Randomized complexity of parametric integration and the role of adaption II. Sobolev spaces

IF 1.8 2区数学 Q1 MATHEMATICS

Journal of Complexity Pub Date : 2024-01-02 DOI:10.1016/j.jco.2023.101823

Stefan Heinrich

引用次数: 0

Abstract

We study the complexity of randomized computation of integrals depending on a parameter, with integrands from Sobolev spaces. That is, for $r, d_{1}, d_{2} \in N$ , $1 \leq p, q \leq \infty$ , $D_{1} = {[0, 1]}^{d_{1}}$ , and $D_{2} = {[0, 1]}^{d_{2}}$ we are given $f \in W_{p}^{r} (D_{1} \times D_{2})$ and we seek to approximate $(S f) (s) = \int_{D_{2}} f (s, t) d t (s \in D_{1}),$ with error measured in the $L_{q} (D_{1})$ -norm. Information is standard, that is, function values of f. Our results extend previous work of Heinrich and Sindambiwe (1999) [10] for $p = q = \infty$ and Wiegand (2006) [15] for $1 \leq p = q < \infty$ . Wiegand's analysis was carried out under the assumption that $W_{p}^{r} (D_{1} \times D_{2})$ is continuously embedded in $C (D_{1} \times D_{2})$ (embedding condition). We also study the case that the embedding condition does not hold. For this purpose a new ingredient is developed – a stochastic discretization technique. In Part I a basic problem of Information-Based Complexity on the power of adaption for linear problems in the randomized setting was solved. Here a further aspect of this problem is settled.

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参数积分的随机复杂性和适应的作用 II.索波列夫空间

我们研究的是随机计算取决于参数的积分的复杂性，积分来自索波列夫空间。也就是说，对于 r,d1,d2∈N，1≤p,q≤∞，D1=[0,1]d1，D2=[0,1]d2，我们给定了 f∈Wpr(D1×D2)，我们寻求逼近(Sf)(s)=∫D2f(s,t)dt(s∈D1)，误差以 Lq(D1)-norm 度量。我们的结果扩展了海因里希和辛丹比韦（《复杂性学报》，15 (1999)，317-341）先前针对 p=q=∞ 和维甘德（Shaker Verlag，2006）针对 1≤p=q<∞ 所做的工作。Wiegand 的分析是在 Wpr(D1×D2) 连续嵌入 C(D1×D2) 的假设条件（嵌入条件）下进行的。我们还研究了嵌入条件不成立的情况。本文以第一部分为基础，研究了矢量均值计算--参数积分的有限维对应物。在第一部分中，解决了基于信息的复杂性的一个基本问题，即随机设置中线性问题的适应能力。这里解决了这个问题的另一个方面。

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.