论文标题
局部近似操作员
Local approximation of operators
论文作者
论文摘要
许多应用程序，例如系统识别，时间序列的分类，部分微分方程中的直接和反问题以及不确定性量化导致了公制空间之间非线性操作员近似的问题，$ \ mathfrak {x} $ {x} $和$ \ mathfrak {y} $。我们使用一定数量的信息研究了在紧凑型子集$ k_ \ mathfrak {x} \ subset \ mathfrak {x} $上确定此类运算符近似程度的问题。如果$ \ MATHCAL {F}：K_ \ MATHFRAK {X} \ to K_ \ Mathfrak {Y} $，这是一种完善的策略，可以近似$ \ Mathcal {f}（f）$ for k_ \ mathfrak {x ode $ f $ fil $ f $ fil $ fin（f），f $ fin（f）f $ fin（f）数字$ d $（$ m $）的实数。加上适当的重建算法（解码器），该问题将$ m $的近似值减少到高尺寸的欧几里得空间的紧凑子集上，同等地，单位sphere $ \ mathbb {s}^d $ embbb in $ \ nim sphere $ \ nim sphere $ \ nim sphey这个问题是具有挑战性的，因为$ d $，$ m $以及$ \ mathbb {s}^d $上近似值的复杂性都是很大的，并且有必要估算准确性，以跟踪所有涉及的近似值的相互依赖性。在本文中，我们建立了有效执行此操作的建设性方法。即，与$ \ Mathbb {s}^d $ as $ \ Mathcal {O}（d^{1/6}）$的近似值有关的常数。我们为操作员研究不同的平滑度课程，还提出了一种使用$ f $的小社区中仅信息的$ \ Mathcal {f}（f）$近似的方法，从而有效地减少了所涉及的参数数量。
Many applications, such as system identification, classification of time series, direct and inverse problems in partial differential equations, and uncertainty quantification lead to the question of approximation of a non-linear operator between metric spaces $\mathfrak{X}$ and $\mathfrak{Y}$. We study the problem of determining the degree of approximation of such operators on a compact subset $K_\mathfrak{X}\subset \mathfrak{X}$ using a finite amount of information. If $\mathcal{F}: K_\mathfrak{X}\to K_\mathfrak{Y}$, a well established strategy to approximate $\mathcal{F}(F)$ for some $F\in K_\mathfrak{X}$ is to encode $F$ (respectively, $\mathcal{F}(F)$) in terms of a finite number $d$ (repectively $m$) of real numbers. Together with appropriate reconstruction algorithms (decoders), the problem reduces to the approximation of $m$ functions on a compact subset of a high dimensional Euclidean space $\mathbb{R}^d$, equivalently, the unit sphere $\mathbb{S}^d$ embedded in $\mathbb{R}^{d+1}$. The problem is challenging because $d$, $m$, as well as the complexity of the approximation on $\mathbb{S}^d$ are all large, and it is necessary to estimate the accuracy keeping track of the inter-dependence of all the approximations involved. In this paper, we establish constructive methods to do this efficiently; i.e., with the constants involved in the estimates on the approximation on $\mathbb{S}^d$ being $\mathcal{O}(d^{1/6})$. We study different smoothness classes for the operators, and also propose a method for approximation of $\mathcal{F}(F)$ using only information in a small neighborhood of $F$, resulting in an effective reduction in the number of parameters involved.
