Tractability of tensor product problems in the average case setting

Abstract

It has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem S = { S d } in the average case setting to be weakly tractable but not polynomially tractable. As a result of the tensor product structure, the eigenvalues of the covariance operator of the induced measure in the one-dimensional problem characterize the complexity of approximating S d , d ≥ 1 , with accuracy ε . If ∑ j = 1 ∞ λ j < 1 and λ 2 > 0 , we know that S is not polynomially tractable iff lim sup j → ∞ λ j j p = ∞ for all p > 1 . Thus we settle the open problem by showing that S is weakly tractable iff ∑ j > n λ j = o ( ln − 2 n ) . In particular, assume that ℓ = lim j → ∞ λ j j ln 3 ( j + 1 ) , exists. Then S is weakly tractable iff ℓ = 0 .