# Reduced Basis Methods: Success, Limitations and Future Challenges

## Main Article Content

## Abstract

Parametric model order reduction using reduced basis methods can be an effective tool for obtaining quickly solvable reduced order model of parametrized partial differential equation problems. With speedups that can reach several orders of magnitude, reduced basis methods enable high fidelity real-time simulations of complex systems and dramatically reduce the computational costs in many-query applications. In this contribution we analyze the methodology, mainly focussing on the theoretical aspects of the approach. In particular we discuss what is known about the convergence properties of these methods: when they succeed and when they are bound to fail. Moreover, we highlight some recent approaches employing nonlinear approximation techniques which aim to overcome the current limitations of reduced basis methods.

## Article Details

How to Cite

OHLBERGER, Mario; RAVE, Stephan.
Reduced Basis Methods: Success, Limitations and Future Challenges.

**Proceedings of the Conference Algoritmy**, [S.l.], p. 1-12, feb. 2016. Available at: <http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/389>. Date accessed: 22 sep. 2017.
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Articles

## References

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[2] M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera, An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 667–672.

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[6] K. Carlberg, Adaptive h-refinement for reduced-order models, Int. j. numer. meth. engng., 102 (2015), pp. 1192–1210.

[7] K. Carlberg, C. Bou-Mosleh, and C. Farhat, Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, Int. j. numer. meth. engng., 86 (2011), pp. 155–181.

[8] S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32 (2010), pp. 2737–2764.

[9] A. Cohen and R. DeVore, Kolmogorov widths under holomorphic mappings, IMA Journal of Numerical Analysis, (2015).

[10] A. Cohen, R. Devore, and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s, Analysis and Applications, 09 (2011), pp. 11–47.

[11] P. Constantine and G. Iaccarino, Reduced order models for parameterized hyperbolic conservations laws with shock reconstruction, Center for Turbulence Research Annual Brief, (2012).

[12] W. Dahmen, C. Plesken, and G. Welper, Double greedy algorithms: Reduced basis methods for transport dominated problems, ESAIM: M2AN, 48 (2014), pp. 623–663.

[13] R. DeVore, G. Petrova, and P. Wojtaszczyk, Greedy algorithms for reduced bases in Banach spaces, Constr. Approx., 37 (2013), pp. 455–466.

[14] M. Dihlmann, M. Drohmann, and B. Haasdonk, Model reduction of parametrized evolution problems using the reduced basis method with adaptive time-partitioning, in Proc. of ADMOS 2011, 2011.

[15] M. Drohmann, B. Haasdonk, and M. Ohlberger, Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation, SIAM J. Sci. Comput., 34 (2012), pp. A937–A969.

[16] J.-F. Gerbeau and D. Lombardi, Approximated lax pairs for the reduced order integration of nonlinear evolution equations, J. Comput. Phys., 265 (2014), pp. 246 – 269.

[17] B. Haasdonk, Convergence rates of the POD-Greedy method, ESAIM: M2AN, 47 (2013), pp. 859–873.

[18] B. Haasdonk, Reduced basis methods for parametrized PDEs – A tutorial introduction for stationary and instationary problems, tech. rep., 2014. Chapter to appear in P. Benner, A. Cohen, M. Ohlberger and K. Willcox: ”Model Reduction and Approximation for Complex Systems”, Springer.

[19] B. Haasdonk, M. Dihlmann, and M. Ohlberger, A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space, Math. Comput. Model. Dyn. Syst., 17 (2011), pp. 423–442.

[20] B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations, ESAIM: M2AN, 42 (2008), pp. 277–302.

[21] B. Haasdonk, M. Ohlberger, and G. Rozza, A reduced basis method for evolution schemes with parameter-dependent explicit operators, Electron. Trans. Numer. Anal., 32 (2008), pp. 145–161.

[22] J. S. Hesthaven, G. Rozza, and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, SpringerBriefs in Mathematics, Springer International Publishing, 2016.

[23] D. Huynh, G. Rozza, S. Sen, and A. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants, C. R. Math. Acad. Sci. Paris, 345 (2007), pp. 473 – 478.

[24] A. Iollo and D. Lombardi, Advection modes by optimal mass transfer, Phys. Rev. E, 89 (2014), p. 022923.

