Frames and Numerical Approximation II: Generalized Sampling
Frames and Numerical Approximation II: Generalized Sampling
复制标题
框架和数值逼近 II：广义采样
DOI：
--
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发表时间：
2018
影响因子：
1.2
通讯作者：
D. Huybrechs
中科院分区：
文献类型：
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作者：
B. Adcock;D. Huybrechs
在之前的一篇论文（Adcock 和 Huybrechs in SIAM Rev 61(3):443–473, 2019）中，我们描述了使用冗余集和框架的函数的数值逼近。与使用基相比，函数表示中的冗余提供了巨大的灵活性，但病态条件通常会妨碍最佳近似值的数值计算。我们表明，尽管存在上述病态条件，正则化近似仍然可以提供高达 ϵdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} 的精度egin{文档}$$sqrt{epsilon}$$end{文档}，其中 ϵdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt}egin{文档}$$epsilon $$end{document} 是一个小的截断阈值。当使用框架时，即通常是冗余的但提供具有有限范数系数的无限表示的完整系统，实际上可以为空间中的所有函数实现这种精度。在这里，我们以两种方式概括该设置。我们假设 f 的信息或样本来自作用于 f 的各种线性算子，而不是与最佳近似投影相关的内积。这使得能够仅基于函数值来分析完全离散的近似值。接下来，我们允许过采样，从而得到最小二乘近似。我们表明，这可以大大提高 ϵdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$epsilon $$end{document} 顺序的准确性，而不是ϵdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt}egin{文档}$$sqrt{epsilon}$$end{文档}。总的来说，我们证明，尽管必须求解病态方程组，但使用冗余表示的数值函数逼近可能会导致高度准确的逼近。
In a previous paper (Adcock and Huybrechs in SIAM Rev 61(3):443–473, 2019) we described the numerical approximation of functions using redundant sets and frames. Redundancy in the function representation offers enormous flexibility compared to using a basis, but ill-conditioning often prevents the numerical computation of best approximations. We showed that, in spite of said ill-conditioning, approximations with regularization may still provide accuracy up to order ϵdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$sqrt{epsilon }$$end{document}, where ϵdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$epsilon $$end{document} is a small truncation threshold. When using frames, i.e. complete systems that are generally redundant but which provide infinite representations with coefficients of bounded norm, this accuracy can actually be achieved for all functions in a space. Here, we generalize that setting in two ways. We assume information or samples from f from a wide class of linear operators acting on f, rather than inner products associated with the best approximation projection. This enables the analysis of fully discrete approximations based, for instance, on function values only. Next, we allow oversampling, leading to least-squares approximations. We show that this leads to much improved accuracy on the order of ϵdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$epsilon $$end{document} rather than ϵdocumentclass[12pt]{minimal} usepackage{amsmath} usepackage{wasysym} usepackage{amsfonts} usepackage{amssymb} usepackage{amsbsy} usepackage{mathrsfs} usepackage{upgreek} setlength{oddsidemargin}{-69pt} egin{document}$$sqrt{epsilon }$$end{document}. Overall, we demonstrate that numerical function approximation using redundant representations may lead to highly accurate approximations in spite of having to solve ill-conditioned systems of equations.