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.
Mots clés : greedy approximation, proper orthogonal decomposition, convergence rates, reduced basis methods
@article{M2AN_2013__47_3_859_0, author = {Haasdonk, Bernard}, title = {Convergence {Rates} of the {POD-Greedy} {Method}}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis }, pages = {859--873}, publisher = {EDP-Sciences}, volume = {47}, number = {3}, year = {2013}, doi = {10.1051/m2an/2012045}, mrnumber = {3056412}, zbl = {1277.65074}, language = {en}, url = {http://www.numdam.org/articles/10.1051/m2an/2012045/} }
TY - JOUR AU - Haasdonk, Bernard TI - Convergence Rates of the POD-Greedy Method JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2013 SP - 859 EP - 873 VL - 47 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2012045/ DO - 10.1051/m2an/2012045 LA - en ID - M2AN_2013__47_3_859_0 ER -
Haasdonk, Bernard. Convergence Rates of the POD-Greedy Method. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 3, pp. 859-873. doi : 10.1051/m2an/2012045. http://www.numdam.org/articles/10.1051/m2an/2012045/
[1] Convergence rates for greedy algorithms in reduced basis methods. IGPM Report, RWTH Aachen 310 (2010). | MR | Zbl
, , , , and ,[2] A priori convergence of the greedy algorithm for the parametrized reduced basis. Math. Model. Numer. Anal. submitted (2009). | Numdam | Zbl
, , , and ,[3] Pattern Classification. Wiley Interscience, 2nd edition (2001). | MR | Zbl
, and ,[4] An hp certified reduced basis method for parametrized parabolic partial differential equations. MCMDS, Math. Comput. Model. Dynamical Systems 17 (2011) 395-422. | MR
, and ,[5] Reduced-basis Approximations and a Posteriori Error Estimation for Parabolic Partial Differential Equations. Ph.D. Thesis. Massachusetts Inst. Techn. (2005). | Numdam | Zbl
,[6] Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: M2AN 42 (2008) 277-302. | Numdam | MR
and ,[7] Error estimates for abstract linear-quadratic optimal control problems using proper orthogonal decomposition. Comput. Optim. Appl. 39 (2008) 319-345. | MR | Zbl
and ,[8] Turbulence, Coherent Structures, Dynamical Systems and Symmetry. Cambridge University Press, Cambridge (1996). | MR | Zbl
, and ,[9] Analysis of a complex of statistical variables into principal components. J. Educational Psychol. (1933). | JFM
,[10] On n-dimensional diameters of compacts in a Hilbert space. Functional Anal. Appl. 2 (1968) 125-132. | MR | Zbl
,[11] Principal Component Analysis. John Wiley and Sons (2002).
,[12] Über lineare Methoden in der Wahrscheinlichkeitsrechnung. Annal. Acad. Sri. Fennicae, Ser. A l . Math. Phys. 37 (1946). | MR | Zbl
,[13] 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. | MR | Zbl
and ,[14] Galerkin proper orthogonal decomposition methods for parabolic problems. Numer. Math. 90 (2001) 117-148. | MR | Zbl
and ,[15] Probability Theory. Van Nostrand, Princeton, NJ (1955). | MR | Zbl
,[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. | MR | Zbl
[17] On n-widths of certain functional classes defined by linear differential operators. Proc. Amer. Math. Soc. 89 (1983) 109-112. | MR | Zbl
,[18] 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. | MR
, and ,[19] Turbulence and the dynamics of coherent structures I. coherent structures. Quart. Appl. Math. 45 (1987) 561-571. | MR | Zbl
,[20] On the best approximation of given classes of functions by arbitrary polynomials. Uspekhi Matematicheskikh Nauk 9 (1954) 133-134. | Zbl
,[21] A new error bound for reduced basis approximation of parabolic partial differential equations. CRAS, Comptes Rendus Math. 350 (2012) 203-207. | MR | Zbl
and ,[22] 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.
, , and ,[23] Model Reduction using Proper Orthogonal Decomposition, Lect. Notes. University of Constance (2011).
,Cité par Sources :