Tractability of multivariate problems for standard and linear information in the worst case setting: part II

We study QPT (quasi-polynomial tractability) in the worst case setting for linear tensor product problems defined over Hilbert spaces. We assume that the domain space is a reproducing kernel Hilbert space so that function values are well defined. We prove QPT for algorithms that use only function values under the three assumptions: 1) the minimal errors for the univariate case decay polynomially fast to zero, 2) the largest singular value for the univariate case is simple and 3) the eigenfunction corresponding to the largest singular value is a multiple of the function value at some point. The first two assumptions are necessary for QPT. The third assumption is necessary for QPT for some Hilbert spaces

Tractability of multivariate analytic problems

In the theory of tractability of multivariate problems one usually studies problems with finite smoothness. Then we want to know which $s$-variate problems can be approximated to within $\varepsilon$ by using, say, polynomially many in $s$ and $\varepsilon^{-1}$ function values or arbitrary linear functionals. There is a recent stream of work for multivariate analytic problems for which we want to answer the usual tractability questions with $\varepsilon^{-1}$ replaced by $1+\log \varepsilon^{-1}$. In this vein of research, multivariate integration and approximation have been studied over Korobov spaces with exponentially fast decaying Fourier coefficients. This is work of J. Dick, G. Larcher, and the authors. There is a natural need to analyze more general analytic problems defined over more general spaces and obtain tractability results in terms of $s$ and $1+\log \varepsilon^{-1}$. The goal of this paper is to survey the existing results, present some new results, and propose further questions for the study of tractability of multivariate analytic questions

The Complexity of Linear Tensor Product Problems in (Anti-) Symmetric Hilbert Spaces

We study linear problems defined on tensor products of Hilbert spaces with an additional (anti-) symmetry property. We construct a linear algorithm that uses finitely many continuous linear functionals and show an explicit formula for its worst case error in terms of the singular values of the univariate problem. Moreover, we show that this algorithm is optimal with respect to a wide class of algorithms and investigate its complexity. We clarify the influence of different (anti-) symmetry conditions on the complexity, compared to the classical unrestricted problem. In particular, for symmetric problems we give characterizations for polynomial tractability and strong polynomial tractability in terms of the amount of the assumed symmetry. Finally, we apply our results to the approximation problem of solutions of the electronic Schr\"odinger equation.Comment: Extended version (53 pages); corrected typos, added journal referenc