@article {2021,
	title = {A weighted POD-reduction approach for parametrized PDE-constrained optimal control problems with random inputs and applications to environmental sciences},
	journal = {Computers and Mathematics with Applications},
	volume = {102},
	year = {2021},
	pages = {261-276},
	doi = {10.1016/j.camwa.2021.10.020},
	url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85117948561\&doi=10.1016\%2fj.camwa.2021.10.020\&partnerID=40\&md5=cb57d59a6975a35315b2cf5d0e3a6001},
	author = {G. Carere and Maria Strazzullo and Francesco Ballarin and Gianluigi Rozza and R. Stevenson}
}
@article {2019,
	title = {A Weighted POD Method for Elliptic PDEs with Random Inputs},
	journal = {Journal of Scientific Computing},
	volume = {81},
	year = {2019},
	pages = {136-153},
	abstract = {<p>In this work we propose and analyze a weighted proper orthogonal decomposition method to solve elliptic partial differential equations depending on random input data, for stochastic problems that can be transformed into parametric systems. The algorithm is introduced alongside the weighted greedy method. Our proposed method aims to minimize the error in a L2 norm and, in contrast to the weighted greedy approach, it does not require the availability of an error bound. Moreover, we consider sparse discretization of the input space in the construction of the reduced model; for high-dimensional problems, provided the sampling is done accordingly to the parameters distribution, this enables a sensible reduction of computational costs, while keeping a very good accuracy with respect to high fidelity solutions. We provide many numerical tests to assess the performance of the proposed method compared to an equivalent reduced order model without weighting, as well as to the weighted greedy approach, in both low and high dimensional problems.</p>},
	doi = {10.1007/s10915-018-0830-7},
	url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85053798049\&doi=10.1007\%2fs10915-018-0830-7\&partnerID=40\&md5=5cad501b6ef1955da55868807079ee5d},
	author = {L.Venturi and F. Ballarin and Gianluigi Rozza}
}
@article {2019,
	title = {Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs},
	journal = {PoliTO Springer Series},
	year = {2019},
	pages = {27-40},
	abstract = {<p>In this manuscript we discuss weighted reduced order methods for stochastic partial differential equations. Random inputs (such as forcing terms, equation coefficients, boundary conditions) are considered as parameters of the equations. We take advantage of the resulting parametrized formulation to propose an efficient reduced order model; we also profit by the underlying stochastic assumption in the definition of suitable weights to drive to reduction process. Two viable strategies are discussed, namely the weighted reduced basis method and the weighted proper orthogonal decomposition method. A numerical example on a parametrized elasticity problem is shown.</p>},
	doi = {10.1007/978-3-030-04870-9_2},
	url = {https://www.scopus.com/inward/record.uri?eid=2-s2.0-85084009379\&doi=10.1007\%2f978-3-030-04870-9_2\&partnerID=40\&md5=446bcc1f331167bbba67bc00fb170150},
	author = {L. Venturi and D. Torlo and F. Ballarin and Gianluigi Rozza}
}
@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}
}
