Integral approximation by kernel smoothing

Let $(X_1,\ldots,X_n)$ be an i.i.d. sequence of random variables in $\mathbb{R}^d$, $d\geq 1$. We show that, for any function $\varphi :\mathbb{R}^d\rightarrow\mathbb{R}$, under regularity conditions, $$n^ {1/2}\Biggl(n^{-1}\sum_{i=1}^n\frac{\varphi(X_i)}{\widehat{f}^(X_i)}- \int \varphi(x)\,dx\Biggr)\stackrel{\mathbb{P}}{\longrightarrow}0,$$ where $\widehat{f}$ is the classical kernel estimator of the density of $X_1$. This result is striking because it speeds up traditional rates, in root $n$, derived from the central limit theorem when $\widehat{f}=f$. Although this paper highlights some applications, we mainly address theoretical issues related to the later result. We derive upper bounds for the rate of convergence in probability. These bounds depend on the regularity of the functions $\varphi$ and $f$, the dimension $d$ and the bandwidth of the kernel estimator $\widehat{f}$. Moreover, they are shown to be accurate since they are used as renormalizing sequences in two central limit theorems each reflecting different degrees of smoothness of $\varphi$. As an application to regression modelling with random design, we provide the asymptotic normality of the estimation of the linear functionals of a regression function. As a consequence of the above result, the asymptotic variance does not depend on the regression function. Finally, we debate the choice of the bandwidth for integral approximation and we highlight the good behavior of our procedure through simulations.Comment: Published at http://dx.doi.org/10.3150/15-BEJ725 in the Bernoulli (http://isi.cbs.nl/bernoulli/) by the International Statistical Institute/Bernoulli Society (http://isi.cbs.nl/BS/bshome.htm). arXiv admin note: text overlap with arXiv:1312.449

Front-tracking finite element methods for a void electro-stress migration problem

Continued research in electronic engineering technology has led to a miniaturisation of integrated circuits. Further reduction in the dimensions of the interconnects is impeded by the presence of small cracks or voids. Subject to high current and elastic stress, voids tend to drift and change shape in the interconnect, leading to a potential mechanical failure of the system. This thesis investigates the temporal evolution of voids moving along conductors, in the presence of surface diffusion, electric loading and elastic stress. We simulate a bulk-interface coupled system, with a moving interface governed by a fourth-order geometric evolution equation and a bulk where the electric potential and the displacement field are computed. We first give a general overview about geometric evolution equations, which define the motion of a hypersurface by prescribing its normal velocity in terms of geometric quantities. We briefly describe the three main approaches that have been proposed in the literature to solve numerically this class of equations, namely parametric approach, level set approach and phase field approach. We then present in detail two methods from the parametric approach category for the void electro-stress migration problem. We first introduce an unfitted method, where bulk and interface grids are totally independent, i.e. no topological compatibility between the two grids has to be enforced over time. We then discuss a fitted method, where the interface grid is at all times part of the boundary of the bulk grid. A detailed analysis, in terms of existence and uniqueness of the finite element solutions, experimental order of convergence (when the exact solution to the free boundary problem is known) and coupling operations (e.g., smoothing/remeshing of the grids, intersection between elements of the two grids), is carried out for both approaches. Several numerical simulations, both two- and three-dimensional, are performed in order to test the accuracy of the methods.Open Acces

Hacia la evaluación de los procesos de interactividad del Dispositivo Hipermedial Dinámico “Telares de la Memoria”

