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Convergence Rates of the POD–Greedy Method

Published online by Cambridge University Press:  17 April 2013

Bernard Haasdonk*
Affiliation:
Institute of Applied Analysis and Numerical Simulation, University of Stuttgart, Germany. haasdonk@mathematik.uni-stuttgart.de
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Abstract

Iterative approximation algorithms are successfully applied in parametric approximation tasks. In particular, reduced basis methods make use of the so-called Greedy algorithm for approximating solution sets of parametrized partial differential equations. Recently, a priori convergence rate statements for this algorithm have been given (Buffa et al. 2009, Binev et al. 2010). The goal of the current study is the extension to time-dependent problems, which are typically approximated using the POD–Greedy algorithm (Haasdonk and Ohlberger 2008). In this algorithm, each greedy step is invoking a temporal compression step by performing a proper orthogonal decomposition (POD). Using a suitable coefficient representation of the POD–Greedy algorithm, we show that the existing convergence rate results of the Greedy algorithm can be extended. In particular, exponential or algebraic convergence rates of the Kolmogorov n-widths are maintained by the POD–Greedy algorithm.

Type
Research Article
Copyright
© EDP Sciences, SMAI, 2013

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References

P. Binev, A. Cohen, W. Dahmen, R. DeVore, G. Petrova and P. Wojtaszczyk, Convergence rates for greedy algorithms in reduced basis methods. IGPM Report, RWTH Aachen 310 (2010).
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. Math. Model. Numer. Anal. submitted (2009).
R.O. Duda, P.E. Hart and D.G. Stork, Pattern Classification. Wiley Interscience, 2nd edition (2001).
Eftang, J.L., Knezevic, D. and Patera, A.T., An hp certified reduced basis method for parametrized parabolic partial differential equations. MCMDS, Math. Comput. Model. Dynamical Systems 17 (2011) 395422. Google Scholar
M.A. Grepl, Reduced-basis Approximations and a Posteriori Error Estimation for Parabolic Partial Differential Equations. Ph.D. Thesis. Massachusetts Inst. Techn. (2005).
Haasdonk, B. and Ohlberger, M., Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: M2AN 42 (2008) 277302. Google Scholar
Hinze, M. and Volkwein, S., Error estimates for abstract linear–quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl. 39 (2008) 319345. Google Scholar
P. Holmes, J. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996).
H. Hotelling, Analysis of a complex of statistical variables into principal components. J. Educational Psychol. (1933).
Ismagilov, R.S., On n-dimensional diameters of compacts in a Hilbert space. Functional Anal. Appl. 2 (1968) 125132. Google Scholar
I.T. Joliffe, Principal Component Analysis. John Wiley and Sons (2002).
K. Karhunen, Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Annal. Acad. Sri. Fennicae, Ser. A l . Math. Phys. 37 (1946).
Knezevic, D.J. and Patera, A.T., A certified reduced basis method for the Fokker-Planck equation of dilute polymeric fluids: FENE dumbbells in extensional flow. SIAM J. Sci. Comput. 32 (2010) 793817. Google Scholar
Kunisch, K. and Volkwein, S., Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117148. Google Scholar
M.M. Loeve, Probability Theory. Van Nostrand, Princeton, NJ (1955).
Maday, Y., Patera, A.T. and Turinici, G., a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptiv partial differential equations. C.R. Acad Sci. Paris, Der. I 335 (2002) 289294. Google Scholar
Makovoz, Y., On n-widths of certain functional classes defined by linear differential operators. Proc. Amer. Math. Soc. 89 (1983) 109112. 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. Arch. Comput. Meth. Engrg. 15 (2008) 229275. Google Scholar
Sirovich, L., Turbulence and the dynamics of coherent structures I. coherent structures. Quart. Appl. Math. 45 (1987) 561571. Google Scholar
Stechkin, S.B., On the best approximation of given classes of functions by arbitrary polynomials. Uspekhi Matematicheskikh Nauk 9 (1954) 133134. Google Scholar
Urban, K. and Patera, A.T., A new error bound for reduced basis approximation of parabolic partial differential equations. CRAS, Comptes Rendus Math. 350 (2012) 203207. Google Scholar
K. Veroy, C. Prud’homme, D.V. Rovas and A.T. Patera, A posteriori error bounds for reduced–basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In Proc. 16th AIAA computational fluid dynamics conference (2003) 2003–3847.
S. Volkwein, Model Reduction using Proper Orthogonal Decomposition, Lect. Notes. University of Constance (2011).