### Abstract

We introduce a new notion of tractability for multivariate problems, namely (s,ln^{κ})-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,ln^{κ})-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,ln^{κ})-weakly tractable when s∈(0,1] and κ∈(0,1]. On the other hand, they are (s,ln^{κ})-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,ln^{κ})-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

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## Cite this

^{κ})-weak tractability of linear problems.

*Journal of Complexity*,

*40*, 1-16. https://doi.org/10.1016/j.jco.2016.10.008