Bayesian quadrature for H1(µ) with Poincaré inequality on a compact interval
Résumé
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 H1(µ). Here, µ belongs to a class of continuous probability distributions on [a, b] ⊂ R and \sum_{i=1}^n w_i δ_{x_i} 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.
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