A new characterization of (s,t) -weak tractability

Paweł Siedlecki

Paweł Siedlecki

Michael Gnewuch, Henryk Woźniakowski

Weiran Ding, Jiansong Li, Yumeng Jia +1 more

This paper is devoted to the study of tractability of the $ L_2 $-approximation and integration from weighted Korobov spaces of increasing smoothness in the worst-case setting. The considered algorithms use information from the class $ \Lambda^{\rm all} $, including all continuous linear functionals, and from the class $ \Lambda^{\rm std} $, including function evaluations. Necessary and sufficient conditions on the weights of the function space for strong polynomial tractability, polynomial tractability, quasi-polynomial tractability, uniform weak tractability, weak tractability and $ (\sigma, \tau) $-weak tractability, are provided. Our results give a comprehensive picture of the weight conditions for all standard notions of algebraic tractability. It may be helpful to study the tractability of nonhomogeneous tensor product spaces.

Arthur G. Werschulz, H. Woźniakowski

М. Г. Степанов, Vladimir Chernyak, Michael Chertkov +1 more

One of the main obstacles to the wider use of the modern error-correction codes is that, due to the complex behavior of their decoding algorithms, no systematic method which would allow characterization of the bit-error-rate (BER) is known. This is especially true at the weak noise where many systems operate and where coding performance is difficult to estimate because of the diminishingly small number of errors. We show how the instanton method of physics allows one to solve the problem of BER analysis in the weak noise range by recasting it as a computationally tractable minimization problem.