The purpose of this article is to investigate (s, t)-weak tractability of multivariate linear problems in the average case setting. The considered algorithms use finitely many evaluations of arbitrary linear functionals. Generally, we obtained matching necessary and sufficient conditions for (s, t)-weak tractability in terms of the corresponding non-increasing sequence of eigenvalues. Specifically, we discussed (s, t)-weak tractability of linear tensor product problems and obtained necessary and sufficient conditions in terms of the corresponding one-dimensional problem. As an example of applications, we discussed also (s, t)-weak tractability of a multivariate approximation problem.
(s, t)-weak tractability linear problem linear tensor product problem Hilbert space average case setting
2010 MR Subject Classification
41A63 65Y20 68Q25
This is a preview of subscription content, log in to check access.
Siedlecki P, Weimar M. Notes on (s, t)-weak tractability: A refined classification of problems with (sub)exponential information complexity. J Approx Theory, 2015, 200: 227–258MathSciNetCrossRefzbMATHGoogle Scholar
Novak E, Woźniakowski H. Tractability of multivariate problems. Volume I: Linear Information//EMS Tracts in Mathematics. Vol 6. Zürich: European Mathematical Society (EMS), 2008Google Scholar
Novak E, Woźniakowski H. Tractability of multivariate problems. Volume II: Standard Information for Functionals//EMS Tracts in Mathematics. Vol 12. Zürich: European Mathematical Society (EMS), 2010Google Scholar
Novak E, Woźniakowski H. Tractability of multivariate problems. Volume III: Standard Information for Operator//EMS Tracts in Mathematics. Vol 18. Zürich: European Mathematical Society (EMS), 2012Google Scholar
Hickernell F J, Wasilkowski G W, Woźniakowski H. Tractability of linear multivariate problems in the average case setting//Keller A, Heinrich S, Niederreiter H. Monte Carlo and Quasi-Monte Carlo Methods 2006. Berlin: Springer, 2008: 461–494CrossRefGoogle Scholar
Lifshits M A, Papageorgiou A, Woźniakowski H. Tractability of multi-parametric Euler and Wiener integrated processes. Probability and Mathematical Statistics, 2012, 32(1): 131–165MathSciNetzbMATHGoogle Scholar
Weimar M. Breaking the curse of dimensionality[D]. Marburg: Philipps University Marburg, 2015, 505: 1–112MathSciNetGoogle Scholar
Lifshits M, Zani M. Approximation of additive random fields based on standard information: Average case and probabilistic settings. J Complexity, 2015, 31: 659–674MathSciNetCrossRefzbMATHGoogle Scholar