Intégrale de Gauss | EPFL Graph Search

En mathématiques, une intégrale de Gauss est l'intégrale d'une fonction gaussienne sur l'ensemble des réels. Sa valeur est reliée à la constante π par la formule : \int_{-\infty}^{+\infty} \mathrm e^{-\alpha, x^2}\mathrm d x=\sqrt{\frac{\pi}{\alpha}}, où α est un paramètre réel strictement positif. Elle intervient dans la définition de la loi de probabilité appelée loi gaussienne, ou loi normale. Cette formule peut être obtenue grâce à une intégrale double et un changement de variable polaire. Sa première démonstration connue est donnée par Pierre-Simon de Laplace. Ainsi on a par exemple, avec les notations classiques : :\int_{-\infty}^{+\infty}\frac1{|\sigma|\sqrt{2\pi}}\mathrm e^{-\frac{(x-\mu)^2}{2\sigma^2}} \mathrm d x=1. Si l'on travaille à n dimensions, la formule se généralise sous la forme suivante : : \int_{\R^n} \mathrm e^{-\alpha,|x|^2} \mathrm d x=\left(\frac{\pi}{\alpha}\right)^{\frac n2}\text{ avec }x= (x_1,\dots,x_n)\text{ et }|x\

Contribution to Integral Equation Techniques for Solving Electromagnetic Problems

Sergio Lopez Pena

Surface Mixed Potential Integral Equation (MPIE) formulations together with the Method of Moments (MoM) are widely used to solve electromagnetic problems. An accurate evaluation of the Green functions (GF) associated to the integral equation and of the coupling integrals needed to fill the MoM matrix are the cornerstone steps in the implementation of integral equation algorithms. This thesis is mainly focused on these two topics. The main intended application of our MPIE-MoM formulation is the analysis of enclosed structures, with the shield being materialized by rectangular cavities with perfect conducting (PEC) walls. GFs for rectangular cavities constitute a classic research topic, where there is still a lot of room for improvements. In this area, three main original results are presented in this thesis. Firstly, the exponential convergence of the modal series is ensured via a sophisticated coordinate permutation method. In second place, a study which allows setting the relationship between cavity resonances, excited modes and GF components' singularities, is fully developed. Finally, a novel hybrid method, to compute the GF static part is introduced. This method combines in a new original way both, the modal and image expansions of the cavity GFs. The discretization of the MPIE via the method of moments leads to a matrix equation. In the Galerkin version of the MoM, the matrix elements are given by four-dimensional integrals over source and observer surface domains of the GFs multiplied by some basis and test functions. These so-called coupling integrals invoke the integration of the GF singularity, which in the MPIE case is of the weak type (1/R). The accurate integration of this singularity is a very challenging topic, which has been tackled following many different strategies. Here, the closed analytical expressions of the 4D integral over rectangular domains of this singularity are presented. The problem related to the integration of the GF singularity on arbitrary shaped domains is solved through a hybrid numerical-analytical technique based on an original integral transformation and using by the first time double exponential (DE) numerical integration rules. The thesis concludes with several numerical examples and benchmarks of practical interest. They ascertain the validity of strategies, concepts and results of this thesis and they strongly hint to the development of future competitive computer tools.

EPFL2010

Feynman-Kac representation for the parabolic Anderson model driven by fractional noise

Kamran Kalbasi

We consider the parabolic Anderson model driven by fractional noise: partial derivative/partial derivative t u(t,x) = k Delta u(t,x) + u(t,x)partial derivative/partial derivative t W(t,x) x is an element of Z(d), t >= 0, where k > 0 is a diffusion constant, Delta is the discrete Laplacian defined by Delta f (x) = 1/2d Sigma vertical bar y-x vertical bar=1 (f(Y) - f (0)), and {W(t,x) ; t > 0} x is an element of Z(d) is a family of independent fractional Brownian motions with Hurst parameter H is an element of (0,1), indexed by Z(d). We make sense of this equation via a Stratonovich integration obtained by approximating the fractional Brownian motions with a family of Gaussian processes possessing absolutely continuous sample paths. We prove that the Feynman-Kac representation u(t, x) = E-infinity [u(0) (X(t)) exp integral(t)(0) W(ds,X(t-s))], (1) is a mild solution to this problem. Here u(0) (y) is the initial value at site y is an element of Z(d), {X(t); t >= 0} is a simple random walk with jump rate f, started at x E Zd and independent of the family {W(t, x) ; t >= 0}(x is an element of Zd) and E-infinity is expectation withrespect to this random walk. We give a unified argument that works for any Hurst parameter H is an element of (0,1). (C) 2015 Elsevier Inc. All rights reserved.

Academic Press Inc Elsevier Science2015

Network Formulations of Mixed-Integer Programs

Marco Di Summa, Friedrich Eisenbrand

We consider mixed-integer sets described by system of linear inequalities in which the constraint matrix A is totally unimodular; the right-hand side is arbitrary vector; and a subset of the variables is required to be integer. We show that the problem of checking nonemptiness of a set of this type is NP-complete, even in the case in which the linear system describes mixed-integer network flows with half-integral requirement on the nodes.

INFORMS2009

Contribution to Integral Equation Techniques for Solving Electromagnetic Problems

Sergio Lopez Pena

EPFL2010