Rigorous Computation for DifferentialEquation Problems
Org:
Roberto Castelli (VU Amsterdam) and
Holger Teismann (Acadia)
[
PDF]
 ROBERTO CASTELLI, VU University Amsterdam
Parameterization of invariant manifolds for periodic orbits of vector fields [PDF]

Invariant sets and connecting orbits provide several information about the long term behaviour of nonlinear system.
If the invariant sets are hyperbolic, the
connecting orbits may be found in the intersection of stable and unstable manifolds.
In this talk we consider periodic orbits as invariant sets and we present an efficient
numerical method for computing FourierTaylor expansion of invariant stable/unstable manifolds.
A fundamental ingredient in our construction is the Floquet theory. Indeed by continuously
exploiting the Floquet normal form, the computation of the FourierTaylor coefficients
results in solving algebraic equations with constant coefficients. The technique does not
require any rigorous integration and permits fast computation of the parameterisation up to
any desired order also for high dimensional manifolds. Moreover the method is well suited
for a subsequent rigorous validation of the computed parametrisation.
Joint work with JeanPhilippe Lessard and Jason D. Mireles James.
 JACEK CYRANKA, University of Warsaw
A construction of two different solutions to an elliptic system [PDF]

Joint work with P.B. Mucha from University of Warsaw. We construct two different solutions to an elliptic system
$$
u \cdot \nabla u + (\Delta)^m u = \lambda F
$$
defined on the two dimensional torus. Here $u=(u^1,u^2)$ is sought as a vector function.
The operator $(\Delta)^m$ is elliptic homogenous of order $2m$.
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
$$
\lambda \sin y \partial_x w + (\Delta)^{m} w.
$$
Roughly speaking the term with $\lambda$ effects in a special stabilization of the norms of the operator.
We shall underline that the special features of this operator were found firstly via numerical analysis.
Our proof is valid for a particular force $F$ and for $\lambda > \lambda_0$, $m> m_0$ 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. Experiments are agreed with the conjecture :
for small $m$, in particular for $m=1$  for the classical Burgers equation with diffusion,
the system does not admit solutions for large $\lambda$.
Jacek Cyranka, Piotr Bogusław Mucha, \emph{A construction of two different solutions to an elliptic system}, arXiv:1502.03363.
 ANDRÉA DESCHÊNES, Université Laval
Stationary coexistence of hexagons and rolls via rigorous computations [PDF]

Over the past few decades, the spontaneous formation of patterns such as spatially periodic rolls, hexagonal cell structures, and spiral waves in spatially extended systems has attracted much attention. In the context of the modified SwiftHohenberg PDE, some of these interesting interfaces can be modelled as stationary fronts between rolls and hexagonal patterns. Via the appropriate change of coordinates introduced in [Doelman, Sandstede, Scheel and Schneider. European J. of Appl. Math. 14 (1), 2003], it is known that computing the stationary fronts reduces to computing heteroclinic orbits between equilibria of a given system of second order ODEs. In this talk, we introduce the computational method that has been used to prove existence of some of these connecting orbits, hence leading to rigorous statements about coexistence of different types of non trivial patterns for the original PDE. This rigorous method combines Chebyshev series, the parameterization method of invariant manifold, fixed point theory and interval arithmetics. This is a joint work with J.B. van den Berg, J.P. Lessard and J.D. Mireles James.
 MIOARA JOLDES, LAASCNRS, Toulouse, France
A Fast and Accurate Power Series Expansion Method to Compute the Probability of Collision for Shortterm Space Encounters [PDF]

We present a new method for computing the probability of collision between two spherical space objects involved in a shortterm encounter under Gaussiandistributed uncertainty. In this model of conjunction, classical assumptions reduce the probability of collision to the integral of a twodimensional Gaussian probability density function over a disk. We derive an analytic expression for the integral, in the form of a product between an exponential term and a convergent power series
with positive coefficients. For this we use Laplace transform and algorithmic properties of Dfinite functions (which are solutions of linear differential equations with polynomial coefficients). Moreover, rigorous analytic bounds on the truncation errors are also derived. This results in a reliable, accurate and efficient algorithm for the risk evaluation. This is a joint work with R. Serra, D. Arzelier, J.B. Lasserre, A.Rondepierre and B. Salvy.
 TOMASZ KACZYNSKI, Université de Sherbrooke
Towards a formal tie between combinatorial and classical vector field dynamics [PDF]

The Forman’s discrete Morse theory is an analogy of the classical Morse theory with, so far, only informal ties. Our goal is to establish a formal tie on the level of induced dynamics. Following the Forman’s 1998 paper on “Combinatorial vector fields and dynamical systems”, we start with a possibly nongradient combinatorial vector field. We construct a flowlike upper semicontinuous acyclicvalued mapping whose dynamics is equivalent to the dynamics of the Forman’s combinatorial vector field, in the sense that isolated invariant sets and index pairs are in onetoone correspondence.
This is a joint work with M. Mrozek and Th. Wanner.
 CHRISTIAN REINHARDT, VU University Amsterdam
Rigorous numerics for nonlinear ODEs using Chebyshev series [PDF]

In this talk we present a rigorous numerical method to solve initial and boundary value problems for nonlinear ODEs based on Chebyshev series. Our method results in a numerical approximate solution of the ODE together with mathematically rigorous error bounds. The main idea of our proposed approach is to first expand the solution of a given differential equation using its classical Chebyshev series, plug the expansion in the equation and obtain an equivalent infinite dimensional problem of the form $f(x) = 0$ to solve in a Banach space of rapidly decaying Chebyshev coefficients. Via a fixed point argument, we obtain the existence of a genuine solution of $f(x) = 0$ nearby a numerical approximate zero of a finite dimensional projection of f. The NewtonKantorovich type argument is carried out by using the radii polynomials, which provide an efficient way of constructing a set on which the contraction mapping theorem is applicable. We illustrate the method with examples.
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