Weighted Reduced Order Methods for Parametrized Partial Differential Equations with Random Inputs
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.
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  library, developed at SISSA mathLab, which is an implementation in FEniCS  of several reduced order modelling techniques; we acknowledge developers and contributors to both libraries.
- 1.Ballarin F, Sartori A, Rozza G (2015) RBniCS—reduced order modelling in FEniCS. http://mathlab.sissa.it/rbnics
- 3.Chen P (2014) Model order reduction techniques for uncertainty quantification problems. Ph.D. Thesis, École polytechnique fédérale de Lausanne EPFLGoogle Scholar
- 8.Hesthaven J, Rozza G, Stamm B (2016) Certified reduced basis methods for parametrized partial differential equations. SpringerGoogle Scholar
- 9.Holtz M (2010) Sparse grid quadrature in high dimensions with applications in finance and insurance. SpringerGoogle Scholar
- 10.Iliescu T, Liu H, Xie X (2017) Regularized reduced order models for a stochastic Burgers equation. arXiv:1701.01155
- 12.Logg A, Mardal KA, Wells G (2012) Automated solution of differential equations by the finite element method: the FEniCS book, vol 84. Springer Science & Business MediaGoogle Scholar
- 15.Quarteroni A, Valli A (1994) Numerical approximation of partial differential equations. SpringerGoogle Scholar
- 16.Rozza G, Huynh D, Nguyen N, Patera A (2009) Real-time reliable simulation of heat transfer phenomena. In: ASME-american society of mechanical engineers-heat transfer summer conference proceedings. San Francisco, CA, USAGoogle Scholar
- 18.Rozza G, Nguyen N, Patera A, Deparis S (2009) Reduced basis methods and a posteriori error estimators for heat transfer problems. In: ASME-american society of mechanical engineers-heat transfer summer conference proceedings. San Francisco, CA, USAGoogle Scholar
- 19.Spannring C (2018) Weighted reduced basis methods for parabolic PDEs with random input data. Ph.D. Thesis, Graduate School CE, Technische Universität DarmstadtGoogle Scholar
- 20.Spannring C, Ullmann S, Lang J (2017) A weighted reduced basis method for parabolic PDEs with random data. In: Recent advances in Computational Engineering, International conference on Computational Engineering. Lecture notes in Computational Science and Engineering, vol 124. Springer, Cham. arXiv:1712.07393
- 22.Venturi L, Ballarin F, Rozza G (2018) A weighted POD method for elliptic PDEs with random inputs. J Sci Comput 1–18. arXiv:1802.08724