(s, Lnκ)-weak Tractability of Linear Problems

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 ln e − 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 .

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