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/

Zendy Open

Presents the pdf analysis as premium feature

PDF Analysis

Insights

Presents the summarisation as premium feature

Summarisation

Presents the keyphrase highlighting as premium feature

Keyphrase Highlighting

Presents the zaia usage as premium feature

ZAIA

Zendy Tools

Zendy Plus

Presents the access of premium content as premium feature

Premium Content

Sampling orthogonal polynomial bases via Monte Carlo is of interest foruncertainty quantification of models with high-dimensional random inputs, usingPolynomial Chaos (PC) expansions. It is known that bounding a probabilisticparameter, referred to as {\it coherence}, yields a bound on the number ofsamples necessary to identify coefficients in a sparse PC expansion viasolution to an $\ell_1$-minimization problem. Utilizing asymptotic results fororthogonal polynomials, we bound the coherence parameter for polynomials ofHermite and Legendre type under the respective natural sampling distribution.In both polynomial bases we identify an importance sampling distribution whichyields a bound with weaker dependence on the order of the approximation. Formore general orthonormal bases, we propose the {\it coherence-optimal}sampling: a Markov Chain Monte Carlo sampling, which directly uses the basisfunctions under consideration to achieve a statistical optimality among allsampling schemes with identical support. We demonstrate these differentsampling strategies numerically in both high-order and high-dimensional,manufactured PC expansions. In addition, the quality of each sampling method iscompared in the identification of solutions to two differential equations, onewith a high-dimensional random input and the other with a high-order PCexpansion. In both cases the coherence-optimal sampling scheme leads to similaror considerably improved accuracy.

Compressive Sampling of Polynomial Chaos Expansions: Convergence Analysis and Sampling Strategies