An algorithmic comparison of the hyper-reduction and the discrete empirical interpolation method for a nonlinear thermal problem. (English) Zbl 1390.80017
Summary: A novel algorithmic discussion of the methodological and numerical differences of competing parametric model reduction techniques for nonlinear problems is presented. First, the Galerkin reduced basis (RB) formulation is presented, which fails at providing significant gains with respect to the computational efficiency for nonlinear problems. Renowned methods for the reduction of the computing time of nonlinear reduced order models are the Hyper-Reduction and the (Discrete) Empirical Interpolation Method (EIM, DEIM). An algorithmic description and a methodological comparison of both methods are provided. The accuracy of the predictions of the hyper-reduced model and the (D)EIM in comparison to the Galerkin RB is investigated. All three approaches are applied to a simple uncertainty quantification of a planar nonlinear thermal conduction problem. The results are compared to computationally intense finite element simulations.
MSC:
| 80M10 | Finite element, Galerkin and related methods applied to problems in thermodynamics and heat transfer |
| 65N30 | Finite element, Rayleigh-Ritz and Galerkin methods for boundary value problems involving PDEs |
Keywords:
model order reduction (MOR); reduced basis model order reduction (RB MOR); uncertainty quantification (UQ); (discrete) empirical interpolation method (EIM; DEIM); hyper-reduction (HR)References:
| [1] | Gubisch, M.; Volkwein, S.; POD for Linear-Quadratic Optimal Control; Model Reduction and Approximation: Theory and Algorithms: Philadelphia, PA, USA 2017; . · Zbl 1302.49039 |
| [2] | Volkwein, S.; Optimal control of a phase-field model using proper orthogonal decomposition; ZAMM: 2001; Volume 81 ,83-97. · Zbl 1007.49019 |
| [3] | Antoulas, A.; ; Approximation of Large-Scale Dynamical Systems: Philadelphia, PA, USA 2005; . · Zbl 1112.93002 |
| [4] | Patera, A.; Rozza, G.; ; Reduced Basis Approximation and a Posteriori Error Estimation for Parametrized Partial Differential Equations: Cambridge, MA, USA 2007; . · Zbl 1304.65251 |
| [5] | Haasdonk, B.; Reduced Basis Methods for Parametrized PDEs—A Tutorial Introduction for Stationary and Instationary Problems; Model Reduction and Approximation: Theory and Algorithms: Philadelphia, PA, USA 2017; . |
| [6] | Barrault, M.; Maday, Y.; Nguyen, N.; Patera, A.; An ’empirical interpolation’ method: Application to efficient reduced-basis discretization of partial differential equations; C. R. Math. Acad. Sci. Paris Ser. I: 2004; Volume 339 ,667-672. · Zbl 1061.65118 |
| [7] | Haasdonk, B.; Ohlberger, M.; Reduced basis method for explicit finite volume approximations of nonlinear conservation laws; Proceedings of the HYP 2008, International Conference on Hyperbolic Problems: Theory, Numerics and Applications: Providence, RI, USA 2009; Volume Volume 67 ,605-614. · Zbl 1407.76084 |
| [8] | Drohmann, M.; Haasdonk, B.; Ohlberger, M.; Reduced Basis Approximation for Nonlinear Parametrized Evolution Equations based on Empirical Operator Interpolation; SIAM J. Sci. Comput.: 2012; Volume 34 ,A937-A969. · Zbl 1259.65133 |
| [9] | Chaturantabut, S.; Sorensen, D.C.; Discrete empirical interpolation for nonlinear model reduction; Proceedings of the 48th IEEE Conference on Decision and Control and the 28th Chinese Control Conference (CDC/CCC 2009): ; ,4316-4321. |
| [10] | Chaturantabut, S.; Sorensen, D.C.; A State Space Error Estimate for POD-DEIM Nonlinear Model Reduction; SIAM J. Numer. Anal.: 2012; Volume 50 ,46-63. · Zbl 1237.93035 |
| [11] | Ryckelynck, D.; A priori hypereduction method: An adaptive approach; J. Comput. Phys.: 2005; Volume 202 ,346-366. · Zbl 1288.65178 |
| [12] | Ryckelynck, D.; Hyper reduction of mechanical models involving internal variables; Int. J. Numer. Methods Eng.: 2009; Volume 77 ,75-89. · Zbl 1195.74299 |
| [13] | Everson, R.; Sirovich, L.; Karhunen-Loève procedure for gappy data; J. Opt. Soc. Am. A: 1995; Volume 12 ,1657-1664. |