[25] S. Kaulmann and B. Haasdonk, Online greedy reduced basis construction using dictionaries, in VI International Conference on Adaptive Modeling and Simulation (ADMOS 2013), J. P. B. Moitinho de Almeida, P. Diez, C. Tiago, and N. Pars, eds., 2013, pp. 365–376.

[26] M. Ohlberger and S. Rave, Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing, C. R. Math. Acad. Sci. Paris, 351 (2013), pp. 901–906.

[27] M. Ohlberger and F. Schindler, Error control for the localized reduced basis multi-scale method with adaptive on-line enrichment, SIAM J. Sci. Comput., (2015, accepted).

[28] A. Pinkus, n-widths in approximation theory, vol. 7 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1985.

[29] A. Quarteroni, A. Manzoni, and F. Negri, Reduced Basis Methods for Partial Differential Equations, La Matematica per il 3+2, Springer International Publishing, 2016.

[30] A. Quarteroni, G. Rozza, and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications, Journal of Mathematics in Industry, 1 (2011).

[31] T. Taddei, S. Perotto, and A. Quarteroni, Reduced basis techniques for nonlinear conservation laws, ESAIM: M2AN, 49 (2015), pp. 787–814.

[32] K. Urban and A. T. Patera, A new error bound for reduced basis approximation of parabolic partial differential equations, C. R. Math. Acad. Sci. Paris, 350 (2012), pp. 203–207.

[33] G. Welper, Transformed snapshot interpolation, arXiv e-prints 1505.01227v1, (2015).

[34] M. Yano, A space-time Petrov-Galerkin certified reduced basis method: Application to the boussinesq equations, SIAM J. Sci. Comput., 36 (2014), pp. A232–A266.

[35] M. Yano, A minimum-residual mixed reduced basis method: Exact residual certification and simultaneous finite-element reduced-basis refinement, ESAIM: M2AN, (2015, accepted).

[36] O. Zahm and A. Nouy, Interpolation of inverse operators for preconditioning parameter dependent equations, arXiv e-prints 1504.07903v3, (2015).

[2] M. Barrault, Y. Maday, N. C. Nguyen, and A. T. Patera, An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, C. R. Math. Acad. Sci. Paris, 339 (2004), pp. 667–672.

[3] P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova, and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods, SIAM J. Math. Anal., 43 (2011), pp. 1457–1472.

[4] J. Brunken, M. Ohlberger, and K. Smetana, Problem adapted hierarchical model reduction for the Fokker-Planck equation, in Proceedings of ALGORITMY, 2016.

[5] A. Buffa, Y. Maday, A. T. Patera, C. Prud’homme, and G. Turinici, A priori convergence of the greedy algorithm for the parametrized reduced basis method, ESAIM: M2AN, 46 (2012), pp. 595–603.

[6] K. Carlberg, Adaptive h-refinement for reduced-order models, Int. j. numer. meth. engng., 102 (2015), pp. 1192–1210.

[7] K. Carlberg, C. Bou-Mosleh, and C. Farhat, Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, Int. j. numer. meth. engng., 86 (2011), pp. 155–181.

[8] S. Chaturantabut and D. C. Sorensen, Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32 (2010), pp. 2737–2764.

[9] A. Cohen and R. DeVore, Kolmogorov widths under holomorphic mappings, IMA Journal of Numerical Analysis, (2015).

[10] A. Cohen, R. Devore, and C. Schwab, Analytic regularity and polynomial approximation of parametric and stochastic elliptic PDE’s, Analysis and Applications, 09 (2011), pp. 11–47.

[11] P. Constantine and G. Iaccarino, Reduced order models for parameterized hyperbolic conservations laws with shock reconstruction, Center for Turbulence Research Annual Brief, (2012).

[12] W. Dahmen, C. Plesken, and G. Welper, Double greedy algorithms: Reduced basis methods for transport dominated problems, ESAIM: M2AN, 48 (2014), pp. 623–663.

[13] R. DeVore, G. Petrova, and P. Wojtaszczyk, Greedy algorithms for reduced bases in Banach spaces, Constr. Approx., 37 (2013), pp. 455–466.

[14] M. Dihlmann, M. Drohmann, and B. Haasdonk, Model reduction of parametrized evolution problems using the reduced basis method with adaptive time-partitioning, in Proc. of ADMOS 2011, 2011.

