Identification of random shapes from images through polynomial chaos expansion of random level set functions

Abstract

To cite this version:G. Stefanou, Anthony Nouy, Alexandre Clement. Identification of random shapes from images through polynomial chaos expansion of random level-set functions. International Journal for Numerical Methods in Engineering, Wiley, 2009, 79 (2) SUMMARYIn this paper, an e cient method is proposed for the identi cation of random shapes in a form suitable for numerical simulation within the eXtended Stochastic Finite Element Method (X-SFEM).The method starts from a collection of images representing di erent outcomes of the random shape to identify. The key-point of the method is to represent the random geometry in an implicit manner using the level-set technique. In this context, the problem of random geometry identi cation is equivalent to the identi cation of a random level-set function, which is a random eld. This random eld is represented on a polynomial chaos basis and various e cient numerical strategies are proposed in order to identify the coe cients of its polynomial chaos decomposition. The performance of these strategies is evaluated through some "manufactured" problems and useful conclusions are provided. The propagation of geometrical uncertainties in structural analysis using the X-SFEM is nally examined.key words:Random geometry; Level-set method; Probabilistic identi cation; Polynomial chaos; Maximum likelihood; Extended stochastic nite element method.

Uncertainty quantification in computational stochastic multiscale analysis of nonlinear elastic materials

“…(24), had to be rejected. The identification of the vector-valued coefficients y 1 ,...,y N for a fixed value of N is done using the maximum likelihood method [36,42] as performed in [2,43,10]. For a given value of y 1 ,...,y N , the estimation of…”

“…However, taking into account a high number of random parameters is of first importance in a stochastic multiscale analysis and we thus propose a different methodology based on polynomial chaos representations. Initiated in [14], the methodology to construct a polynomial chaos expansion of random fields has been intensely developed to solve stochastic partial differential equations [3,15,16,13,24,22,29,32,31,35,38,11] but also for the identification of random fields using experimental data and classical inference techniques [17,2] or maximum likelihood estimation [9,10,43,18]. A new methodology has been recently introduced to deal with the identification of polynomial chaos representations in high-dimension [39,41].…”

An extended stochastic finite element method for solving stochastic partial differential equations on random domains

“…In this article, we consider that the probabilistic model of the geometry is given. Let us note that in [27], a method has been introduced for the identication of a random geometry from sample images. It is based on the identication of a polynomial chaos expansion of the random level-set function ϕ whose samples are obtained from image recovery techniques.…”

Recent Developments in Spectral Stochastic Methods for the Numerical Solution of Stochastic Partial Differential Equations

“…$ This hypothesis is not veried for classical formulations of PDE dened on random domains [95,25,92,122].…”

“…This representation can be used as a complement to Karhunen-Loève expansions (115) or (122). Indeed, it allows for a representation of random variables which appear in these expansions: Hα(ξ(θ)).…”