# Non-intrusive uncertainty quantification using reduced cubature rules

## TL;DR

This paper introduces a novel method for generating nested quadrature and cubature rules with positive weights, enhancing uncertainty quantification by reducing nodes while maintaining accuracy and positivity, applicable in high-dimensional problems.

## Contribution

The paper presents a reduction procedure for creating nested quadrature rules with positive weights, applicable to multi-dimensional sparse rules, improving upon existing methods like Smolyak rules.

## Key findings

- Generated sparse cubature rules with positive weights.
- Achieved competitive accuracy with fewer nodes.
- Demonstrated effectiveness in fluid dynamics and mathematical test problems.

## Abstract

For the purpose of uncertainty quantification with collocation, a method is proposed for generating families of one-dimensional nested quadrature rules with positive weights and symmetric nodes. This is achieved through a reduction procedure: we start with a high-degree quadrature rule with positive weights and remove nodes while preserving symmetry and positivity. This is shown to be always possible, by a lemma depending primarily on Carath\'eodory's theorem. The resulting one-dimensional rules can be used within a Smolyak procedure to produce sparse multi-dimensional rules, but weight positivity is lost then. As a remedy, the reduction procedure is directly applied to multi-dimensional tensor-product cubature rules. This allows to produce a family of sparse cubature rules with positive weights, competitive with Smolyak rules. Finally the positivity constraint is relaxed to allow more flexibility in the removal of nodes. This gives a second family of sparse cubature rules, in which iteratively as many nodes as possible are removed. The new quadrature and cubature rules are applied to test problems from mathematics and fluid dynamics. Their performance is compared with that of the tensor-product and standard Clenshaw-Curtis Smolyak cubature rule.