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. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.

A Weighted POD Method for Elliptic PDEs with Random Inputs / Venturi, Luca; Ballarin, Francesco; Rozza, Gianluigi. - In: JOURNAL OF SCIENTIFIC COMPUTING. - ISSN 0885-7474. - 81:1(2019), pp. 136-153. [10.1007/s10915-018-0830-7]

A Weighted POD Method for Elliptic PDEs with Random Inputs

Ballarin, Francesco;Rozza, Gianluigi
2019-01-01

Abstract

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. © 2018, Springer Science+Business Media, LLC, part of Springer Nature.
81
1
136
153
https://doi.org/10.1007/s10915-018-0830-7
https://arxiv.org/abs/1802.08724
Venturi, Luca; Ballarin, Francesco; Rozza, Gianluigi
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Utilizza questo identificativo per citare o creare un link a questo documento: https://hdl.handle.net/20.500.11767/103642
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