Approximation | EPFL Graph Search

Une approximation est une représentation imprécise ayant toutefois un lien étroit avec la quantité ou l’objet qu’elle reflète : approximation d’un nombre (de π par 3,14, de la vitesse instantanée d’un véhicule par sa vitesse moyenne entre deux points), d’une fonction mathématique, d’une solution d’un problème d’optimisation, d’une forme géométrique, d’une loi physique. Lorsqu’une partie de l’information nécessaire fait défaut, une approximation peut se substituer à une représentation exacte. Cependant, même si cette dernière est connue, une approximation est parfois préférable par le fait qu’elle simplifie l’analyse sans générer de trop grandes erreurs. Par exemple, les physiciens rapprochent souvent la forme de la Terre à celle d’une sphère, même si des représentations plus précises sont possibles : plusieurs phénomènes physiques (telle la pesanteur) sont en effet plus faciles à étudier en supposant une sphère à la place d’une forme plus complexe. Le choix d’un degré d’approximation

Reduced Basis Method for 3D Problems governed by Parametrized PDEs and Applications

Alberto Trezzini

In this thesis we will deal with the creation of a Reduced Basis (RB) approximation of parametrized Partial Differential Equations (PDE) for three-dimensional problems. The the idea behind RB is to decouple the generation and projection stages (Ofﬂine/Online computational proce- dures) of the approximation process in order to solve parametrized (PDE) in a fast, cheap and reliable way. The RB method, especially applied to 3D problems, allows great computational savings with respect to the clas- sical Galerkin Finite Element (FE) Method. The standard FE method is typically ill suited to (i) iterative contexts like in optimization, sensitivity analysis and many queries in general and (ii) real time evaluation. We consider both coercive and noncoercive PDEs. For each class we discuss the steps to set up a RB approximation, either from an analytical and a numerical point of view. Then we present the applications of the RB method to three different problems of engineering interest and applica- bility: (i) a steady thermal conductivity problem in heat transfer; (ii) a linear elasticity problem; (iii) Stokes ﬂows with emphasis on geometrical and physical parameters.

2010

Approximation des moments par l'utilisation de la théorie des fonctions analytiques

Khalid Ohmiti

In this thesis, I define and explain the notion of punctual analytical uniform development (DUAP) versus moments or cumulants approximation of punctual uniforms analyticals 's class statistics. Hence, I derive "truncated" DUAP's version for numerical computation and implementation which called finite DUAP (F-DUAP). Using F-DUAP approximation lead to an error which was estimated. Due to nature's one of axiom of UAP statistics, the concept extension of DUAP method, to an other class statistic is limited. So, a new local theoretical concept was defined named analytical uniform development (DUA). This generalization let all derived DUAP's theorems become more general. Automatic differentiation and F-DUAP allow the implementation of DUAP or DUA method's on computer: I write the CUMAD and CUMADG codes that make the methods of a practical use. By, the programme CUMAD, I valid the utility of DUAP method, when I applied it to the approximation to moments of "weighted sum of squares statistic".

EPFL2007

Isogeometric Analysis for High Order Geometric Partial Differential Equations with Applications

Andrea Bartezzaghi

In this thesis, we consider the numerical approximation of high order geometric Partial Differential Equations (PDEs). We first consider high order PDEs defined on surfaces in the 3D space that are represented by single-patch tensor product NURBS. Then, we spatially discretize the PDEs by means of NURBS-based Isogeometric Analysis (IGA) in the framework of the Galerkin method. With this aim, we consider the construction of periodic NURBS function spaces with high degree of global continuity, even on closed surfaces. As benchmark problems for the proposed discretization, we propose Laplace-Beltrami problems of the fourth and sixth orders, as well as the corresponding eigenvalue problems, and we analyze the impact of the continuity of the basis functions on the accuracy as well as on computational costs. The numerical solution of two high order phase field problems on both open and closed surfaces is also considered: the fourth order Cahn-Hilliard equation and the sixth order crystal equation, both discretized in time with the generalized-alpha method. We then consider the numerical approximation of geometric PDEs, derived, in particular, from the minimization of shape energy functionals by L^2-gradient flows. We analyze the mean curvature and the Willmore gradient flows, leading to second and fourth order PDEs, respectively. These nonlinear geometric PDEs are discretized in time with Backward Differentiation Formulas (BDF), with a semi-implicit formulation based on an extrapolation of the geometry, leading to a linear problem to be solved at each time step. Results about the numerical approximation of the two geometric flows on several geometries are analyzed. Then, we study how the proposed mathematical framework can be employed to numerically approximate the equilibrium shapes of lipid bilayer biomembranes, or vesicles, governed by the Canham-Helfrich curvature model. We propose two numerical schemes for enforcing the conservation of the area and volume of the vesicles, and report results on benchmark problems. Then, the approximation of the equilibrium shapes of biomembranes with different values of reduced volume is presented. Finally, we consider the dynamics of a vesicle, e.g. a red blood cell, immersed in a fluid, e.g. the plasma. In particular, we couple the curvature-driven model for the lipid membrane with the incompressible Navier-Stokes equations governing the fluid. We consider a segregated approach, with a formulation based on the Resistive Immersed Surface method applied to NURBS geometries. After analyzing benchmark fluid simulations with immersed NURBS objects, we report numerical results for the investigation of the dynamics of a vesicle under different flow conditions.

EPFL2017

Isogeometric Analysis for High Order Geometric Partial Differential Equations with Applications

Andrea Bartezzaghi

EPFL2017