. Using Drummond's sequence transformation, we prove linear
complexity recurrence relations for degree-graded sequences of numerators and
denominators in the rational approximations to the divergent asymptotic series.
These relations are important to ensure that the algorithms are efficient, and
also increase the numerical stability compared with the conventional algorithm
with quadratic complexity
Cauchy Problem of the non-self-adjoint Gauss-Laguerre semigroups and uniform bounds of generalized Laguerre polynomials
We propose a new approach to construct the eigenvalue expansion in a weighted
Hilbert space of the solution to the Cauchy problem associated to
Gauss-Laguerre invariant Markov semigroups that we introduce. Their generators
turn out to be natural non-self-adjoint and non-local generalizations of the
Laguerre differential operator. Our methods rely on intertwining relations that
we establish between these semigroups and the classical Laguerre semigroup and
combine with techniques based on non-harmonic analysis. As a by-product we also
provide regularity properties for the semigroups as well as for their heat
kernels. The biorthogonal sequences that appear in their eigenvalue expansion
can be expressed in terms of sequences of polynomials, and they generalize the
Laguerre polynomials. By means of a delicate saddle point method, we derive
uniform asymptotic bounds that allow us to get an upper bound for their norms
in weighted Hilbert spaces. We believe that this work opens a way to construct
spectral expansions for more general non-self-adjoint Markov semigroups.Comment: 33 page
Continuity of the von Neumann entropy
A general method for proving continuity of the von Neumann entropy on subsets
of positive trace-class operators is considered. This makes it possible to
re-derive the known conditions for continuity of the entropy in more general
forms and to obtain several new conditions. The method is based on a particular
approximation of the von Neumann entropy by an increasing sequence of concave
continuous unitary invariant functions defined using decompositions into finite
rank operators. The existence of this approximation is a corollary of a general
property of the set of quantum states as a convex topological space called the
strong stability property. This is considered in the first part of the paper.Comment: 42 pages, the minor changes have been made, the new applications of
the continuity condition have been added. To appear in Commun. Math. Phy
Existence and instability of steady states for a triangular cross-diffusion system: a computer-assisted proof
In this paper, we present and apply a computer-assisted method to study
steady states of a triangular cross-diffusion system. Our approach consist in
an a posteriori validation procedure, that is based on using a fxed point
argument around a numerically computed solution, in the spirit of the
Newton-Kantorovich theorem. It allows us to prove the existence of various non
homogeneous steady states for different parameter values. In some situations,
we get as many as 13 coexisting steady states. We also apply the a posteriori
validation procedure to study the linear stability of the obtained steady
states, proving that many of them are in fact unstable
Orthogonal polynomials of equilibrium measures supported on Cantor sets
We study the orthogonal polynomials associated with the equilibrium measure,
in logarithmic potential theory, living on the attractor of an Iterated
Function System. We construct sequences of discrete measures, that converge
weakly to the equilibrium measure, and we compute their Jacobi matrices via
standard procedures, suitably enhanced for the scope. Numerical estimates of
the convergence rate to the limit Jacobi matrix are provided, that show
stability and efficiency of the whole procedure. As a secondary result, we also
compute Jacobi matrices of equilibrium measures on finite sets of intervals,
and of balanced measures of Iterated Function Systems.
These algorithms can reach large orders: we study the asymptotic behavior of
the orthogonal polynomials and we show that they can be used to efficiently
compute Green's functions and conformal mappings of interest in constructive
function theory.Comment: 28 pages, 15 figure
Pseudo-Spectrum of the Resistive Magneto-hydrodynamics Operator: Resolving the Resistive Alfven Paradox
The `Alfv\'en Paradox' is that as resistivity decreases, the discrete
eigenmodes do not converge to the generalized eigenmodes of the ideal Alfv\'en
continuum. To resolve the paradox, the -pseudospectrum of the RMHD
operator is considered. It is proven that for any , the -
pseudospectrum contains the Alfv\'en continuum for sufficiently small
resistivity. Formal are constructed using the
formal Wentzel-Kramers-Brillouin-Jeffreys solutions, and it is shown that the
entire stable half-annulus of complex frequencies with
is resonant to order ,
i.e.~belongs to the . The resistive eigenmodes are
exponentially ill-conditioned as a basis and the condition number is
proportional to , where is the magnetic Reynolds
number
Isogeometric Least-squares Collocation Method with Consistency and Convergence Analysis
In this paper, we present the isogeometric least-squares collocation (IGA-L)
method, which determines the numerical solution by making the approximate
differential operator fit the real differential operator in a least-squares
sense. The number of collocation points employed in IGA-L can be larger than
that of the unknowns. Theoretical analysis and numerical examples presented in
this paper show the superiority of IGA-L over state-of-the-art collocation
methods. First, a small increase in the number of collocation points in IGA-L
leads to a large improvement in the accuracy of its numerical solution. Second,
IGA-L method is more flexible and more stable, because the number of
collocation points in IGA-L is variable. Third, IGA-L is convergent in some
cases of singular parameterization. Moreover, the consistency and convergence
analysis are also developed in this paper
A construction of two different solutions to an elliptic system
The paper aims at constructing two different solutions to an elliptic system
defined on the two dimensional torus.
It can be viewed as an elliptic regularization of the stationary Burgers 2D
system.
A motivation to consider the above system comes from an examination of
unusual propetries of the linear operator
arising from a linearization
of the equation about the dominant part of .
We argue that the skew-symmetric part of the operator provides in some sense
a smallness of norms of the linear operator inverse.
Our analytical proof is valid for a particular force and for , sufficiently large. The main steps of the proof concern
finite dimension approximation of the system and concentrate on analysis of
features of large matrices, which resembles standard numerical analysis. Our
analytical results are illustrated by numerical simulations
Non-linear eigenvalue problems and applications to photonic crystals
We establish new analytic and numerical results on a general class of
rational operator Nevanlinna functions that arise e.g. in modelling photonic
crystals. The capability of these dielectric nano-structured materials to
control the flow of light depends on specific features of their eigenvalues.
Our results provide a complete spectral analysis including variational
principles and two-sided estimates for all eigenvalues along with numerical
implementations. They even apply to multi-pole Lorentz models of permittivity
functions and to the eigenvalues between the poles where classical min-max
variational principles fail completely. In particular, we show that our
abstract two-sided eigenvalue estimates are optimal and we derive explicit
bounds on the band gap above a Lorentz pole. A high order finite element method
is used to compute the two-sided estimates of a selection of eigenvalues for
several concrete Lorentz models, e.g. polaritonic materials and multi-pole
models.Comment: 40 page
First-order limits, an analytical perspective
In this paper we present a novel approach to graph (and structural) limits
based on model theory and analysis. The role of Stone and Gelfand dualities is
displayed prominently and leads to a general theory, which we believe is
naturally emerging. This approach covers all the particular examples of
structural convergence and it put the whole in new context. As an application,
it leads to new intermediate examples of structural convergence and to a "grand
conjecture" dealing with sparse graphs. We survey the recent developments
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