A unified treatment of tractability for approximation problems defined on Hilbert spaces

IF 1.8 2区数学 Q1 MATHEMATICS Journal of Complexity Pub Date : 2024-04-20 DOI:10.1016/j.jco.2024.101856

Onyekachi Emenike , Fred J. Hickernell , Peter Kritzer

引用次数: 0

Abstract

A large literature specifies conditions under which the information complexity for a sequence of numerical problems defined for dimensions $1, 2, \dots$ 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.

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对定义在希尔伯特空间上的近似问题可操作性的统一处理

大量文献阐述了这样的条件，即以维数定义的一系列数值问题的信息复杂度以适度的速度增长，即问题序列是......。在此，我们重点研究可用信息空间由所有线性函数组成的情况，问题被定义为希尔伯特空间之间的线性算子映射。我们统一了已知可操作性结果的证明，并对一些现有结果进行了概括。这些概括表达为五个定理，它们用解算子奇异值的函数之和提供了（强）可操作性的等价条件。

本文章由计算机程序翻译，如有差异，请以英文原文为准。

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来源期刊

Journal of Complexity 工程技术-计算机：理论方法

CiteScore

3.10

自引率

17.60%

发文量

审稿时长

>12 weeks

期刊介绍： The multidisciplinary Journal of Complexity publishes original research papers that contain substantial mathematical results on complexity as broadly conceived. Outstanding review papers will also be published. In the area of computational complexity, the focus is on complexity over the reals, with the emphasis on lower bounds and optimal algorithms. The Journal of Complexity also publishes articles that provide major new algorithms or make important progress on upper bounds. Other models of computation, such as the Turing machine model, are also of interest. Computational complexity results in a wide variety of areas are solicited. Areas Include: • Approximation theory • Biomedical computing • Compressed computing and sensing • Computational finance • Computational number theory • Computational stochastics • Control theory • Cryptography • Design of experiments • Differential equations • Discrete problems • Distributed and parallel computation • High and infinite-dimensional problems • Information-based complexity • Inverse and ill-posed problems • Machine learning • Markov chain Monte Carlo • Monte Carlo and quasi-Monte Carlo • Multivariate integration and approximation • Noisy data • Nonlinear and algebraic equations • Numerical analysis • Operator equations • Optimization • Quantum computing • Scientific computation • Tractability of multivariate problems • Vision and image understanding.