Elsevier

Journal of Complexity

Volume 40, June 2017, Pages 1-16
Journal of Complexity

(s,lnκ)-weak tractability of linear problems

Dedicated to the memory of Joseph F. Traub (1932–2015)
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Abstract

We introduce a new notion of tractability for multivariate problems, namely (s,lnκ)-weak tractability for positive s and κ. This allows us to study the information complexity of a d-variate problem with respect to different powers of d and the bits of accuracy lnε1. We consider the worst case error for the absolute and normalized error criteria. We provide necessary and sufficient conditions for (s,lnκ)-weak tractability for general linear problems and linear tensor product problems defined over Hilbert spaces. In particular, we show that non-trivial linear tensor product problems cannot be (s,lnκ)-weakly tractable when s(0,1] and κ(0,1]. On the other hand, they are (s,lnκ)-weakly tractable for κ>1 and s>1 if the univariate eigenvalues of the linear tensor product problem enjoy a polynomial decay. Finally, we study (s,lnκ)-weak tractability for the remaining combinations of the values of s and κ.

Keywords

Multivariate problem
Complexity
Tractability

Cited by (0)

Communicated by E. Novak.