无限维全纯函数的最优逼近II:从i点样例中恢复

IF 1.8 2区 数学 Q1 MATHEMATICS
Ben Adcock , Nick Dexter , Sebastian Moraga
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引用次数: 0

摘要

在过去的几十年里,由于无限维全纯函数与参数微分方程和计算不确定性量化的相关性,人们对它们进行了详细的研究。由于构造替代模型对物理过程的复杂数学模型的实际重要性,从有限多样本中逼近这些函数是特别有趣的。在之前的工作中,我们研究了无限维超立方体[−1,1]N上所谓的banach值,(b,ε)-全纯函数的近似,来自m(潜在自适应)样本。特别地,我们推导了这类函数的自适应m-宽度的下界,这表明无论采样-恢复对如何,形式为m1/2−1/p的某些代数速率都是最好的。在这项工作中,我们通过关注实际情况来继续这项调查,其中样本是与问题的潜在概率度量相同且独立的逐点评估。具体来说,对于hilbert值(b,ε)-全纯函数,我们证明了对于张量积Jacobi测度可以达到相同的速率(直到一个小的多对数或代数因子)。我们的重建地图是基于最小二乘和压缩感知程序使用相应的正交雅可比多项式。在这样做的过程中,我们加强和推广了过去的工作,这些工作仅为一致和切比雪夫测度(以及相应的多项式)导出了较弱的非一致保证。我们还将各种最佳s项多项式近似误差界推广到任意雅可比多项式展开式。总的来说,我们证明了从潜在的概率度量中提取的i.i.d点样本对于无限维全纯函数的恢复是接近最优的。
本文章由计算机程序翻译,如有差异,请以英文原文为准。
Optimal approximation of infinite-dimensional holomorphic functions II: Recovery from i.i.d. pointwise samples
Infinite-dimensional, holomorphic functions have been studied in detail over the last several decades, due to their relevance to parametric differential equations and computational uncertainty quantification. The approximation of such functions from finitely-many samples is of particular interest, due to the practical importance of constructing surrogate models to complex mathematical models of physical processes. In a previous work, [5] we studied the approximation of so-called Banach-valued, (b,ε)-holomorphic functions on the infinite-dimensional hypercube [1,1]N from m (potentially adaptive) samples. In particular, we derived lower bounds for the adaptive m-widths for classes of such functions, which showed that certain algebraic rates of the form m1/21/p are the best possible regardless of the sampling-recovery pair. In this work, we continue this investigation by focusing on the practical case where the samples are pointwise evaluations drawn identically and independently from the underlying probability measure for the problem. Specifically, for Hilbert-valued (b,ε)-holomorphic functions, we show that the same rates can be achieved (up to a small polylogarithmic or algebraic factor) for tensor-product Jacobi measures. Our reconstruction maps are based on least squares and compressed sensing procedures using the corresponding orthonormal Jacobi polynomials. In doing so, we strengthen and generalize past work that has derived weaker nonuniform guarantees for the uniform and Chebyshev measures (and corresponding polynomials) only. We also extend various best s-term polynomial approximation error bounds to arbitrary Jacobi polynomial expansions. Overall, we demonstrate that i.i.d. pointwise samples drawn from an underlying probability measure are near-optimal for the recovery of infinite-dimensional, holomorphic functions.
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来源期刊
Journal of Complexity
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
57
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.
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