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

par Olivier Roustant (IMT, INSA Toulouse)

mardi 7 févr. 2023, 11:15 → 12:15 Europe/Paris

Salle K. Johnson, 1er étage (1R3)

Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form  ∫_a^bf(x)dμ(x) = ∑_i=1ⁿ w_if(x_i)  where f belongs to H1(μ). Here, μ belongs to a class of continuous probability distributions on [a,b]⊂R and ∑_i=1ⁿ w_iδ_xi 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 (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. 

 

Joint work with Nora Lüthen (ETH Zürich) and Fabrice Gamboa (IMT)