数学物理学报(英文版) ›› 2019, Vol. 39 ›› Issue (4): 1033-1052.doi: 10.1007/s10473-019-0409-x

• 论文 • 上一篇    下一篇

(S, T)-WEAK TRACTABILITY OF MULTIVARIATE LINEAR PROBLEMS IN THE AVERAGE CASE SETTING

刘永平1, 许贵桥2   

  1. 1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China;
    2. Department of Mathematics, Tianjin Normal University, Tianjin 300387, China
  • 收稿日期:2017-10-24 修回日期:2018-07-02 出版日期:2019-08-25 发布日期:2019-09-12
  • 作者简介:Yongping LIU,E-mail:ypliu@bnu.edu.cn;Guiqiao XU,E-mail:Xuguiqiao@aliyun.com
  • 基金资助:
    This work was supported by the National Natural Science Foundation of China (11471043, 11671271) and the Beijing Natural Science Foundation (1172004).

(S, T)-WEAK TRACTABILITY OF MULTIVARIATE LINEAR PROBLEMS IN THE AVERAGE CASE SETTING

Yongping LIU1, Guiqiao XU2   

  1. 1. School of Mathematical Sciences, Beijing Normal University, Beijing 100875, China;
    2. Department of Mathematics, Tianjin Normal University, Tianjin 300387, China
  • Received:2017-10-24 Revised:2018-07-02 Online:2019-08-25 Published:2019-09-12
  • Supported by:
    This work was supported by the National Natural Science Foundation of China (11471043, 11671271) and the Beijing Natural Science Foundation (1172004).

摘要: 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.

关键词: (s,t)-weak tractability, linear problem, linear tensor product problem, Hilbert space, average case setting

Abstract: 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.

Key words: (s,t)-weak tractability, linear problem, linear tensor product problem, Hilbert space, average case setting

中图分类号: 

  • 41A63