[15] M. Drohmann, B. Haasdonk, and M. Ohlberger, Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation, SIAM J. Sci. Comput., 34 (2012), pp. A937–A969.

[16] J.-F. Gerbeau and D. Lombardi, Approximated lax pairs for the reduced order integration of nonlinear evolution equations, J. Comput. Phys., 265 (2014), pp. 246 – 269.

[17] B. Haasdonk, Convergence rates of the POD-Greedy method, ESAIM: M2AN, 47 (2013), pp. 859–873.

[18] B. Haasdonk, Reduced basis methods for parametrized PDEs – A tutorial introduction for stationary and instationary problems, tech. rep., 2014. Chapter to appear in P. Benner, A. Cohen, M. Ohlberger and K. Willcox: ”Model Reduction and Approximation for Complex Systems”, Springer.

[19] B. Haasdonk, M. Dihlmann, and M. Ohlberger, A training set and multiple bases generation approach for parameterized model reduction based on adaptive grids in parameter space, Math. Comput. Model. Dyn. Syst., 17 (2011), pp. 423–442.

[20] B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations, ESAIM: M2AN, 42 (2008), pp. 277–302.

[21] B. Haasdonk, M. Ohlberger, and G. Rozza, A reduced basis method for evolution schemes with parameter-dependent explicit operators, Electron. Trans. Numer. Anal., 32 (2008), pp. 145–161.

[22] J. S. Hesthaven, G. Rozza, and B. Stamm, Certified Reduced Basis Methods for Parametrized Partial Differential Equations, SpringerBriefs in Mathematics, Springer International Publishing, 2016.

[23] D. Huynh, G. Rozza, S. Sen, and A. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants, C. R. Math. Acad. Sci. Paris, 345 (2007), pp. 473 – 478.

[24] A. Iollo and D. Lombardi, Advection modes by optimal mass transfer, Phys. Rev. E, 89 (2014), p. 022923.

[25] S. Kaulmann and B. Haasdonk, Online greedy reduced basis construction using dictionaries, in VI International Conference on Adaptive Modeling and Simulation (ADMOS 2013), J. P. B. Moitinho de Almeida, P. Diez, C. Tiago, and N. Pars, eds., 2013, pp. 365–376.

[26] M. Ohlberger and S. Rave, Nonlinear reduced basis approximation of parameterized evolution equations via the method of freezing, C. R. Math. Acad. Sci. Paris, 351 (2013), pp. 901–906.

[27] M. Ohlberger and F. Schindler, Error control for the localized reduced basis multi-scale method with adaptive on-line enrichment, SIAM J. Sci. Comput., (2015, accepted).

[28] A. Pinkus, n-widths in approximation theory, vol. 7 of Ergebnisse der Mathematik und ihrer Grenzgebiete (3), Springer-Verlag, Berlin, 1985.

[29] A. Quarteroni, A. Manzoni, and F. Negri, Reduced Basis Methods for Partial Differential Equations, La Matematica per il 3+2, Springer International Publishing, 2016.

[30] A. Quarteroni, G. Rozza, and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications, Journal of Mathematics in Industry, 1 (2011).

[31] T. Taddei, S. Perotto, and A. Quarteroni, Reduced basis techniques for nonlinear conservation laws, ESAIM: M2AN, 49 (2015), pp. 787–814.

[32] K. Urban and A. T. Patera, A new error bound for reduced basis approximation of parabolic partial differential equations, C. R. Math. Acad. Sci. Paris, 350 (2012), pp. 203–207.

[33] G. Welper, Transformed snapshot interpolation, arXiv e-prints 1505.01227v1, (2015).

[34] M. Yano, A space-time Petrov-Galerkin certified reduced basis method: Application to the boussinesq equations, SIAM J. Sci. Comput., 36 (2014), pp. A232–A266.

[35] M. Yano, A minimum-residual mixed reduced basis method: Exact residual certification and simultaneous finite-element reduced-basis refinement, ESAIM: M2AN, (2015, accepted).

[36] O. Zahm and A. Nouy, Interpolation of inverse operators for preconditioning parameter dependent equations, arXiv e-prints 1504.07903v3, (2015).