01594nas a2200145   4500008004100000245006300041210006100104300001200165490000700177520106000184100001601244700001701260700002101277856015001298       2019                        eng d00aA Weighted POD Method for Elliptic PDEs with Random Inputs0 aWeighted POD Method for Elliptic PDEs with Random Inputs  a136-1530 v813 a<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>1 a.Venturi, L1 aBallarin, F.1 aRozza, Gianluigi  uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85053798049&doi=10.1007%2fs10915-018-0830-7&partnerID=40&md5=5cad501b6ef1955da55868807079ee5d01267nas a2200145   4500008004100000245010200041210006900143300001000212520067900222100001600901700001400917700001700931700002100948856015200969       2019                        eng d00aWeighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs0 aWeighted Reduced Order Methods for Parametrized Partial Differen  a27-403 a<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>1 aVenturi, L.1 aTorlo, D.1 aBallarin, F.1 aRozza, Gianluigi  uhttps://www.scopus.com/inward/record.uri?eid=2-s2.0-85084009379&doi=10.1007%2f978-3-030-04870-9_2&partnerID=40&md5=446bcc1f331167bbba67bc00fb170150