The curse of dimensionality for the class of monotone functions and for the class of convex functions |
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Authors: | Aicke Hinrichs Erich Novak Henryk Woźniakowski |
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Institution: | 1. Mathematisches Institut, Universität Jena, Ernst-Abbe-Platz 2, 07740 Jena, Germany;2. Department of Computer Science, Columbia University, New York, NY 10027, USA;3. Institute of Applied Mathematics, University of Warsaw, ul. Banacha 2, 02-097 Warszawa, Poland |
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Abstract: | We study the integration and approximation problems for monotone or convex bounded functions that depend on variables, where can be arbitrarily large. We consider the worst case error for algorithms that use finitely many function values. We prove that these problems suffer from the curse of dimensionality. That is, one needs exponentially many (in ) function values to achieve an error . |
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