879 research outputs found
Normal form for travelling kinks in discrete Klein-Gordon lattices
We study travelling kinks in the spatial discretizations of the nonlinear
Klein--Gordon equation, which include the discrete lattice and the
discrete sine--Gordon lattice. The differential advance-delay equation for
travelling kinks is reduced to the normal form, a scalar fourth-order
differential equation, near the quadruple zero eigenvalue. We show numerically
non-existence of monotonic kinks (heteroclinic orbits between adjacent
equilibrium points) in the fourth-order equation. Making generic assumptions on
the reduced fourth-order equation, we prove the persistence of bounded
solutions (heteroclinic connections between periodic solutions near adjacent
equilibrium points) in the full differential advanced-delay equation with the
technique of center manifold reduction. Existence and persistence of multiple
kinks in the discrete sine--Gordon equation are discussed in connection to
recent numerical results of \cite{ACR03} and results of our normal form
analysis
Periodic travelling waves in convex Klein-Gordon chains
We study Klein-Gordon chains with attractive nearest neighbour forces and
convex on-site potential, and show that there exists a two-parameter family of
periodic travelling waves (wave trains) with unimodal and even profile
functions. Our existence proof is based on a saddle-point problem with
constraints and exploits the invariance properties of an improvement operator.
Finally, we discuss the numerical computation of wave trains.Comment: 12 pages, 3 figure
Normalizations with exponentially small remainders for nonautonomous analytic periodic vector fields
In this paper we deal with analytic nonautonomous vector fields with a
periodic time-dependancy, that we study near an equilibrium point. In a first
part, we assume that the linearized system is split in two invariant subspaces
E0 and E1. Under light diophantine conditions on the eigenvalues of the linear
part, we prove that there is a polynomial change of coordinates in E1 allowing
to eliminate up to a finite polynomial order all terms depending only on the
coordinate u0 of E0 in the E1 component of the vector field. We moreover show
that, optimizing the choice of the degree of the polynomial change of
coordinates, we get an exponentially small remainder. In the second part, we
prove a normal form theorem with exponentially small remainder. Similar
theorems have been proved before in the autonomous case : this paper
generalizes those results to the nonautonomous periodic case
Global sensitivity analysis of computer models with functional inputs
Global sensitivity analysis is used to quantify the influence of uncertain
input parameters on the response variability of a numerical model. The common
quantitative methods are applicable to computer codes with scalar input
variables. This paper aims to illustrate different variance-based sensitivity
analysis techniques, based on the so-called Sobol indices, when some input
variables are functional, such as stochastic processes or random spatial
fields. In this work, we focus on large cpu time computer codes which need a
preliminary meta-modeling step before performing the sensitivity analysis. We
propose the use of the joint modeling approach, i.e., modeling simultaneously
the mean and the dispersion of the code outputs using two interlinked
Generalized Linear Models (GLM) or Generalized Additive Models (GAM). The
``mean'' model allows to estimate the sensitivity indices of each scalar input
variables, while the ``dispersion'' model allows to derive the total
sensitivity index of the functional input variables. The proposed approach is
compared to some classical SA methodologies on an analytical function. Lastly,
the proposed methodology is applied to a concrete industrial computer code that
simulates the nuclear fuel irradiation
Small divisor problem in the theory of three-dimensional water gravity waves
We consider doubly-periodic travelling waves at the surface of an infinitely
deep perfect fluid, only subjected to gravity and resulting from the
nonlinear interaction of two simply periodic travelling waves making an angle
between them. \newline Denoting by the dimensionless
bifurcation parameter ( is the wave length along the direction of the
travelling wave and is the velocity of the wave), bifurcation occurs for
. For non-resonant cases, we first give a large family of
formal three-dimensional gravity travelling waves, in the form of an expansion
in powers of the amplitudes of two basic travelling waves. "Diamond waves" are
a particular case of such waves, when they are symmetric with respect to the
direction of propagation.\newline \emph{The main object of the paper is the
proof of existence} of such symmetric waves having the above mentioned
asymptotic expansion. Due to the \emph{occurence of small divisors}, the main
difficulty is the inversion of the linearized operator at a non trivial point,
for applying the Nash Moser theorem. This operator is the sum of a second order
differentiation along a certain direction, and an integro-differential operator
of first order, both depending periodically of coordinates. It is shown that
for almost all angles , the 3-dimensional travelling waves bifurcate
for a set of "good" values of the bifurcation parameter having asymptotically a
full measure near the bifurcation curve in the parameter plane Comment: 119
Uncertainty and sensitivity analysis of functional risk curves based on Gaussian processes
A functional risk curve gives the probability of an undesirable event as a
function of the value of a critical parameter of a considered physical system.
In several applicative situations, this curve is built using phenomenological
numerical models which simulate complex physical phenomena. To avoid cpu-time
expensive numerical models, we propose to use Gaussian process regression to
build functional risk curves. An algorithm is given to provide confidence
bounds due to this approximation. Two methods of global sensitivity analysis of
the models' random input parameters on the functional risk curve are also
studied. In particular, the PLI sensitivity indices allow to understand the
effect of misjudgment on the input parameters' probability density functions
Global Sensitivity Analysis of Stochastic Computer Models with joint metamodels
The global sensitivity analysis method, used to quantify the influence of
uncertain input variables on the response variability of a numerical model, is
applicable to deterministic computer code (for which the same set of input
variables gives always the same output value). This paper proposes a global
sensitivity analysis methodology for stochastic computer code (having a
variability induced by some uncontrollable variables). The framework of the
joint modeling of the mean and dispersion of heteroscedastic data is used. To
deal with the complexity of computer experiment outputs, non parametric joint
models (based on Generalized Additive Models and Gaussian processes) are
discussed. The relevance of these new models is analyzed in terms of the
obtained variance-based sensitivity indices with two case studies. Results show
that the joint modeling approach leads accurate sensitivity index estimations
even when clear heteroscedasticity is present
Latin hypercube sampling with inequality constraints
In some studies requiring predictive and CPU-time consuming numerical models,
the sampling design of the model input variables has to be chosen with caution.
For this purpose, Latin hypercube sampling has a long history and has shown its
robustness capabilities. In this paper we propose and discuss a new algorithm
to build a Latin hypercube sample (LHS) taking into account inequality
constraints between the sampled variables. This technique, called constrained
Latin hypercube sampling (cLHS), consists in doing permutations on an initial
LHS to honor the desired monotonic constraints. The relevance of this approach
is shown on a real example concerning the numerical welding simulation, where
the inequality constraints are caused by the physical decreasing of some
material properties in function of the temperature
Derivative-based global sensitivity measures: general links with Sobol' indices and numerical tests
The estimation of variance-based importance measures (called Sobol' indices)
of the input variables of a numerical model can require a large number of model
evaluations. It turns to be unacceptable for high-dimensional model involving a
large number of input variables (typically more than ten). Recently, Sobol and
Kucherenko have proposed the Derivative-based Global Sensitivity Measures
(DGSM), defined as the integral of the squared derivatives of the model output,
showing that it can help to solve the problem of dimensionality in some cases.
We provide a general inequality link between DGSM and total Sobol' indices for
input variables belonging to the class of Boltzmann probability measures, thus
extending the previous results of Sobol and Kucherenko for uniform and normal
measures. The special case of log-concave measures is also described. This link
provides a DGSM-based maximal bound for the total Sobol indices. Numerical
tests show the performance of the bound and its usefulness in practice
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