Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, p. 259-283

We propose two new algorithms to improve greedy sampling of high-dimensional functions. While the techniques have a substantial degree of generality, we frame the discussion in the context of methods for empirical interpolation and the development of reduced basis techniques for high-dimensional parametrized functions. The first algorithm, based on a saturation assumption of the error in the greedy algorithm, is shown to result in a significant reduction of the workload over the standard greedy algorithm. In a further improved approach, this is combined with an algorithm in which the train set for the greedy approach is adaptively sparsified and enriched. A safety check step is added at the end of the algorithm to certify the quality of the sampling. Both these techniques are applicable to high-dimensional problems and we shall demonstrate their performance on a number of numerical examples.

DOI : https://doi.org/10.1051/m2an/2013100
Classification:  41A05,  41A46,  65N15,  65N30
Keywords: greedy algorithm, reduced basis method, empirical interpolation method
@article{M2AN_2014__48_1_259_0,
     author = {Hesthaven, Jan S. and Stamm, Benjamin and Zhang, Shun},
     title = {Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {48},
     number = {1},
     year = {2014},
     pages = {259-283},
     doi = {10.1051/m2an/2013100},
     zbl = {1292.41001},
     mrnumber = {3177844},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2014__48_1_259_0}
}
Hesthaven, Jan S.; Stamm, Benjamin; Zhang, Shun. Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 48 (2014) no. 1, pp. 259-283. doi : 10.1051/m2an/2013100. http://www.numdam.org/item/M2AN_2014__48_1_259_0/

[1] R.E. Bank and A. Weiser, Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985) 303-320. | MR 777265 | Zbl 0569.65079

[2] M. Barrault, N.C. Nguyen, Y. Maday and A.T. Patera, An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations. C.R. Acad. Sci. Paris, Ser. I 339 (2004) 667-672. | MR 2103208 | Zbl 1061.65118

[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) 1457-1472. | MR 2821591 | Zbl 1229.65193

[4] A. Buffa, Y. Maday, A. Patera, C. Prud'Homme and G. Turinici, A priori convergence of the greedy algorithm for the parametrized reduced basis. M2AN 46 (2012) 595-603. Special Issue in honor of David Gottlieb. | Numdam | Zbl 1272.65084

[5] T. Bui-Thanh, Model-Constrained Optimization Methods for Reduction of Parameterized Large-Scale Systems, MIT Thesis (2007).

[6] T. Bui-Thanh, K. Willcox and O. Ghattas, Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J. Sci. Comput. 30 (2008) 3270-3288. | MR 2452388 | Zbl 1196.37127

[7] Y. Chen, J.S. Hesthaven, Y. Maday and J. Rodriguez, A monotonic evaluation of lower bounds for inf-sup stability constants in the frame of reduced basis approximations. C.R. Acad. Sci. Paris, Ser. I 346 (2008) 1295-1300. | MR 2473311 | Zbl 1152.65109

[8] Y. Chen, J.S. Hesthaven, Y. Maday and J. Rodriguez, Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2d Maxwells problem. ESAIM: M2AN 43 (2009) 1099-1116. | Numdam | MR 2588434 | Zbl 1181.78019

[9] Y. Chen, J.S. Hesthaven, Y. Maday and J. Rodriguez, Certified reduced basis methods and output bounds for the harmonic maxwell equations. SIAM J. Sci. Comput. 32 (2010) 970-996. | MR 2639602 | Zbl 1213.78011

[10] J.L. Eftang, A.T. Patera and E.M. Ronquist, An “hp” certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput. 32 (2010) 3170-3200. | MR 2746617 | Zbl 1228.35097

[11] J.L. Eftang and B. Stamm, Parameter multi-domain hp empirical interpolation. Int. J. Numer. Meth. Engng. 90 (2012) 412-428. | MR 2911317 | Zbl 1242.65255

