Spectral decomposition of H1(μ) and Poincaré inequality on a compact interval — Application to kernel quadrature

Abstract

Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form ${\int }_a^{b}f\left(x\right)d\mu \left(x\right)={\sum }_{i=1}^n{w}_{i}f\left(x_i\right)$ where $f$ belongs to $H^1\left(\mu \right)$. Here, $\mu$ belongs to a class of continuous probability distributions on $[a,b]\subset R$ and ${\sum }_{i=1}^n{w}_i{\delta }_{x_i}$ is a discrete probability distribution on $[a,b]$. We show that $H^1\left(\mu \right)$ 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 $\mu$ 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 $\mu$, 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 $O\left(n^{-1}\right)$ for large $n$.