论文标题
最佳抽样和基督教在一般领域起作用
Optimal sampling and Christoffel functions on general domains
论文作者
论文摘要
我们考虑在给定采样点的评估中重建l^2（d，μ）$中未知功能的问题$ x^1，\ dots，x^m \ in D $，其中$ d \ subset \ subset \ mathbb r^d $是一般域和$μ$ $ $ a aboritibality Materiable Materiable Matery。近似是从线性空间$ v_n $的利息中挑选的，其中$ n = \ dim（v_n）$。最近的结果表明，某些加权最小二乘方法的实现接近最佳近似，采样预算$ m $与$ n $成正比，最多可进行对数因子$ \ ln（2n/\ varepsilon）$，其中$ \ varepsilon> 0 $是失败的可能性。采样点应根据精心挑选的概率度量$σ$随机选择，其密度由逆基督佛尔函数给出，该函数均取决于$ v_n $和$μ$。虽然当$ d $和$μ$具有张量产品结构时，这种方法会极大地促进，但对于具有任意几何形状的域$ d $，它变得有问题，因为最佳度量取决于$ l^2（d，μ）中的$ v_n $的正常基础，即使对简单的多种机构空间也不明确。因此，根据此措施进行采样实际上是不可行的。在本文中，我们讨论了实用的抽样策略，该策略是使用可以在离线阶段计算的扰动度量的$ \widetildeσ$，而不涉及$ u $的测量。我们表明，在近乎最佳的采样预算中，由此产生的加权最小二乘法实现了几乎最佳的近似，我们讨论了多级方法，这些方法可以在迭代划分时保留累积采样预算的最佳性。这些策略依赖于基督教反诉功能的A-Priori上限的知识。我们为多元代数多项式和通用域$ d $建立了$ v_n $的范围。
We consider the problem of reconstructing an unknown function $u\in L^2(D,μ)$ from its evaluations at given sampling points $x^1,\dots,x^m\in D$, where $D\subset \mathbb R^d$ is a general domain and $μ$ a probability measure. The approximation is picked from a linear space $V_n$ of interest where $n=\dim(V_n)$. Recent results have revealed that certain weighted least-squares methods achieve near best approximation with a sampling budget $m$ that is proportional to $n$, up to a logarithmic factor $\ln(2n/\varepsilon)$, where $\varepsilon>0$ is a probability of failure. The sampling points should be picked at random according to a well-chosen probability measure $σ$ whose density is given by the inverse Christoffel function that depends both on $V_n$ and $μ$. While this approach is greatly facilitated when $D$ and $μ$ have tensor product structure, it becomes problematic for domains $D$ with arbitrary geometry since the optimal measure depends on an orthonormal basis of $V_n$ in $L^2(D,μ)$ which is not explicitly given, even for simple polynomial spaces. Therefore sampling according to this measure is not practically feasible. In this paper, we discuss practical sampling strategies, which amount to using a perturbed measure $\widetilde σ$ that can be computed in an offline stage, not involving the measurement of $u$. We show that near best approximation is attained by the resulting weighted least-squares method at near-optimal sampling budget and we discuss multilevel approaches that preserve optimality of the cumulated sampling budget when the spaces $V_n$ are iteratively enriched. These strategies rely on the knowledge of a-priori upper bounds on the inverse Christoffel function. We establish such bounds for spaces $V_n$ of multivariate algebraic polynomials, and for general domains $D$.
