  For the purpose of uncertainty propagation a new quadrature rule technique is
proposed that has positive weights, has high degree, and is constructed using
only samples that describe the probability distribution of the uncertain
parameters. Moreover, nodes can be added to the quadrature rule, resulting into
a sequence of nested rules. The rule is constructed by iterating over the
samples of the distribution and exploiting the null space of the
Vandermonde-system that describes the nodes and weights, in order to select
which samples will be used as nodes in the quadrature rule. The main novelty of
the quadrature rule is that it can be constructed using any number of
dimensions, using any basis, in any space, and using any distribution. It is
demonstrated both theoretically and numerically that the rule always has
positive weights and therefore has high convergence rates for sufficiently
smooth functions. The convergence properties are demonstrated by approximating
the integral of the Genz test functions. The applicability of the quadrature
rule to complex uncertainty propagation cases is demonstrated by determining
the statistics of the flow over an airfoil governed by the Euler equations,
including the case of dependent uncertain input parameters. The new quadrature
rule significantly outperforms classical sparse grid methods.

Numerical Analysis;Numerical Analysis