论文标题
贝叶斯正交$ h^1（μ）$带有庞加莱不平等，紧凑的间隔
Bayesian quadrature for $H^1(μ)$ with Poincaré inequality on a compact interval
论文作者
论文摘要
通过对复杂系统的不确定性量化的动机，我们旨在寻找$ \ int_a^b f（x）dμ（x）= \ sum_ {i = 1}^n w_i f（x_i）$的正交公式，其中$ f $属于$ h^1（μ）$。在这里，$μ$属于$ [a，b] \ subset \ mathbb {r} $和$ \ sum_ {i = 1}^nw_iδ_{x_i} $是$ [a，b] $的离散概率分布的$ [a，b] \ subset \ mathbb {r} $和$ \ sum_ {i = 1}^nw_iδ_{x_i} $。我们表明，$ h^1（μ）$是带有连续内核$ k $的繁殖内核希尔伯特空间，它可以将正交问题重新制定为贝叶斯（或内核）正交问题。尽管$ k $一般并不容易封闭形式，但我们在其频谱分解与庞加莱不平等相关的频谱分解之间建立了对应关系，而庞加莱的不平等现象则是$ t $系统的共同特征性函数（Karlin and Studden，1966）。然后可以在第一个本征函数跨越的有限维代理空间中解决正交问题。该解决方案由广义的高斯正交正交给出，我们称之为Poincaré正交。我们得出了庞加莱正交重量和相关最坏情况误差的几个结果。当$μ$是统一分布时，结果是明确的：庞加莱正交等于中点（矩形）正交规则。它的节点与特征功能的零相吻合，而最坏的误差则缩放为$ \ frac {b-a} {2 \ sqrt {3}} n^{ - 1} $对于大$ n $。与$ h^1（0,1）$的已知结果相比，这表明庞加莱正交在渐近上是最佳的。对于一般的$μ$，我们根据有限元素和线性编程提供有效的数值程序。数值实验提供了有用的见解：节点几乎均匀地间隔，重量接近节点处的概率密度，而最差的案例误差约为$ o（n^{ - 1}）$。
Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form $\int_a^b f(x) dμ(x) = \sum_{i=1}^n w_i f(x_i)$ where $f$ belongs to $H^1(μ)$. Here, $μ$ belongs to a class of continuous probability distributions on $[a, b] \subset \mathbb{R}$ and $\sum_{i=1}^n w_i δ_{x_i}$ is a discrete probability distribution on $[a, b]$. We show that $H^1(μ)$ is a reproducing kernel Hilbert space with a continuous kernel $K$, which allows to reformulate the quadrature question as a Bayesian (or kernel) quadrature problem. Although $K$ has not an easy closed form in general, we establish a correspondence between its spectral decomposition and the one associated to Poincaré inequalities, whose common eigenfunctions form a $T$-system (Karlin and Studden, 1966). The quadrature problem can then be solved in the finite-dimensional proxy space spanned by the first eigenfunctions. The solution is given by a generalized Gaussian quadrature, which we call Poincaré quadrature. We derive several results for the Poincaré quadrature weights and the associated worst-case error. When $μ$ is the uniform distribution, the results are explicit: the Poincaré quadrature is equivalent to the midpoint (rectangle) quadrature rule. Its nodes coincide with the zeros of an eigenfunction and the worst-case error scales as $\frac{b-a}{2\sqrt{3}}n^{-1}$ for large $n$. By comparison with known results for $H^1(0,1)$, this shows that the Poincaré quadrature is asymptotically optimal. For a general $μ$, we provide an efficient numerical procedure, based on finite elements and linear programming. Numerical experiments provide useful insights: nodes are nearly evenly spaced, weights are close to the probability density at nodes, and the worst-case error is approximately $O(n^{-1})$ for large $n$.
