# Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs

• Luca Venturi
• Davide Torlo
• Francesco Ballarin
• Gianluigi Rozza
Chapter
Part of the PoliTO Springer Series book series (PTSS)

## Abstract

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.

## Notes

### 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 [1] library, developed at SISSA mathLab, which is an implementation in FEniCS [12] of several reduced order modelling techniques; we acknowledge developers and contributors to both libraries.

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## Authors and Affiliations

• Luca Venturi
• 1
• 2
• Davide Torlo
• 1
• 3
• Francesco Ballarin
• 1
• Gianluigi Rozza
• 1
1. 1.mathLab, Mathematics AreaSISSATriesteItaly
2. 2.Courant Institute of Mathematical SciencesNew York UniversityNew YorkUS
3. 3.Institut für MathematikUniversität ZürichZürichSwitzerland