01557nas a2200133   4500008004100000245009400041210006900135260001700204520109300221100001501314700002201329700002101351856005101372       2014                        en  d00aA weighted empirical interpolation method: A priori convergence analysis and applications0 aweighted empirical interpolation method A priori convergence ana  bEDP Sciences3 aWe 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.1 aChen, Peng1 aQuarteroni, Alfio1 aRozza, Gianluigi  uhttp://urania.sissa.it/xmlui/handle/1963/3502101375nas a2200145   4500008004100000245010300041210006900144300001600213490000700229520080500236100001501041700002201056700002101078856013001099       2013                        eng d00aA weighted reduced basis method for elliptic partial differential equations with random input data0 aweighted reduced basis method for elliptic partial differential   a3163–31850 v513 aIn 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.1 aChen, Peng1 aQuarteroni, Alfio1 aRozza, Gianluigi  uhttps://www.math.sissa.it/publication/weighted-reduced-basis-method-elliptic-partial-differential-equations-random-input-data