Convergence rates of the pod—greedy method

Mathematical Modelling and Numerical Analysis

Volumn 47, Issue 3, 2013, Pages 859-873

  Haasdonk, Bernard ^a  

^a UNIVERSITY OF STUTTGART   (Germany)

Author keywords

Convergence rates; Greedy approximation; Proper orthogonal decomposition; Reduced Basis methods

Indexed keywords

EID: 84996141276     PISSN: 0764583X     EISSN: 12903841     Source Type: Journal    
DOI: 10.1051/m2an/2012045     Document Type: Article

References (23)

1. 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).
2. 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).
3. R.O. Duda, P.E. Hart and D.G. Stork, Pattern Classification. Wiley Interscience, 2nd edition (2001).
4. J. L. Eftang, D. Knezevic and A.T. Patera, An hp certified reduced basis method for parametrized parabolic partial differential equations. MCMDS, Math. Comput. Model. Dynamical Systems 17 (2011) 395-422.
5. M.A. Grepl, Reduced-basis Approximations and a Posteriori Error Estimation for Parabolic Partial Differential Equations. Ph.D. Thesis. Massachusetts Inst. Techn. (2005).
6. B. Haasdonk and M. Ohlberger, Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: M2AN 42 (2008) 277-302.
7. M. Hinze and S. Volkwein, Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl. 39 (2008) 319-345.
8. P. Holmes, J. Lumley and G. Berkooz, Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996).
9. H. Hotelling, Analysis of a complex of statistical variables into principal components. J. Educational Psychol. (1933).
10. R. S. Ismagilov, On n-dimensional diameters of compacts in a Hilbert space. Functional Anal. Appl. 2 (1968) 125-132.
11. I. T. Joliffe, Principal Component Analysis. John Wiley and Sons (2002).
12. K. Karhunen, Uber lineare Methoden in der Wahrscheinlichkeitsrechnung. Annal. Acad. Sri. Fennicae, Ser. A l. Math. Phys. 37 (1946).
13. D.J. Knezevic and A.T. Patera, 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) 793-817.
14. K. Kunisch and S. Volkwein, Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117-148.
15. M.M. Loeve, Probability Theory. Van Nostrand, Princeton, NJ (1955).
16. Y. Maday, A.T. Patera and G. Turinici, 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) 289-294.
17. Y. Makovoz, On n-widths of certain functional classes defined by linear differential operators. Proc. Amer. Math. Soc. 89 (1983) 109-112.
18. 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. Arch. Comput. Meth. Engrg. 15 (2008) 229-275.
19. L. Sirovich, Turbulence and the dynamics of coherent structures I. coherent structures. Quart. Appl. Math. 45 (1987) 561-571.
20. S. B. Stechkin, On the best approximation of given classes of functions by arbitrary polynomials. Uspekhi Matematicheskikh Nauk 9 (1954) 133-134.
21. K. Urban and A.T. Patera, A new error bound for reduced basis approximation of parabolic partial differential equations. CRAS, Comptes Rendus Math. 350 (2012) 203-207.
22. 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.
23. S. Volkwein, Model Reduction using Proper Orthogonal Decomposition, Lect. Notes. University of Constance (2011).