  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.