El artículo aborda una problemática clave de Investigación y Desarrollo que se estudia en el marco del Programa interdisciplinario “Dispositivos Hipermediales Dinámicos”, referida al seguimiento y evaluación de los procesos de interactividad observables en redes sociales con fines educativos, investigativos y/o de producción, mediatizadas a través de entornos web colaborativos. Luego de presentar el marco teórico y metodológico del Dispositivo Hipermedial Dinámico (DHD) y fundamentar el concepto de “Interactividad-DHD”, se exponen características de un desarrollo de software original denominado “SEPI-DHD” y su aplicación inicial al análisis de los procesos de interactividad virtual de “Telares de la Memoria”. El mencionado caso, que trata sobre la escritura abierta y colaborativa de la memoria colectiva plural, se desarrolla desde el 2010 en la comuna de Wheelwright (Santa Fe, Argentina) y promueve la apropiación activa del patrimonio cultural. Dicha experiencia físico-virtual, se concibe en su diseño, desarrollo e implementación como un proceso de aprendizaje emergente hacia la producción de “civitas”. Sobre lo realizado se concluye que, la utilización de “SEPI-DHD”, aporta datos cuali-cuantitativos válidos para efectuar una evaluación más integral de los procesos de interactividad del DHD “Telares de la Memoria” en función del sostenimiento de su crecimiento escalar y sustentabilidad participativa.IRICE, CONICET - UN

Numerical Quadrature for Singular Integrals on Fractals

We present and analyse numerical quadrature rules for evaluating regular and singular integrals on self-similar fractal sets. The integration domain Γ⊂R^{n} is assumed to be the compact attractor of an iterated function system of contracting similarities satisfying the open set condition. Integration is with respect to any “invariant” (also known as “balanced” or “self-similar”) measure supported on Γ, including in particular the Hausdorff measure H^{d} restricted to Γ, where d is the Hausdorff dimension of Γ. Both single and double integrals are considered. Our focus is on composite quadrature rules in which integrals over Γ are decomposed into sums of integrals over suitable partitions of Γ into self-similar subsets. For certain singular integrands of logarithmic or algebraic type, we show how in the context of such a partitioning the invariance property of the measure can be exploited to express the singular integral exactly in terms of regular integrals. For the evaluation of these regular integrals, we adopt a composite barycentre rule, which for sufficiently regular integrands exhibits second-order convergence with respect to the maximum diameter of the subsets. As an application we show how this approach, combined with a singularity-subtraction technique, can be used to accurately evaluate the singular double integrals that arise in Hausdorff-measure Galerkin boundary element methods for acoustic wave scattering by fractal screens

Quantum Control Landscapes

Numerous lines of experimental, numerical and analytical evidence indicate that it is surprisingly easy to locate optimal controls steering quantum dynamical systems to desired objectives. This has enabled the control of complex quantum systems despite the expense of solving the Schrodinger equation in simulations and the complicating effects of environmental decoherence in the laboratory. Recent work indicates that this simplicity originates in universal properties of the solution sets to quantum control problems that are fundamentally different from their classical counterparts. Here, we review studies that aim to systematically characterize these properties, enabling the classification of quantum control mechanisms and the design of globally efficient quantum control algorithms.Comment: 45 pages, 15 figures; International Reviews in Physical Chemistry, Vol. 26, Iss. 4, pp. 671-735 (2007

SuSpect: a Fortran Code for the Supersymmetric and Higgs Particle Spectrum in the MSSM

We present the Fortran code SuSpect version 2.3, which calculates the Supersymmetric and Higgs particle spectrum in the Minimal Supersymmetric Standard Model (MSSM). The calculation can be performed in constrained models with universal boundary conditions at high scales such as the gravity (mSUGRA), anomaly (AMSB) or gauge (GMSB) mediated breaking models, but also in the non-universal MSSM case with R-parity and CP conservation. Care has been taken to treat important features such as the renormalization group evolution of parameters between low and high energy scales, the consistent implementation of radiative electroweak symmetry breaking and the calculation of the physical masses of the Higgs bosons and supersymmetric particles taking into account the dominant radiative corrections. Some checks of important theoretical and experimental features, such as the absence of non desired minima, large fine-tuning in the electroweak symmetry breaking condition, as well as agreement with precision measurements can be performed. The program is user friendly, simple to use, self-contained and can easily be linked with other codes; it is rather fast and flexible, thus allowing scans of the parameter space with several possible options and choices for model assumptions and approximations.Comment: 44 pages, 1 figure. Program updated and text shortened. The program can be found at http://www.lpta.univ-montp2.fr/~kneur/Suspec