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

Authors

O. Roustant, N. Lüthen, F. Gamboa

Abstract

Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas for functions f belonging to H¹(μ). Here, μ belongs to a class of continuous probability distributions on [a,b] and the quadrature formula 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 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 (b−a)/(2√3)n⁻¹ 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(n⁻¹) for large n.