Spectral decomposition of H1(μ) and Poincaré inequality on a compact interval - Application to kernel quadrature
Résumé
Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form $\int_a^b f(x) d\mu(x) = \sum_{i=1}^n w_i f(x_i)$ where f belongs to $H^1(\mu)$. 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 $H^1(\mu)$ 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 $O(n^{-1})$ for large $n$.
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main.pdf (1.58 Mo)
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PoincareBasisTruncExp.pdf (30.61 Ko)
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PoincareBasisUnif.pdf (30.21 Ko)
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densities_100reps.pdf (867.53 Ko)
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location_n9.pdf (57.85 Ko)
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node_dist_deg9_100reps.pdf (3.85 Ko)
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random_densities_wce.pdf (7.18 Ko)
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vis_random_8.pdf (55.63 Ko)
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vis_trunc_exp_8.pdf (54.9 Ko)
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vis_uniform_3.pdf (28.73 Ko)
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vis_uniform_5.pdf (31.17 Ko)
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wasserstein_dist_n9.pdf (5.97 Ko)
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wce_random_densities_N20_Ktrunc100.pdf (14.83 Ko)
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wce_uniform_N20_increasing_ktrunc_numerical.pdf (26.03 Ko)
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weight_dist_deg9_100reps.pdf (3.4 Ko)
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| Origine | Fichiers produits par l'(les) auteur(s) |
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| Origine | Fichiers produits par l'(les) auteur(s) |
|---|