Spectral decomposition of [formula omitted] 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 ∫ a b f (x) d μ (x) = ∑ 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 ] ⊂ R and ∑ 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 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 b − a 2 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. [ABSTRACT FROM AUTHOR]