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 lnε−1. 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.
Original language | English (US) |
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Pages (from-to) | 1-16 |
Number of pages | 16 |
Journal | Journal of Complexity |
Volume | 40 |
DOIs | |
State | Published - Jun 1 2017 |
All Science Journal Classification (ASJC) codes
- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- Mathematics(all)
- Control and Optimization
- Applied Mathematics
Keywords
- Complexity
- Multivariate problem
- Tractability