Partie bornée | EPFL Graph Search

En mathématiques, la notion de partie bornée (ou, par raccourci, de borné) étend celle d'intervalle borné de réels à d'autres structures, notamment en topologie et en théorie des ordres. Selon les cas, la définition privilégie l'existence de bornes ponctuelles ou la négation de l'éloignement à l'infini. Une fonction bornée est une fonction dont l' est bornée dans l'ensemble d'arrivée. Un opérateur borné est un opérateur linéaire dont les images de bornés sont bornées également. Dans le cadre des espaces vectoriels normés, cette définition est équivalente à celle d'opérateur continu. La donnée de parties bornées sur un ensemble indépendamment de toute autre structure est appelée . Topologie Dans un espace métrique Une partie d'un espace métrique est dite bornée si la distance entre ses points est majorée par un réel fixé, autrement dit si son diamètre est fini. En ce sens, les bornés de la droite réelle sont bien les parties majorées et minorées, c'est-à-dire

An Adaptive Sparse Grid Algorithm for Elliptic PDEs with Lognormal Diffusion Coefficient

Fabio Nobile, Lorenzo Tamellini, Francesco Tesei

In this work we build on the classical adaptive sparse grid algorithm (T. Gerstner and M. Griebel, Dimension-adaptive tensor-product quadrature), obtaining an enhanced version capable of using non-nested collocation points, and supporting quadrature and interpolation on unbounded sets. We also consider several profit indicators that are suitable to drive the adaptation process. We then use such algorithm to solve an important test case in Uncertainty Quantification problem, namely the Darcy equation with lognormal permeability random field, and compare the results with those obtained with the quasi-optimal sparse grids based on profit estimates, which we have proposed in our previous works (cf. e.g. Convergence of quasi-optimal sparse grids approximation of Hilbert-valued functions: application to random elliptic PDEs). To treat the case of rough permeability fields, in which a sparse grid approach may not be suitable, we propose to use the adaptive sparse grid quadrature as a control variate in a Monte Carlo simulation. Numerical results show that the adaptive sparse grids have performances similar to those of the quasi-optimal sparse grids and are very effective in the case of smooth permeability fields. Moreover, their use as control variate in a Monte Carlo simulation allows to tackle efficiently also problems with rough coefficients, significantly improving the performances of a standard Monte Carlo scheme.

Springer2016

MATHICSE Technical Report : An adaptive sparse grid algorithm for elliptic PDEs with lognormal diffusion coefficient

Fabio Nobile, Lorenzo Tamellini, Francesco Tesei

In this work we build on the classical adaptive sparse grid algorithm (T. Gerstner and M. Griebel, Dimension-adaptive tensor-product quadrature), obtaining an enhanced version capable of using non-nested collocation points, and supporting quadrature and interpolation on unbounded sets. We also consider several profit indicators that are suitable to drive the adaptation process.We then use such algorithm to solve an important test case in Uncertainty Quantification problem, namely the Darcy equation with lognormal permeability random field, and compare the results with those obtained with the quasi-optimal sparse grids based on profit estimates, which we have proposed in our previous works (cf. e.g. Convergence of quasi-optimal sparse grids approximation of Hilbert-valued functions: application to random elliptic PDEs). To treat the case of rough permeability fields, in which a sparse grid approach may not be suitable, we propose to use the adaptive sparse grid quadrature as a control variate in a Monte Carlo simulation. Numerical results show that the adaptive sparse grids have performances similar to those of the quasi-optimal sparse grids and are very effective in the case of smooth permeability fields. Moreover, their use as control variate in a Monte Carlo simulation allows to tackle efficiently also problems with rough coefficients, significantly improving the performances of a standard Monte Carlo scheme.

MATHICSE2015

MATHICSE Technical Report : Quasi-Toeplitz matrix arithmetic : a Matlab toolbox

Stefano Massei

A Quasi Toeplitz (QT) matrix is a semi-infinite matrix of the kind

A = T(a) + E

where

T(a) = (aj-i)_{i,j\in \mathbb{Z}_{+}}

E = (e_{i,j})_{i,j\in\mathbb{Z}_{+}}

is compact and the norms

\| a\|_{\mathcal{W}} =\sum_{i\in\mathbb{Z}} |a|_j

and

\|E\|_{2}

are finite. These properties allow to approximate any QT-matrix, within any given precision,by means of a finite number of parameters.QT-matrices, equipped with the norm

\|A\|_{\mathcal{QT}} = \alpha\|a\|_{\mathcal{W}} + \|E\|_{2}, for

\alpha \geq (1 + \sqrt{5})/2$,are a Banach algebra with the standard arithmetic operations. We provide an algorithmicdescription of these operations on the finite parametrization of QT-matrices, and we developa MATLAB toolbox implementing them in a transparent way. The toolbox is then extended toperform arithmetic operations on matrices of finite size that have a Toeplitz plus low-rankstructure. This enables the development of algorithms for Toeplitz and quasi-Toeplitz matriceswhose cost does not necessarily increase with the dimension of the problem.Some examples of applications to computing matrix functions and to solving matrix equa-tions are presented, and confirm the effectiveness of the approach.

MATHICSE2018