1Blurb:: Order for tensor-products of Gaussian quadrature rules
2
3Description::
4
5Multidimensional integration by a tensor-product of Gaussian
6quadrature rules (specified with \c quadrature_order, and, optionally,
7\c dimension_preference). The default rule selection is to employ \c
8non_nested Gauss rules including Gauss-Hermite (for normals or
9transformed normals), Gauss-Legendre (for uniforms or transformed
10uniforms), Gauss-Jacobi (for betas), Gauss-Laguerre (for
11exponentials), generalized Gauss-Laguerre (for gammas), and
12numerically-generated Gauss rules (for other distributions when using
13an Extended basis). For the case of \c p_refinement or the case of an
14explicit \c nested override, Gauss-Hermite rules are replaced with
15Genz-Keister nested rules and Gauss-Legendre rules are replaced with
16Gauss-Patterson nested rules, both of which exchange lower integrand
17precision for greater point reuse.  By specifying a \c
18dimension_preference, where higher preference leads to higher order
19polynomial resolution, the tensor grid may be rendered
20anisotropic. The dimension specified to have highest preference will
21be set to the specified \c quadrature_order and all other dimensions
22will be reduced in proportion to their reduced preference; any
23non-integral portion is truncated.  To synchronize with tensor-product
24integration, a tensor-product expansion is used, where the order
25\f$p_i\f$ of the expansion in each dimension is selected to be half of
26the integrand precision available from the rule in use, rounded
27down. In the case of non-nested Gauss rules with integrand precision
28\f$2m_i-1\f$, \f$p_i\f$ is one less than the quadrature order
29\f$m_i\f$ in each dimension (a one-dimensional expansion contains the
30same number of terms, \f$p+1\f$, as the number of Gauss points). The
31total number of terms, \e N, in a tensor-product expansion involving
32\e n uncertain input variables is \f[N ~=~ 1 + P ~=~ \prod_{i=1}^{n}
33(p_i + 1)\f] In some advanced use cases (e.g., multifidelity UQ),
34multiple grid resolutions can be employed; for this reason, the \c
35quadrature_order specification supports an array input.
36
37A corresponding sequence specification is documented at, e.g.,
38\ref method-multifidelity_polynomial_chaos-quadrature_order_sequence and
39\ref method-multifidelity_stoch_collocation-quadrature_order_sequence
40
41Topics::
42Examples::
43Theory::
44Faq::
45See_Also::
46method-multifidelity_polynomial_chaos-quadrature_order_sequence and
47method-multifidelity_stoch_collocation-quadrature_order_sequence
48