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 \(L^2\) 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.
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Notes
We assume that \(\mathcal {U}\) is a quadrature rule for integration with respect to dy. If a quadrature rule \(\mathcal {U}_{\rho }\) for integration with respect to the weighted measure \(\rho dy\) is used instead, that is suffices to take \(w_i = \omega _i\).
When using tensor product quadrature rule, we can not impose the cardinality of \(\varXi _t\) a priori. We also note that when we use the Clenshaw–Curtis approximation, the majority of the points in \(\varXi _t\) lies on \(\partial \varGamma \): these point are completely negligible, since \(\rho |_{\partial \varGamma } \equiv 0\). So, in this case, we need to take a considerably larger value for \(n_t\) to reach the desired cardinality of nodes in the interior \(\mathring{\varGamma }\).
The standard algorithms are obtained imposing unitary weights and sampling from uniform distribution.
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Acknowledgements
We acknowledge the support by European Union Funding for Research and Innovation—Horizon 2020 Program—in the framework of European Research Council Executive Agency: H2020 ERC Consolidator Grant 2015 AROMA-CFD project 681447 “Advanced Reduced Order Methods with Applications in Computational Fluid Dynamics”. We also acknowledge the INDAM-GNCS projects “Metodi numerici avanzati combinati con tecniche di riduzione computazionale per PDEs parametrizzate e applicazioni” and “Numerical methods for model order reduction of PDEs”. The computations in this work have been performed with RBniCS [2] library, developed at SISSA mathLab, which is an implementation in FEniCS [27] of several reduced order modelling techniques; we acknowledge developers and contributors to both libraries.
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Venturi, L., Ballarin, F. & Rozza, G. A Weighted POD Method for Elliptic PDEs with Random Inputs. J Sci Comput 81, 136–153 (2019). https://doi.org/10.1007/s10915-018-0830-7
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DOI: https://doi.org/10.1007/s10915-018-0830-7