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Efficient greedy algorithms for high-dimensional parameter spaces with applications to empirical interpolation and reduced basis methods∗

Published online by Cambridge University Press:  10 January 2014

Jan S. Hesthaven ,
Benjamin Stamm  and
Shun Zhang
Jan S. Hesthaven
Affiliation:
Division of Applied Mathematics, Box F, Brown University, 182 George St., Providence, RI 02912, USA. Jan.Hesthaven@Brown.edu
Benjamin Stamm
Affiliation:
CNRS, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France; stamm@ann.jussieu.fr UPMC Univ Paris 06, UMR 7598, Laboratoire Jacques-Louis Lions, F-75005, Paris, France
Shun Zhang
Affiliation:
Department of Mathematics, City University of Hong Kong, Kowloon Tong, Hong Kong, China; shun.zhang@cityu.edu.hk
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Abstract

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.

Keywords

Greedy algorithmreduced basis methodempirical interpolation method
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 48 , Issue 1 , January 2014 , pp. 259 - 283
Copyright
© EDP Sciences, SMAI, 2014

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References

Bank, R.E. and Weiser, A., Some a posteriori error estimators for elliptic partial differential equations. Math. Comput. 44 (1985) 303–320. Google Scholar
Barrault, M., Nguyen, N.C., Maday, Y. and Patera, A.T., 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. Google Scholar
Binev, P., Cohen, A., Dahmen, W., DeVore, R., Petrova, G. and Wojtaszczyk, P., Convergence rates for greedy algorithms in reduced basis methods. SIAM J. Math. Anal. 43 (2011) 1457–1472. Google Scholar
Buffa, A., Maday, Y., Patera, A., Prud’homme, C. and Turinici, G., A priori convergence of the greedy algorithm for the parametrized reduced basis. M2AN 46 (2012) 595–603. Special Issue in honor of David Gottlieb. Google Scholar
T. Bui-Thanh, Model-Constrained Optimization Methods for Reduction of Parameterized Large-Scale Systems, MIT Thesis (2007).
Bui-Thanh, T., Willcox, K. and Ghattas, O., Model reduction for large-scale systems with high-dimensional parametric input space. SIAM J. Sci. Comput. 30 (2008) 3270–3288. Google Scholar
Chen, Y., Hesthaven, J.S., Maday, Y. and Rodriguez, J., 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. Google Scholar
Chen, Y., Hesthaven, J.S., Maday, Y. and Rodriguez, J., Improved successive constraint method based a posteriori error estimate for reduced basis approximation of 2d Maxwells problem. ESAIM: M2AN 43 (2009) 1099–1116. Google Scholar
Chen, Y., Hesthaven, J.S., Maday, Y. and Rodriguez, J., Certified reduced basis methods and output bounds for the harmonic maxwell equations. SIAM J. Sci. Comput. 32 (2010) 970–996. Google Scholar
Eftang, J.L., Patera, A.T. and Ronquist, E.M., An “hp” certified reduced basis method for parametrized elliptic partial differential equations. SIAM J. Sci. Comput. 32 (2010) 3170–3200. Google Scholar
Eftang, J.L. and Stamm, B., Parameter multi-domain hp empirical interpolation. Int. J. Numer. Meth. Engng. 90 (2012) 412–428. Google Scholar
Fares, B., Hesthaven, J.S., Maday, Y. and Stamm, B., The reduced basis method for the electric field integral equation. J. Comput. Phys. 230 (2011) 5532–5555. Google Scholar
Grepl, M.A., Maday, Y., Nguyen, N. C. and Patera, A.T., Efficient reduced-basis treatment of nonaffine and nonlinear partial differential equations. Math. Model. Numer. Anal. 41 (2007) 575–605. Google Scholar
Grepl, M.A. and Patera, A.T., A posteriori error bounds for reduced-basis approximations of parametrized parabolic partial differential equations. M2AN 39 (2005) 157–181.
Haasdonk, B., Dihlmann, M. and Ohlberger, M., 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. Google Scholar
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.
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.
Huynh, D.B.P., Rozza, G., Sen, S. and Patera, A.T., 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. Google Scholar
Maday, Y., Nguyen, N.C., Patera, A.T. and Pau, G.S.H., A general multipurpose interpolation procedure: the magic points. Commun. Pure Appl. Anal. 8 (2009) 383–404. Google Scholar
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.
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.
Quarteroni, A., Rozza, G. and Manzoni, A., Certified reduced basis approximation for parametrized partial differential equations and applications. J. Math. Ind. 1 (2011) 3. Google Scholar
S. Repin, A Posteriori Estimates for Partial Differential Equations, Walter de Gruyter, Berlin (2008).
Rozza, G. and Veroy, K., On the stability of the reduced basis method for Stokes equations in parametrized domains. Comput. Methods Appl. Mech. Eng. 196 (2007) 1244–1260. Google Scholar
Rozza, G., Huynh, D.B.P. and Patera, A.T., 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. Google Scholar
Sen, S., Reduced-basis approximation and a posteriori error estimation for many-parameter heat conduction problems. Numerical Heat Transfer, Part B: Fundamentals 54 (2008) 369–389. Google Scholar
V.N. Temlyakov, Greedy Approximation. Acta Numerica (2008) 235–409.
K. Veroy, Reduced-Basis Methods Applied to Problems in Elasticity: Analysis and Applications, MIT Thesis (2003).
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.
S. Zhang, Efficient greedy algorithms for successive constraints methods with high-dimensional parameters, Brown Division of Applied Math Scientific Computing Tech Report 2011-23.