5,866 research outputs found
Positive trigonometric polynomials for strong stability of difference equations
We follow a polynomial approach to analyse strong stability of linear
difference equations with rationally independent delays. Upon application of
the Hermite stability criterion on the discrete-time homogeneous characteristic
polynomial, assessing strong stability amounts to deciding positive
definiteness of a multivariate trigonometric polynomial matrix. This latter
problem is addressed with a converging hierarchy of linear matrix inequalities
(LMIs). Numerical experiments indicate that certificates of strong stability
can be obtained at a reasonable computational cost for state dimension and
number of delays not exceeding 4 or 5
Discontinuous Galerkin finite element methods for time-dependent Hamilton--Jacobi--Bellman equations with Cordes coefficients
We propose and analyse a fully-discrete discontinuous Galerkin time-stepping
method for parabolic Hamilton--Jacobi--Bellman equations with Cordes
coefficients. The method is consistent and unconditionally stable on rather
general unstructured meshes and time-partitions. Error bounds are obtained for
both rough and regular solutions, and it is shown that for sufficiently smooth
solutions, the method is arbitrarily high-order with optimal convergence rates
with respect to the mesh size, time-interval length and temporal polynomial
degree, and possibly suboptimal by an order and a half in the spatial
polynomial degree. Numerical experiments on problems with strongly anisotropic
diffusion coefficients and early-time singularities demonstrate the accuracy
and computational efficiency of the method, with exponential convergence rates
under combined - and -refinement.Comment: 40 pages, 3 figures, submitted; extended version with supporting
appendi
Metastable energy strata in numerical discretizations of weakly nonlinear wave equations
The quadratic nonlinear wave equation on a one-dimensional torus with small
initial values located in a single Fourier mode is considered. In this
situation, the formation of metastable energy strata has recently been
described and their long-time stability has been shown. The topic of the
present paper is the correct reproduction of these metastable energy strata by
a numerical method. For symplectic trigonometric integrators applied to the
equation, it is shown that these energy strata are reproduced even on long time
intervals in a qualitatively correct way.Comment: 28 pages, 9 figure
Super-Resolution of Positive Sources: the Discrete Setup
In single-molecule microscopy it is necessary to locate with high precision
point sources from noisy observations of the spectrum of the signal at
frequencies capped by , which is just about the frequency of natural
light. This paper rigorously establishes that this super-resolution problem can
be solved via linear programming in a stable manner. We prove that the quality
of the reconstruction crucially depends on the Rayleigh regularity of the
support of the signal; that is, on the maximum number of sources that can occur
within a square of side length about . The theoretical performance
guarantee is complemented with a converse result showing that our simple convex
program convex is nearly optimal. Finally, numerical experiments illustrate our
methods.Comment: 31 page, 7 figure
A Levinson-Galerkin algorithm for regularized trigonometric approximation
Trigonometric polynomials are widely used for the approximation of a smooth
function from a set of nonuniformly spaced samples
. If the samples are perturbed by noise, controlling
the smoothness of the trigonometric approximation becomes an essential issue to
avoid overfitting and underfitting of the data. Using the polynomial degree as
regularization parameter we derive a multi-level algorithm that iteratively
adapts to the least squares solution of optimal smoothness. The proposed
algorithm computes the solution in at most operations (
being the polynomial degree of the approximation) by solving a family of nested
Toeplitz systems. It is shown how the presented method can be extended to
multivariate trigonometric approximation. We demonstrate the performance of the
algorithm by applying it in echocardiography to the recovery of the boundary of
the Left Ventricle
Equilibria of `Discrete' Integrable Systems and Deformations of Classical Orthogonal Polynomials
The Ruijsenaars-Schneider systems are `discrete' version of the
Calogero-Moser (C-M) systems in the sense that the momentum operator p appears
in the Hamiltonians as a polynomial in e^{\pm\beta' p} (\beta' is a deformation
parameter) instead of an ordinary polynomial in p in the hierarchies of C-M
systems. We determine the polynomials describing the equilibrium positions of
the rational and trigonometric Ruijsenaars-Schneider systems based on classical
root systems. These are deformation of the classical orthogonal polynomials,
the Hermite, Laguerre and Jacobi polynomials which describe the equilibrium
positions of the corresponding Calogero and Sutherland systems. The
orthogonality of the original polynomials is inherited by the deformed ones
which satisfy three-term recurrence and certain functional equations. The
latter reduce to the celebrated second order differential equations satisfied
by the classical orthogonal polynomials.Comment: 45 pages. A few typos in section 6 are correcte
Secular dynamics of a planar model of the Sun-Jupiter-Saturn-Uranus system; effective stability into the light of Kolmogorov and Nekhoroshev theories
We investigate the long-time stability of the Sun-Jupiter-Saturn-Uranus
system by considering a planar secular model, that can be regarded as a major
refinement of the approach first introduced by Lagrange. Indeed, concerning the
planetary orbital revolutions, we improve the classical circular approximation
by replacing it with a solution that is invariant up to order two in the
masses; therefore, we investigate the stability of the secular system for
rather small values of the eccentricities. First, we explicitly construct a
Kolmogorov normal form, so as to find an invariant KAM torus which approximates
very well the secular orbits. Finally, we adapt the approach that is at basis
of the analytic part of the Nekhoroshev's theorem, so as to show that there is
a neighborhood of that torus for which the estimated stability time is larger
than the lifetime of the Solar System. The size of such a neighborhood,
compared with the uncertainties of the astronomical observations, is about ten
times smaller.Comment: 31 pages, 2 figures. arXiv admin note: text overlap with
arXiv:1010.260
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