Blurb:: Order for tensor-products of Gaussian quadrature rules

Description::

Multidimensional integration by a tensor-product of Gaussian
quadrature rules (specified with \c quadrature_order, and, optionally,
\c dimension_preference). The default rule selection is to employ \c
non_nested Gauss rules including Gauss-Hermite (for normals or
transformed normals), Gauss-Legendre (for uniforms or transformed
uniforms), Gauss-Jacobi (for betas), Gauss-Laguerre (for
exponentials), generalized Gauss-Laguerre (for gammas), and
numerically-generated Gauss rules (for other distributions when using
an Extended basis). For the case of \c p_refinement or the case of an
explicit \c nested override, Gauss-Hermite rules are replaced with
Genz-Keister nested rules and Gauss-Legendre rules are replaced with
Gauss-Patterson nested rules, both of which exchange lower integrand
precision for greater point reuse.  By specifying a \c
dimension_preference, where higher preference leads to higher order
polynomial resolution, the tensor grid may be rendered
anisotropic. The dimension specified to have highest preference will
be set to the specified \c quadrature_order and all other dimensions
will be reduced in proportion to their reduced preference; any
non-integral portion is truncated.  To synchronize with tensor-product
integration, a tensor-product expansion is used, where the order
\f$p_i\f$ of the expansion in each dimension is selected to be half of
the integrand precision available from the rule in use, rounded
down. In the case of non-nested Gauss rules with integrand precision
\f$2m_i-1\f$, \f$p_i\f$ is one less than the quadrature order
\f$m_i\f$ in each dimension (a one-dimensional expansion contains the
same number of terms, \f$p+1\f$, as the number of Gauss points). The
total number of terms, \e N, in a tensor-product expansion involving
\e n uncertain input variables is \f[N ~=~ 1 + P ~=~ \prod_{i=1}^{n}
(p_i + 1)\f] In some advanced use cases (e.g., multifidelity UQ),
multiple grid resolutions can be employed; for this reason, the \c
quadrature_order specification supports an array input.

A corresponding sequence specification is documented at, e.g.,
\ref method-multifidelity_polynomial_chaos-quadrature_order_sequence and
\ref method-multifidelity_stoch_collocation-quadrature_order_sequence

Topics::
Examples::
Theory::
Faq::
See_Also::
method-multifidelity_polynomial_chaos-quadrature_order_sequence and
method-multifidelity_stoch_collocation-quadrature_order_sequence
