A unified treatment of tractability for approximation problems defined on Hilbert spaces
Onyekachi Emenike, Fred J. Hickernell, Peter Kritzer
TL;DR
The paper addresses tractability for approximation problems defined on Hilbert spaces with full linear information by linking the information complexity $\\text{comp}(\\varepsilon,d)$ to tractability via a general function $T$ of $\\varepsilon^{-1}$, $d$, and parameters $\\boldsymbol{p}$. It develops five equivalent theorems that reduce (strong) tractability to summability conditions over the singular values $\\lambda_{i,d}$ of the solution operators, unifying a broad range of algebraic, exponential, separable, and quasi-polynomial notions. The framework also extends to restricted domains and introduces sub-$h$ tractability, offering a flexible toolkit for verifying tractability without restricting to tensor-product structures. The results provide practical criteria for verifying tractability in diverse settings and highlight the key influence of the decay of singular values on computational feasibility in high dimensions.
Abstract
A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions $1, 2, \ldots$ grows at a moderate rate, i.e., the sequence of problems is tractable. Here, we focus on the situation where the space of available information consists of all linear functionals and the problems are defined as linear operator mappings between Hilbert spaces. We unify the proofs of known tractability results and generalize a number of existing results. These generalizations are expressed as five theorems that provide equivalent conditions for (strong) tractability in terms of sums of functions of the singular values of the solution operators.