quadrature_order
Order for tensor-products of Gaussian quadrature rules
Specification
Alias: None
Arguments: INTEGER
Child Keywords:
Required/Optional |
Description of Group |
Dakota Keyword |
Dakota Keyword Description |
|---|---|---|---|
Optional |
A set of weights specifying the realtive importance of each uncertain variable (dimension) |
||
Optional (Choose One) |
Quadrature Rule Nesting |
Enforce use of nested quadrature rules if available |
|
Enforce use of non-nested quadrature rules |
|||
Description
Multidimensional integration by a tensor-product of Gaussian
quadrature rules (specified with quadrature_order, and, optionally,
dimension_preference). The default rule selection is to employ
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 p_refinement or the case of an
explicit 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
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 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
\(p_i\) 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
\(2m_i-1\) , \(p_i\) is one less than the quadrature order
\(m_i\) in each dimension (a one-dimensional expansion contains the
same number of terms, \(p+1\) , as the number of Gauss points). The
total number of terms, N, in a tensor-product expansion involving
n uncertain input variables is
.. math:: N ~=~ 1 + P ~=~ prod_{i=1}^{n}
(p_i + 1)
In some advanced use cases (e.g., multifidelity UQ),
multiple grid resolutions can be employed; for this reason, the
quadrature_order specification supports an array input.
A corresponding sequence specification is documented at, e.g., method-multifidelity_polynomial_chaos-quadrature_order_sequence and method-multifidelity_stoch_collocation-quadrature_order_sequence
