紧凑区间上 H1(μ) 的谱分解和 Poincaré 不等式 - 核正交的应用

IF 0.9 3区 数学 Q2 MATHEMATICS Journal of Approximation Theory Pub Date : 2024-04-09 DOI:10.1016/j.jat.2024.106041
Olivier Roustant , Nora Lüthen , Fabrice Gamboa
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引用次数: 0

摘要

受复杂系统不确定性量化的激励,我们的目标是找到形式为∫abf(x)dμ(x)=∑i=1nwif(xi)的正交公式,其中 f 属于 H1(μ)。这里,μ 属于[a,b]⊂R 上的一类连续概率分布,∑i=1nwiδxi 是[a,b]上的离散概率分布。我们证明,H1(μ) 是一个具有连续核 K 的重现核希尔伯特空间,因此可以将正交问题重新表述为核(或贝叶斯)正交问题。虽然 K 在一般情况下并不容易封闭,但我们在其谱分解和与波恩卡莱不等式相关的谱分解之间建立了对应关系,波恩卡莱不等式的公共特征函数构成了一个 T 系统(Karlin 和 Studden,1966 年)。然后,正交问题就可以在第一特征函数所跨越的有限维代理空间中求解。我们推导出 Poincaré 正交权重和相关最坏情况误差的几个结果。当 μ 为均匀分布时,结果是明确的:Poincaré 正交等价于中点(矩形)正交规则。它的节点与特征函数的零点重合,最坏情况下的误差在大 n 时按 b-a23n-1 的比例缩放。通过与 H1(0,1) 的已知结果进行比较,这表明 Poincaré 正交是渐近最优的。对于一般的 μ,我们提供了一种基于有限元和线性规划的高效数值计算程序。数值实验提供了有益的启示:节点间距接近均匀,权重接近节点处的概率密度,对于大 n,最坏情况误差约为 O(n-1)。
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Spectral decomposition of H1(μ) and Poincaré inequality on a compact interval — Application to kernel quadrature

Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form abf(x)dμ(x)=i=1nwif(xi) where f belongs to H1(μ). Here, μ belongs to a class of continuous probability distributions on [a,b]R and i=1nwiδxi is a discrete probability distribution on [a,b]. We show that H1(μ) is a reproducing kernel Hilbert space with a continuous kernel K, which allows to reformulate the quadrature question as a kernel (or Bayesian) 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 ba23n1 for large n. By comparison with known results for H1(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(n1) for large n.

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来源期刊
CiteScore
1.90
自引率
11.10%
发文量
55
审稿时长
6-12 weeks
期刊介绍: The Journal of Approximation Theory is devoted to advances in pure and applied approximation theory and related areas. These areas include, among others: • Classical approximation • Abstract approximation • Constructive approximation • Degree of approximation • Fourier expansions • Interpolation of operators • General orthogonal systems • Interpolation and quadratures • Multivariate approximation • Orthogonal polynomials • Padé approximation • Rational approximation • Spline functions of one and several variables • Approximation by radial basis functions in Euclidean spaces, on spheres, and on more general manifolds • Special functions with strong connections to classical harmonic analysis, orthogonal polynomial, and approximation theory (as opposed to combinatorics, number theory, representation theory, generating functions, formal theory, and so forth) • Approximation theoretic aspects of real or complex function theory, function theory, difference or differential equations, function spaces, or harmonic analysis • Wavelet Theory and its applications in signal and image processing, and in differential equations with special emphasis on connections between wavelet theory and elements of approximation theory (such as approximation orders, Besov and Sobolev spaces, and so forth) • Gabor (Weyl-Heisenberg) expansions and sampling theory.
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Optimization-aided construction of multivariate Chebyshev polynomials In search of a higher Bochner theorem Positive orthogonalizing weights on the unit circle for the generalized Bessel polynomials Editorial Board On the representability of a continuous multivariate function by sums of ridge functions
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