| [14] | Astrid, P.; Weiland, S.; Willcox, K.; Backx, T.; Missing point estimation in models described by proper orthogonal decomposition; IEEE Trans. Autom. Control: 2008; Volume 53 ,2237-2251. · Zbl 1367.93110 |
| [15] | Carlberg, K.; Bou-Mosleh, C.; Farhat, C.; Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations; Int. J. Numer. Methods Eng.: 2011; Volume 86 ,155-181. · Zbl 1235.74351 |
| [16] | Farhat, C.; Avery, P.; Chapman, T.; Cortial, J.; Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency; Int. J. Numer. Methods Eng.: 2014; Volume 98 ,625-662. · Zbl 1352.74348 |
| [17] | Fritzen, F.; Hodapp, M.; The Finite Element Square Reduced (FE2R) method with GPU acceleration: Towards three-dimensional two-scale simulations; Int. J. Numer. Methods Eng.: 2016; Volume 107 ,853-881. · Zbl 1352.74352 |
| [18] | Fritzen, F.; Xia, L.; Leuschner, M.; Breitkopf, P.; Topology optimization of multiscale elastoviscoplastic structures; Int. J. Numer. Methods Eng.: 2016; Volume 106 ,430-453. · Zbl 1352.74239 |
| [19] | Dimitriu, G.; Ştefănescu, R.; Navon, I.M.; Comparative numerical analysis using reduced-order modeling strategies for nonlinear large-scale systems; J. Comput. Appl. Math.: 2017; Volume 310 ,32-43. · Zbl 1348.92122 |
| [20] | Bathe, K.I.; ; Finite-Elemente-Methoden: Berlin/Heidelberg, Germany 2002; . |
| [21] | Zienkiewicz, O.; Taylor, R.; Zhu, J.; ; Finite Element Method: Oxford, UK 2006; . |
| [22] | Kelley, C.T.; ; Iterative Methods for Linear and Nonlinear Equations: Philadelphia, PA, USA 1995; . · Zbl 0832.65046 |
| [23] | Sirovich, L.; Turbulence and the dynamics of coherent structures part I: Coherent structures; Q. Appl. Math.: 1987; Volume 65 ,561-571. · Zbl 0676.76047 |
| [24] | Joliffe, I.; ; Principal Component Analysis: Hoboken, NJ, USA 2002; . · Zbl 1011.62064 |
| [25] | Ştefănescu, R.; Sandu, A.; Navon, I.M.; Comparison of POD reduced order strategies for the nonlinear 2D shallow water equations; Int. J. Numer. Methods Fluids: 2014; Volume 76 ,497-521. |
| [26] | Fritzen, F.; Hodapp, M.; Leuschner, M.; GPU accelerated computational homogenization based on a variational approach in a reduced basis framework; Comput. Methods Appl. Mech. Eng.: 2014; Volume 278 ,186-217. · Zbl 1423.74881 |
| [27] | Haasdonk, B.; Ohlberger, M.; Rozza, G.; A Reduced Basis Method for Evolution Schemes with Parameter-Dependent Explicit Operators; ETNA Electron. Trans. Numer. Anal.: 2008; Volume 32 ,145-161. · Zbl 1391.76413 |
| [28] | Wirtz, D.; Sorensen, D.; Haasdonk, B.; A Posteriori Error Estimation for DEIM Reduced Nonlinear Dynamical Systems; SIAM J. Sci. Comput.: 2014; Volume 36 ,A311-A338. · Zbl 1312.65127 |
| [29] | Ryckelynck, D.; Missoum Benziane, D.; Multi-level a priori hyper reduction of mechanical models involving internal variables; Comput. Methods Appl. Mech. Eng.: 2010; Volume 199 ,1134-1142. · Zbl 1227.74093 |
| [30] | Ryckelynck, D.; Vincent, F.; Cantournet, S.; Multidimensional a priori hyper-reduction of mechanical models involving internal variables; Comput. Methods Appl. Mech. Eng.: 2012; Volume 225 ,28-43. · Zbl 1253.74132 |
| [31] | Ryckelynck, D.; Gallimard, L.; Jules, S.; Estimation of the validity domain of hyper-reduction approximations in generalized standard elastoviscoplasticity; Adv. Model. Simul. Eng. Sci.: 2015; Volume 2 ,6. |
| [32] | Maday, Y.; Nguyen, N.; Patera, A.; Pau, G.; ; A General, Multi-Purpose Interpolation Procedure: The Magic Points: Paris, France 2007; . · Zbl 1184.65020 |
| [33] | Römer, U.; Schöps, S.; Weiland, T.; Stochastic Modeling and Regularity of the Nonlinear Elliptic curl-curl Equation; SIAM/ASA J. Uncertain. Quantif.: 2016; Volume 4 ,952-979. · Zbl 1355.78014 |
| [34] | Xiu, D.; ; Numerical Methods for Stochastic Computations: A Spectral Method Approach: Princeton, NI, USA 2010; . · Zbl 1210.65002 |
| [35] | Haasdonk, B.; Urban, K.; Wieland, B.; Reduced Basis Methods for parameterized partial differential equations with stochastic influences using the Karhunen-Loeve expansion; SIAM/ASA J. Uncertain. Quantif.: 2013; Volume 1 ,79-105. · Zbl 1281.35100 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
