On the tractability of linear tensor product problems in the worst case

Abstract

a b s t r a c tIt has been an open problem to derive a necessary and sufficient condition for a linear tensor product problem to be weakly tractable in the worst case. The complexity of linear tensor product problems in the worst case depends on the eigenvalues {λ i } i∈N of a certain operator. It is known that if λ 1 = 1 and λ 2 ∈ (0, 1) then λ n = o((ln n) −2 ), as n → ∞, is a necessary condition for a problem to be weakly tractable. We show that this is a sufficient condition as well.

Quasi-polynomial tractability

“…The sufficiency has recently been proved by Papageorgiou and Petras [10], improving the slightly weaker result of [3,8]. In [3,8] also the necessity was proved.…”

“…On the other hand, for some of them we do have weak tractability. In particular, this is the case for all linear tensor product problems for which the corresponding eigenvalues λ n for the univariate case go to zero faster than [ln n] −2 ; see [10]. This means that the information complexity n(ε, d) of such multivariate problems goes to infinity faster than any polynomial but slower than an exponential function in ε −1 and d. The question that we study here is that of characterizing more precisely the behavior of n(ε, d).…”

Notes on $(s,t)$-weak tractability: A refined classification of problems with (sub)exponential information complexity

“…In what follows we significantly extend the characterization of weak tractability for linear tensor product problems (as it can be found, e.g., in [9, Theorem 5.5] and [12]) to the case of (s, t)weak tractability. For this purpose, we first derive conditions which are necessary and sufficient for the limiting case min{s, t} = 0.…”

“…Given δ > 0 we can choose β as well as c > 0 small enough such that both entries of the maximum in (12) are smaller than δ for all pairs (ε, d) = (ε k , d k ) with k ∈ N and ε k ≤ c. It might happen that there remains a subsequence with…”

“…While Lemma 3.6 relates the decay of the sequence λ (1) = (λ j ) j∈N to the growth behavior of the univariate information complexity n abs (ε, S 1 ) as ε → 0, Lemma 3.7 below deals with an upper estimate for the multidimensional case. Its proof is based on combinatorial arguments similar to those used in [12] and in the proof of [9, Theorem 5.5], respectively.…”