On greedy algorithm approximating Kolmogorov widths in Banach spaces. (English) Zbl 1310.41031
Summary: The greedy algorithm to produce \(n\)-dimensional subspaces \(X_n\) to approximate a compact set \(\mathcal{F}\) contained in a Hilbert space was introduced in the context of reduced basis method in [Y. Maday et al., J. Sci. Comput. 17, No. 1–4, 437–446 (2002; Zbl 1014.65115); C. R., Math., Acad. Sci. Paris 335, No. 3, 289–294 (2002; Zbl 1009.65066)]. The same algorithm works for a general Banach space and in this context was studied in [R. DeVore et al., Constr. Approx. 37, No. 3, 455–466 (2013; Zbl 1276.41021)]. In this paper we study the case \(\mathcal{F} \subset L_p\). If Kolmogorov diameters \(d_n(\mathcal{F})\) of \(\mathcal{F}\) decay as \(n^{- \alpha}\) we give an almost optimal estimate for the decay of \(\sigma_n : = \operatorname{dist}(\mathcal{F}, X_n)\). We also give some direct estimates of the form \(\sigma_n \leq C_n d_n(\mathcal{F})\).
MSC:
| 41A65 | Abstract approximation theory (approximation in normed linear spaces and other abstract spaces) |
| 46B06 | Asymptotic theory of Banach spaces |
References:
| [1] | Albiac, F.; Kalton, N. J., Topics in Banach Space Theory, Grad. Texts in Math., vol. 233 (2006), Springer-Verlag · Zbl 1094.46002 |
| [2] | Binev, P.; Cohen, A.; Dahmen, W.; DeVore, R.; Petrova, G.; Wojtaszczyk, P., Convergence rates for greedy algorithms in reduced basis method, SIAM J. Math. Anal., 43, 3, 1457-1473 (2011) · Zbl 1229.65193 |
| [3] | Buffa, A.; Maday, Y.; Patera, A. T.; Prud’homme, C.; Turinici, G., A priori convergence of the greedy algorithm for the parametrized reduced basis, Modél. Math. Anal. Numér., 46, 595-603 (2012) · Zbl 1272.65084 |
| [4] | DeVore, R.; Petrova, G.; Wojtaszczyk, P., Greedy algorithms for reduced bases in Banach spaces, Constr. Approx., 37, 3, 455-466 (2013) · Zbl 1276.41021 |
| [5] | Haasdonk, B.; Salomon, J.; Wohlmuth, B., A reduced basis method for parametrized variational inequalities, SIAM J. Numer. Anal., 50, 2, 2656-2676 (2012) · Zbl 1261.65069 |
| [6] | Herrero, H.; Maday, Y.; Pla, F., RB (reduced basis) for RB (Rayleigh-Bénard), Comput. Methods Appl. Mech. Engrg., 261-262, 132-141 (2013) · Zbl 1286.76084 |
| [7] | Hewitt, E.; Stromberg, K., Real and Abstract Analysis (1969), Springer: Springer Berlin · Zbl 0225.26001 |
| [8] | Iwen, M. A.; Krahmer, F., Fast subspace approximation via greedy least square · Zbl 1406.65012 |
| [9] | Johnson, W. B.; Lindenstrauss, J., Basic concepts in the geometry of Banach spaces, (Johnson, W. B.; Lindenstrauss, J., Handbook of the Geometry of Banach Spaces, vol. I (2001), Elsevier: Elsevier Amsterdam), 1-84 · Zbl 1011.46009 |
| [10] | Joichi, J. T., Normed linear spaces equivalent to inner product spaces, Proc. Amer. Math. Soc., 17, 2, 423-426 (1966) · Zbl 0193.09001 |
| [11] | Lorentz, G. G.; Golitschek, M.; Makovoz, Y., Constructive Approximation, Advanced Problems, Grundlehren Math. Wiss., vol. 304 (1996), Springer-Verlag: Springer-Verlag Berlin · Zbl 0910.41001 |
| [12] | Maday, Y.; Patera, A. T.; Turinici, G., A priori convergence theory for reduced-basis approximations of single-parametric elliptic partial differential equations, J. Sci. Comput., 17, 437-446 (2002) · Zbl 1014.65115 |
| [13] | Maday, Y.; Patera, A. T.; Turinici, G., Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations, C. R. Math. Acad. Sci. Paris, 335, 2289-2294 (2002) · Zbl 1009.65066 |
| [14] | Pincus, A., \(n\)-Widths in Approximation Theory (1985), Springer-Verlag · Zbl 0551.41001 |
| [15] | Repovš, D.; Semenov, P. V., Continuous Selections of Multivalued Mappings (1998), Kluwer Academic Publishers: Kluwer Academic Publishers Dordrecht · Zbl 0915.54001 |
| [16] | Tomczak-Jaegermann, N., Banach-Mazur Distances and Finite-Dimensional Operator Ideals, Pitman Monogr. Surveys Pure Appl. Math., vol. 38 (1989) · Zbl 0721.46004 |
| [17] | Wojtaszczyk, P., Banach Spaces for Analysts, Cambridge Stud. Adv. Math., vol. 25 (1991), Cambridge University Press: Cambridge University Press Cambridge · Zbl 0724.46012 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
