Abstract
In this paper we study problems of optimal recovery in the case of a nonsymmetric convex class of functions. We concentrate on the reconstruction of linear functionals, i.e., the problem of optimal numerical integration. We begin with a class of convex functions. We prove that adaption cannot help in the worst case, but considerably helps in the case of Monte Carlo methods.
We give examples for linear problems on a convex set where (deterministic) adaptive methods are much better than nonadaptive ones.
This is a preview of subscription content, log in via an institution.
Buying options
Tax calculation will be finalised at checkout
Purchases are for personal use only
Learn about institutional subscriptionsPreview
Unable to display preview. Download preview PDF.
References
N. S. Bakhvalov (1959): On approximate computation of integrals. Vestnik MGU, Ser. Math. Mech. Astron. Phys. Chem. 4, 3–18. [Russian]
N. S. Bakhvalov (1971): On the optimality of linear methods for operator approximation in convex classes of functions. USSR Comput. Math. Math. Phys. 11, 244–249. [Russian original: Zh. vychisl. Mat. mat. Fiz. 11, 244–249]
H. Braß (1982): Zur Quadraturtheorie konvexer Funktionen. In: G. Hämmerlin (ed.), Numerical integration, pp. 34–47, ISNM 57.
K.-J. Förster, K. Petras (1990): On a problem proposed by H. Braß concerning the remainder term in quadrature for convex functions. Numer. Math. 57, 763–777.
S. Gal, C. A. Micchelli (1980): Optimal sequential and non-sequential procedures for evaluating a functional. Appl. Anal. 10, 105–120.
I. A. Glinkin (1983): Optimal algorithms for integrating convex functions. Math. USSR Comput. Maths. Math. Phys. 23(2), 6–12. [Russian original: Zh. vychisl. Mat. mat. Fiz. 23(2), 267–277]
I. A. Glinkin (1984): Best quadrature formula in the class of convex functions. Math. Notes 35, 368–374. [Russian original: Mat. Zametki 35, 697–707]
R. Gorenflo, S. Vessella (1991): Abel Integral Equations. Lecture Notes in Mathematics 1461, Springer-Verlag.
G. Heindl (1982): Optimal quadrature of convex functions. In “Numerical Integration”, ISNM 57, 138–147.
I. P. Huerta (1986): Adaption helps for some nonconvex classes. J. Complexity 2, 333–352.
A. Iserles, G. Söderlind (1992): Global bounds on numerical error for ordinary differential equations. Numerical Analysis Reports, University of Cambridge, England, to appear in J. Complexity
J. Kiefer (1957): Optimum sequential search and approximation methods under minimum regularity assumptions. J. Soc. Indust. Appl. Math. 5, 105–136.
M. A. Kon, E. Novak (1989): On the adaptive and continuous information problems. J. Complexity 5, 345–362.
M. A. Kon, E. Novak (1990): The adaption problem for approximating linear operators. Bull. Amer. Math. Soc. 23, 159–165.
N. P. Korneichuk (1993): Optimization of active algorithms for recovery of functions from the class Lip-α. To appear in J. Complexity
C. A. Micchelli, T. J. Rivlin (1977): A survey of optimal recovery. In: Optimal estimation in approximation theory, C. A. Micchelli and T. J. Rivlin (eds.), Plenum Press, New York.
C. A. Micchelli, T. J. Rivlin (1985): Lectures in optimal recovery. Lecture Notes in Math. 1129, Springer.
E. Novak (1988a): Deterministic and stochastic error bounds in numerical analysis. Lecture Notes in Mathematics 1349, Springer.
E. Novak (1988b): Stochastic properties of quadrature formulas. Numer. Math. 53, 609–620.
E. Novak (1992): Quadrature formulas for monotone functions. Proc. Amer. Math. Soc. 115, 59–68.
E. Novak (1993): Optimal recovery and n-widths for convex classes of functions. In preparation.
A. Papageorgiou (1992): Integration of monotone functions of several variables. To appear in J. Complexity
K. Petras (1993): Quadrature theory of convex functions. These proceedings.
K. Ritter, G. W. Wasilkowski, H. Wońniakowski (1993): On multivariate integration for stochastic processes. These proceedings.
A. Schnorr (1992): Optimale Verfahren zur Integration konvexer Funktionen. Diplomarbeit, Mathematisches Institut Erlangen.
G. Sonnevend (1983): An optimal sequential algorithm for the uniform approximation of convex functions on [0,1]2. Appl. Math. Optirn. 10, 127–142.
A. G. Sukharev (1986): On the existence of optimal affine methods for approximating linear functionals. J. Complexity 2, 317–322.
V. M. Tikhomirov (1990): Approximation theory. In: Encyclopaedia of Mathematical Sciences, Vol. 14, Springer-Verlag. [The Russian original was published by Publisher VINITI, Moscow 1987, R. V. Gamkrelidze, ed.]
J. F. Traub, G. W. Wasilkowski, H. Wožniakowski (1988): Information-based complexity. Academic Press.
J. F. Traub, H. Wońniakowski (1980): A general theory of optimal algorithms. Academic Press.
G. W. Wasilkowski, F. Gao (1992): On the power of adaptive information for functions with singularities. Math, of Comp. 58, 285–304.
G. W. Wasilkowski, H. Wožniakowski (1984): Can adaption help on the average?, Numer. Math. 44, 169–190.
D. Zwick (1988): Optimal quadrature for convex functions and generalizations. In: Numerical Integration III, H. Braß, G. Hämmerlin (eds.), ISNM 85, 310–315.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 1993 Springer Basel AG
About this chapter
Cite this chapter
Novak, E. (1993). Quadrature Formulas For Convex Classes of Functions. In: Brass, H., Hämmerlin, G. (eds) Numerical Integration IV. ISNM International Series of Numerical Mathematics, vol 112. Birkhäuser, Basel. https://doi.org/10.1007/978-3-0348-6338-4_22
Download citation
DOI: https://doi.org/10.1007/978-3-0348-6338-4_22
Publisher Name: Birkhäuser, Basel
Print ISBN: 978-3-0348-6340-7
Online ISBN: 978-3-0348-6338-4
eBook Packages: Springer Book Archive