English (United Kingdom)

https://curated-unify.zendy.io/wp-json/zendy-region/v1/featured_content/oa?rat=en

https://curated-unify.zendy.io/wp-json/zendy-region/v1/highlighted_journal/

Global: Zendy Plus annual

Presents the access of premium content as premium feature

Premium Content

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the summarisation as premium feature

Summarisation

Insights

Presents the pdf analysis as premium feature

PDF Analysis

Presents the zaia usage as premium feature

ZAIA

Global: Zendy Plus monthly

Global: Zendy Tools annual

Global: Zendy Tools monthly

Global: Zendy Free Plan

Motivated by uncertainty quantification of complex systems, we aim at findingquadrature formulas of the form $\int_a^b f(x) d\mu(x) = \sum_{i=1}^n w_if(x_i)$ where $f$ belongs to $H^1(\mu)$. Here, $\mu$ belongs to a class ofcontinuous probability distributions on $[a, b] \subset \mathbb{R}$ 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 acontinuous kernel $K$, which allows to reformulate the quadrature question as aBayesian (or kernel) quadrature problem. Although $K$ has not an easy closedform in general, we establish a correspondence between its spectraldecomposition and the one associated to Poincar\'e inequalities, whose commoneigenfunctions form a $T$-system (Karlin and Studden, 1966). The quadratureproblem can then be solved in the finite-dimensional proxy space spanned by thefirst eigenfunctions. The solution is given by a generalized Gaussianquadrature, which we call Poincar\'e quadrature. We derive several results forthe Poincar\'e quadrature weights and the associated worst-case error. When$\mu$ is the uniform distribution, the results are explicit: the Poincar\'equadrature is equivalent to the midpoint (rectangle) quadrature rule. Its nodescoincide 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 resultsfor $H^1(0,1)$, this shows that the Poincar\'e quadrature is asymptoticallyoptimal. For a general $\mu$, we provide an efficient numerical procedure,based on finite elements and linear programming. Numerical experiments provideuseful insights: nodes are nearly evenly spaced, weights are close to theprobability density at nodes, and the worst-case error is approximately$O(n^{-1})$ for large $n$.

Bayesian quadrature for $H^1(\mu)$ with Poincar\'e inequality on a compact interval