@article {2014,
	title = {A weighted empirical interpolation method: A priori convergence analysis and applications},
	number = {ESAIM: Mathematical Modelling and Numerical Analysis;volume 48; issue 4; pages 943-953},
	year = {2014},
	publisher = {EDP Sciences},
	abstract = {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-Lo{\`e}ve 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.},
	doi = {10.1051/m2an/2013128},
	url = {http://urania.sissa.it/xmlui/handle/1963/35021},
	author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza}
}
@article {ChenQuarteroniRozza2013a,
	title = {A weighted reduced basis method for elliptic partial differential equations with random input data},
	journal = {SIAM Journal on Numerical Analysis},
	volume = {51},
	number = {6},
	year = {2013},
	pages = {3163{\textendash}3185},
	abstract = {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.},
	doi = {10.1137/130905253},
	author = {Peng Chen and Alfio Quarteroni and Gianluigi Rozza}
}
