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

Olivier Roustant, Nora Lüthen, Fabrice Gamboa

Abstract: Motivated by uncertainty quantification of complex systems, we aim at finding quadrature formulas of the form ∫abf(x)dμ(x)=∑i=1nwif(xi)<math><mrow is="true"><msubsup is="true"><mrow is="true"><mo is="true">∫</mo></mrow><mrow is="true"><mi is="true">a</mi></mrow><mrow is="true"><mi is="true">b</mi></mrow></msubsup><mi is="true">f</mi><mrow is="true"><mo is="true">(</mo><mi is="true">x</mi><mo is="true">)</mo></mrow><mi is="true">d</mi><mi is="true">μ</mi><mrow is="true"><mo is="true">(</mo><mi is="true">x</mi><mo is="true">)</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">=</mo><msubsup is="true"><mrow is="true"><mo is="true">∑</mo></mrow><mrow is="true"><mi is="true">i</mi><mo is="true">=</mo><mn is="true">1</mn></mrow><mrow is="true"><mi is="true">n</mi></mrow></msubsup><msub is="true"><mrow is="true"><mi is="true">w</mi></mrow><mrow is="true"><mi is="true">i</mi></mrow></msub><mi is="true">f</mi><mrow is="true"><mo is="true">(</mo><msub is="true"><mrow is="true"><mi is="true">x</mi></mrow><mrow is="true"><mi is="true">i</mi></mrow></msub><mo is="true">)</mo></mrow></mrow></math> where f<math><mi is="true">f</mi></math> belongs to H1(μ)<math><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msup><mrow is="true"><mo is="true">(</mo><mi is="true">μ</mi><mo is="true">)</mo></mrow></mrow></math>. Here, μ<math><mi is="true">μ</mi></math> belongs to a class of continuous probability distributions on [a,b]⊂R<math><mrow is="true"><mrow is="true"><mo is="true">[</mo><mi is="true">a</mi><mo is="true">,</mo><mi is="true">b</mi><mo is="true">]</mo></mrow><mo linebreak="goodbreak" linebreakstyle="after" is="true">⊂</mo><mi mathvariant="double-struck" is="true">R</mi></mrow></math> and ∑i=1nwiδxi<math><mrow is="true"><msubsup is="true"><mrow is="true"><mo is="true">∑</mo></mrow><mrow is="true"><mi is="true">i</mi><mo is="true">=</mo><mn is="true">1</mn></mrow><mrow is="true"><mi is="true">n</mi></mrow></msubsup><msub is="true"><mrow is="true"><mi is="true">w</mi></mrow><mrow is="true"><mi is="true">i</mi></mrow></msub><msub is="true"><mrow is="true"><mi is="true">δ</mi></mrow><mrow is="true"><msub is="true"><mrow is="true"><mi is="true">x</mi></mrow><mrow is="true"><mi is="true">i</mi></mrow></msub></mrow></msub></mrow></math> is a discrete probability distribution on [a,b]<math><mrow is="true"><mo is="true">[</mo><mi is="true">a</mi><mo is="true">,</mo><mi is="true">b</mi><mo is="true">]</mo></mrow></math>. We show that H1(μ)<math><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msup><mrow is="true"><mo is="true">(</mo><mi is="true">μ</mi><mo is="true">)</mo></mrow></mrow></math> is a reproducing kernel Hilbert space with a continuous kernel K<math><mi is="true">K</mi></math>, which allows to reformulate the quadrature question as a kernel (or Bayesian) quadrature problem. Although K<math><mi is="true">K</mi></math> 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<math><mi is="true">T</mi></math>-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<math><mi is="true">μ</mi></math> 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−a23n−1<math><mrow is="true"><mfrac is="true"><mrow is="true"><mi is="true">b</mi><mo is="true">−</mo><mi is="true">a</mi></mrow><mrow is="true"><mn is="true">2</mn><msqrt is="true"><mrow is="true"><mn is="true">3</mn></mrow></msqrt></mrow></mfrac><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">1</mn></mrow></msup></mrow></math> for large n<math><mi is="true">n</mi></math>. By comparison with known results for H1(0,1)<math><mrow is="true"><msup is="true"><mrow is="true"><mi is="true">H</mi></mrow><mrow is="true"><mn is="true">1</mn></mrow></msup><mrow is="true"><mo is="true">(</mo><mn is="true">0</mn><mo is="true">,</mo><mn is="true">1</mn><mo is="true">)</mo></mrow></mrow></math>, this shows that the Poincaré quadrature is asymptotically optimal. For a general μ<math><mi is="true">μ</mi></math>, 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)<math><mrow is="true"><mi is="true">O</mi><mrow is="true"><mo is="true">(</mo><msup is="true"><mrow is="true"><mi is="true">n</mi></mrow><mrow is="true"><mo is="true">−</mo><mn is="true">1</mn></mrow></msup><mo is="true">)</mo></mrow></mrow></math> for large n<math><mi is="true">n</mi></math>.