Isogeometric analysis and proper orthogonal decomposition for parabolic problems

Alfio Quarteroni, Shengfeng Zhu
Springer Heidelberg, 2017
Article

Résumé

We investigate the combination of Isogeometric Analysis (IGA) and proper orthogonal decomposition (POD) based on the Galerkin method for model order reduction of linear parabolic partial differential equations. For the proposed fully discrete scheme, the associated numerical error features three components due to spatial discretization by IGA, time discretization with the -scheme, and eigenvalue truncation by POD. First, we prove a priori error estimates of the spatial IGA semi-discrete scheme. Then, we show stability and prove a priori error estimates of the space-time discrete scheme and the fully discrete IGA--POD Galerkin scheme. Numerical tests are provided to show efficiency and accuracy of NURBS-based IGA for model order reduction in comparison with standard finite element-based POD techniques.

Official source

https://infoscience.epfl.ch/record/226961?ln=fr

À propos de ce résultat

Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.

Concepts associés

Chargement

Publications associées

Chargement

Isogeometric Analysis And Proper Orthogonal Decomposition For The Acoustic Wave Equation

Alfio Quarteroni, Shengfeng Zhu

Discretization methods such as finite differences or finite elements were usually employed to provide high fidelity solution approximations for reduced order modeling of parameterized partial differential equations. In this paper, a novel discretization technique-Isogeometric Analysis (IGA) is used in combination with proper orthogonal decomposition (POD) for model order reduction of the time parameterized acoustic wave equations. We propose a new fully discrete IGA-Newmark-POD approximation and we analyze the associated numerical error, which features three components due to spatial discretization by IGA, time discretization with the Newmark scheme, and modes truncation by POD. We prove stability and convergence. Numerical examples are presented to show the effectiveness and accuracy of IGA-based POD techniques for the model order reduction of the acoustic wave equation.

Edp Sciences S A2017

Chargement

We present and analyze a novel wavelet-Fourier technique for the numerical treatment of multidimensional advection–diffusion–reaction equations based on the COmpRessed SolvING (CORSING) paradigm. Combining the Petrov–Galerkin technique with the compressed sensing approach the proposed method is able to approximate the largest coefficients of the solution with respect to a biorthogonal wavelet basis. Namely, we assemble a compressed discretization based on randomized subsampling of the Fourier test space and we employ sparse recovery techniques to approximate the solution to the partial differential equation (PDE). In this paper we provide the first rigorous recovery error bounds and effective recipes for the implementation of the CORSING technique in the multidimensional setting. Our theoretical analysis relies on new estimates for the local a-coherence, which measures interferences between wavelet and Fourier basis functions with respect to the metric induced by the PDE operator. The stability and robustness of the proposed scheme are shown by numerical illustrations in the one-, two- and three-dimensional cases.

2020

Chargement

MATHICSE Technical Report : Isogeometric analysis of geometric partial differential equations

Andrea Bartezzaghi, Alfio Quarteroni

We consider the numerical approximation of geometric Partial Differential Equations (PDEs) defined on surfaces in the 3D space. In particular, we focus on the geometric PDEs deriving from the minimization of an energy functional by L2-gradient ow. We analyze two energy functionals: the area, which leads to the mean curvature ow, a nonlinear second order PDE, and the Willmore energy, leading to the Willmore ow, a nonlinear fourth order PDE. We consider surfaces represented by single-patch NURBS and discretize the PDEs by means of NURBS-based Isogeometric Analysis in the framework of the Galerkin method. To approximate the high order geometric PDEs we use high order continuous NURBS basis functions. Instead, for the time discretization of the nonlinear geometric PDEs, we use Backward Differentiation Formulas (BDF) with extrapolation of the geometric quantities involved in the weak formulation of the problem; in this manner, we solve a linear problem at each time step. We report numerical results concerning the mean curvature and Willmore ows on different geometries of interest and we show the accuracy and efficiency of the proposed approximation scheme.

MATHICSE2016

Chargement