[12] B. Fares, J.S. Hesthaven, Y. Maday and B. Stamm, The reduced basis method for the electric field integral equation. J. Comput. Phys. 230 (2011) 5532-5555. | MR 2799523 | Zbl 1220.78045

[13] M.A. Grepl, Y. Maday, N. C. Nguyen and A.T. Patera, Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Math. Model. Numer. Anal. 41 (2007) 575-605. | Numdam | MR 2355712 | Zbl 1142.65078

[14] M.A. Grepl and A.T. Patera, A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. M2AN 39 (2005) 157-181. | Numdam | MR 2136204 | Zbl 1079.65096

[15] B. Haasdonk, M. Dihlmann and M. Ohlberger, A training set and multiple basis functions generation approach for parametrized model reduction based on adaptive grids in parameter space. Math. Comput. Modell. Dyn. Syst. 17 (2011) 423-442. | MR 2823471 | Zbl pre06287795

[16] B. Haasdonk and M. Ohlberger, Basis construction for reduced basis methods by adaptive parameter grids, in Proc. International Conference on Adaptive Modeling and Simulation 2007 (2007) 116-119.

[17] J.S. Hesthaven and S. Zhang, On the use of ANOVA expansions in reduced basis methods for high-dimensional parametric partial differential equations, Brown Division of Applied Math Scientific Computing Tech Report 2011-31.

[18] D.B.P. Huynh, G. Rozza, S. Sen and A.T. Patera, A successive constraint linear optimization method for lower bounds of parametric coercivity and inf-sup stability constants. C.R. Acad. Sci. Paris, Ser. I 345 (2007) 473-478. | MR 2367928 | Zbl 1127.65086

[19] Y. Maday, N.C. Nguyen, A.T. Patera and G.S.H. Pau, A general multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383-404. | MR 2449115 | Zbl 1184.65020

[20] Y. Maday and B. Stamm, Locally adaptive greedy approximations for anisotropic parameter reduced basis spaces, arXiv: math.NA, Apr 2012, accepted in SIAM Journal on Scientific Computing. | MR 3123826 | Zbl 1285.65009

[21] A.T. Patera and G. Rozza, Reduced Basis Approximation and A Posteriori Error Estimation for Parametrized Partial Differential Equations, Version 1.0, Copyright MIT 2006, to appear in (tentative rubric) MIT Pappalardo Graduate Monographs in Mechanical Engineering. | Zbl pre05344486

[22] A. Quarteroni, G. Rozza and A. Manzoni, Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1 (2011) 3. | MR 2824231 | Zbl 1273.65148

[23] S. Repin, A Posteriori Estimates for Partial Differential Equations, Walter de Gruyter, Berlin (2008). | MR 2458008 | Zbl 1162.65001

[24] G. Rozza and K. Veroy, On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196 (2007) 1244-1260. | MR 2281777 | Zbl 1173.76352

[25] G. Rozza, D.B.P. Huynh and A.T. Patera, Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations - Application to transport and continuum mechanics. Archives Comput. Methods Engrg. 15 (2008) 229-275. | MR 2430350 | Zbl pre05344486

[26] S. Sen, Reduced-basis approximation and a posteriori error estimation for many-parameter heat conduction problems. Numerical Heat Transfer, Part B: Fundamentals 54 (2008) 369-389.

[27] V.N. Temlyakov, Greedy Approximation. Acta Numerica (2008) 235-409. | MR 2436013 | Zbl 1178.65050

[28] K. Veroy, Reduced-Basis Methods Applied to Problems in Elasticity: Analysis and Applications, MIT Thesis (2003).

[29] K. Veroy, C. Prudhomme, D.V. Rovas and A. Patera, A posteriori error bounds for reduced basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations, in Proc. 16th AIAA Comput. Fluid Dynamics Conf. (2003). Paper 2003-3847.

[30] S. Zhang, Efficient greedy algorithms for successive constraints methods with high-dimensional parameters, Brown Division of Applied Math Scientific Computing Tech Report 2011-23.