532 research outputs found
Non Parametric Models with Instrumental Variables
This paper gives a survey of econometric models characterized by a relation between observable and unobservable random elements where these unobservable terms are assumed to be independent of another set of observable variables called instrumental variables. This kind of specification is usefull to address the question of endogeneity or of selection bias for example. These models are treated non parametrically and in all the example we consider the functional parameter of interest is defined as the solution of a linear or non linear integral equation. The estimation procedure then requires to solve a (generally ill-posed) inverse problem. We illustrate the main questions (construction of the equation, identification, numerical solution, asymptotic properties, selection of the regularization parameter) by the different models we present.
Strong unicity of best approximations : a numerical aspect
The set of functions in C(T) which have a strongly unique best approximation from a given finite-dimensional subspace is denoted by SU(G). Since strong unicity plays an important role in numerical computations and since there the functions are only known up to some error, it is natural to ask what are the functions from the interior of SU(G). A complete characterization of those functions is given and the result is applied to weak Chebyshev and spline subspaces
A plethora of generalised solitary gravity-capillary water waves
The present study describes, first, an efficient algorithm for computing
capillary-gravity solitary waves solutions of the irrotational Euler equations
with a free surface and, second, provides numerical evidences of the existence
of an infinite number of generalised solitary waves (solitary waves with
undamped oscillatory wings). Using conformal mapping, the unknown fluid domain,
which is to be determined, is mapped into a uniform strip of the complex plane.
In the transformed domain, a Babenko-like equation is then derived and solved
numerically.Comment: 20 pages, 7 figures, 45 references. Other author's papers can be
downloaded at http://www.denys-dutykh.com
Global stabilization of a Korteweg-de Vries equation with saturating distributed control
This article deals with the design of saturated controls in the context of
partial differential equations. It focuses on a Korteweg-de Vries equation,
which is a nonlinear mathematical model of waves on shallow water surfaces. Two
different types of saturated controls are considered. The well-posedness is
proven applying a Banach fixed point theorem, using some estimates of this
equation and some properties of the saturation function. The proof of the
asymptotic stability of the closed-loop system is separated in two cases: i)
when the control acts on all the domain, a Lyapunov function together with a
sector condition describing the saturating input is used to conclude on the
stability, ii) when the control is localized, we argue by contradiction. Some
numerical simulations illustrate the stability of the closed-loop nonlinear
partial differential equation. 1. Introduction. In recent decades, a great
effort has been made to take into account input saturations in control designs
(see e.g [39], [15] or more recently [17]). In most applications, actuators are
limited due to some physical constraints and the control input has to be
bounded. Neglecting the amplitude actuator limitation can be source of
undesirable and catastrophic behaviors for the closed-loop system. The standard
method to analyze the stability with such nonlinear controls follows a two
steps design. First the design is carried out without taking into account the
saturation. In a second step, a nonlinear analysis of the closed-loop system is
made when adding the saturation. In this way, we often get local stabilization
results. Tackling this particular nonlinearity in the case of finite
dimensional systems is already a difficult problem. However, nowadays, numerous
techniques are available (see e.g. [39, 41, 37]) and such systems can be
analyzed with an appropriate Lyapunov function and a sector condition of the
saturation map, as introduced in [39]. In the literature, there are few papers
studying this topic in the infinite dimensional case. Among them, we can cite
[18], [29], where a wave equation equipped with a saturated distributed
actuator is studied, and [12], where a coupled PDE/ODE system modeling a
switched power converter with a transmission line is considered. Due to some
restrictions on the system, a saturated feedback has to be designed in the
latter paper. There exist also some papers using the nonlinear semigroup theory
and focusing on abstract systems ([20],[34],[36]). Let us note that in [36],
[34] and [20], the study of a priori bounded controller is tackled using
abstract nonlinear theory. To be more specific, for bounded ([36],[34]) and
unbounded ([34]) control operators, some conditions are derived to deduce, from
the asymptotic stability of an infinite-dimensional linear system in abstract
form, the asymptotic stability when closing the loop with saturating
controller. These articles use the nonlinear semigroup theory (see e.g. [24] or
[1]). The Korteweg-de Vries equation (KdV for short)Comment: arXiv admin note: text overlap with arXiv:1609.0144
Standard finite elements for the numerical resolution of the elliptic Monge-Ampere equation: Aleksandrov solutions
We prove a convergence result for a natural discretization of the Dirichlet
problem of the elliptic Monge-Ampere equation using finite dimensional spaces
of piecewise polynomial C0 or C1 functions. Standard discretizations of the
type considered in this paper have been previous analyzed in the case the
equation has a smooth solution and numerous numerical evidence of convergence
were given in the case of non smooth solutions. Our convergence result is valid
for non smooth solutions, is given in the setting of Aleksandrov solutions, and
consists in discretizing the equation in a subdomain with the boundary data
used as an approximation of the solution in the remaining part of the domain.
Our result gives a theoretical validation for the use of a non monotone finite
element method for the Monge-Amp\`ere equation
Non Parametric Models with Instrumental Variables
This paper gives a survey of econometric models characterized by a relation between observable and unobservable random elements where these unobservable terms are assumed to be independent of another set of observable variables called instrumental variables. This kind of specification is usefull to address the question of endogeneity or of selection bias for example. These models are treated non parametrically and in all the example we consider the functional parameter of interest is defined as the solution of a linear or non linear integral equation. The estimation procedure then requires to solve a (generally ill-posed) inverse problem. We illustrate the main questions (construction of the equation, identification, numerical solution, asymptotic properties, selection of the regularization parameter) by the different models we present
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