Generalized Tractability for Multivariate Problems Part II: Linear Tensor Product Problems, Linear Information, and Unrestricted Tractability

We continue the study of generalized tractability initiated in our previous paper “Generalized tractability for multivariate problems, Part I: Linear tensor product problems and linear information”, J. Complexity, 23, 262-295 (2007). We study linear tensor product problems for which we can compute linear information which is given by arbitrary continuous linear functionals. We want to approximate an operator Sd given as the d-fold tensor product of a compact linear operator S1 for d = 1, 2, . . . , with ‖S1‖ = 1 and S1 has at least two positive singular values. Let n(ε, Sd) be the minimal number of information evaluations needed to approximate Sd to within ε ∈ [0, 1]. We study generalized tractability by verifying when n(ε, Sd) can be bounded by a multiple of a power of T (ε−1, d) for all (ε−1, d) ∈ Ω ⊆ [1,∞)×N. Here, T is a tractability function which is non-decreasing in both variables and grows slower than exponentially to infinity. We study the exponent of tractability which is the smallest power of T (ε−1, d) whose multiple bounds n(ε, Sd). We also study weak tractability, i.e., when limε−1+d→∞,(ε−1,d)∈Ω ln n(ε, Sd)/(ε + d) = 0.