TY - JOUR
T1 - A weighted empirical interpolation method: A priori convergence analysis and applications
Y1 - 2014
A1 - Peng Chen
A1 - Alfio Quarteroni
A1 - Gianluigi Rozza
AB - We extend the classical empirical interpolation method [M. Barrault, Y. Maday, N.C. Nguyen and A.T. Patera, An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations. Compt. Rend. Math. Anal. Num. 339 (2004) 667-672] to a weighted empirical interpolation method in order to approximate nonlinear parametric functions with weighted parameters, e.g. random variables obeying various probability distributions. A priori convergence analysis is provided for the proposed method and the error bound by Kolmogorov N-width is improved from the recent work [Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general, multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404]. We apply our method to geometric Brownian motion, exponential Karhunen-Loève expansion and reduced basis approximation of non-affine stochastic elliptic equations. We demonstrate its improved accuracy and efficiency over the empirical interpolation method, as well as sparse grid stochastic collocation method.
PB - EDP Sciences
UR - http://urania.sissa.it/xmlui/handle/1963/35021
U1 - 35253
U2 - Mathematics
U4 - 1
U5 - MAT/05
ER -
TY - JOUR
T1 - A weighted reduced basis method for elliptic partial differential equations with random input data
JF - SIAM Journal on Numerical Analysis
Y1 - 2013
A1 - Peng Chen
A1 - Alfio Quarteroni
A1 - Gianluigi Rozza
AB - In this work we propose and analyze a weighted reduced basis method to solve elliptic partial differential equations (PDEs) with random input data. The PDEs are first transformed into a weighted parametric elliptic problem depending on a finite number of parameters. Distinctive importance of the solution at different values of the parameters is taken into account by assigning different weights to the samples in the greedy sampling procedure. A priori convergence analysis is carried out by constructive approximation of the exact solution with respect to the weighted parameters. Numerical examples are provided for the assessment of the advantages of the proposed method over the reduced basis method and the stochastic collocation method in both univariate and multivariate stochastic problems.
VL - 51
ER -