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