.. _method-stoch_collocation-sparse_grid_level:

"""""""""""""""""
sparse_grid_level
"""""""""""""""""


Level to use in sparse grid integration or interpolation


.. toctree::
   :hidden:
   :maxdepth: 1

   method-stoch_collocation-sparse_grid_level-dimension_preference
   method-stoch_collocation-sparse_grid_level-nodal
   method-stoch_collocation-sparse_grid_level-hierarchical
   method-stoch_collocation-sparse_grid_level-restricted
   method-stoch_collocation-sparse_grid_level-unrestricted
   method-stoch_collocation-sparse_grid_level-nested
   method-stoch_collocation-sparse_grid_level-non_nested


**Specification**

- *Alias:* None

- *Arguments:* INTEGER


**Child Keywords:**

+-------------------------+--------------------+--------------------------+---------------------------------------------+
| Required/Optional       | Description of     | Dakota Keyword           | Dakota Keyword Description                  |
|                         | Group              |                          |                                             |
+=========================+====================+==========================+=============================================+
| Optional                                     | `dimension_preference`__ | A set of weights specifying the realtive    |
|                                              |                          | importance of each uncertain variable       |
|                                              |                          | (dimension)                                 |
+-------------------------+--------------------+--------------------------+---------------------------------------------+
| Optional (Choose One)   | Quadrature Rule    | `restricted`__           | Restrict the growth rates for nested and    |
|                         | Growth             |                          | non-nested rules can be synchronized for    |
|                         |                    |                          | consistency.                                |
|                         |                    +--------------------------+---------------------------------------------+
|                         |                    | `unrestricted`__         | Overide the default restriction of growth   |
|                         |                    |                          | rates for nested and non-nested rules that  |
|                         |                    |                          | are by defualt synchronized for             |
|                         |                    |                          | consistency.                                |
+-------------------------+--------------------+--------------------------+---------------------------------------------+
| Optional (Choose One)   | Quadrature Rule    | `nested`__               | Enforce use of nested quadrature rules if   |
|                         | Nesting            |                          | available                                   |
|                         |                    +--------------------------+---------------------------------------------+
|                         |                    | `non_nested`__           | Enforce use of non-nested quadrature rules  |
+-------------------------+--------------------+--------------------------+---------------------------------------------+

.. __: method-stoch_collocation-sparse_grid_level-dimension_preference.html
__ method-stoch_collocation-sparse_grid_level-restricted.html
__ method-stoch_collocation-sparse_grid_level-unrestricted.html
__ method-stoch_collocation-sparse_grid_level-nested.html
__ method-stoch_collocation-sparse_grid_level-non_nested.html



**Description**


Multi-dimensional integration by the Smolyak sparse grid method
(specified with sparse_grid_level and, optionally,
dimension_preference). The underlying one-dimensional integration
rules are the same as for the tensor-product quadrature case; however,
the default rule selection is nested for sparse grids (Genz-Keister
for normals/transformed normals and Gauss-Patterson for
uniforms/transformed uniforms). This default can be overridden with an
explicit non_nested specification (resulting in Gauss-Hermite for
normals/transformed normals and Gauss-Legendre for
uniforms/transformed uniforms). As for tensor quadrature, the
dimension_preference specification enables the use of anisotropic
sparse grids (refer to the PCE description in the User's Manual for
the anisotropic index set constraint definition). Similar to
anisotropic tensor grids, the dimension with greatest preference will
have resolution at the full sparse_grid_level and all other dimension
resolutions will be reduced in proportion to their reduced
preference. For PCE with either isotropic or anisotropic sparse grids,
a summation of tensor-product expansions is used, where each
anisotropic tensor-product quadrature rule underlying the sparse grid
construction results in its own anisotropic tensor-product expansion
as described in case 1. These anisotropic tensor-product expansions
are summed into a sparse PCE using the standard Smolyak summation
(again, refer to the User's Manual for additional details). As for
quadrature_order, the sparse_grid_level specification admits an array
input for enabling specification of multiple grid resolutions used by
certain advanced solution methodologies.

A corresponding sequence specification is documented at, e.g.,
:dakkw:`method-multifidelity_polynomial_chaos-sparse_grid_level_sequence` and
:dakkw:`method-multifidelity_stoch_collocation-sparse_grid_level_sequence`