(s,ln^κ)-weak tractability of linear problems

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.