.. _method-polynomial_chaos-expansion_order:
"""""""""""""""
expansion_order
"""""""""""""""
The (initial) order of a polynomial expansion
.. toctree::
:hidden:
:maxdepth: 1
method-polynomial_chaos-expansion_order-dimension_preference
method-polynomial_chaos-expansion_order-basis_type
method-polynomial_chaos-expansion_order-collocation_points
method-polynomial_chaos-expansion_order-collocation_ratio
method-polynomial_chaos-expansion_order-expansion_samples
method-polynomial_chaos-expansion_order-import_build_points_file
**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 | `basis_type`__ | Specify the type of basis truncation to be |
| | | used for a Polynomial Chaos Expansion. |
+-------------------------+--------------------+------------------------------+-----------------------------------------------+
| Required (Choose One) | Required (Choose | `collocation_points`__ | Number of collocation points used to estimate |
| | One) | | expansion coefficients |
| | +------------------------------+-----------------------------------------------+
| | | `collocation_ratio`__ | Set the number of points used to build a PCE |
| | | | via regression to be proportional to the |
| | | | number of terms in the expansion. |
| | +------------------------------+-----------------------------------------------+
| | | `expansion_samples`__ | Number of simulation samples used to estimate |
| | | | the expected value of a set of PCE |
| | | | coefficients |
+-------------------------+--------------------+------------------------------+-----------------------------------------------+
| Optional | `import_build_points_file`__ | File containing points you wish to use to |
| | | build a surrogate |
+----------------------------------------------+------------------------------+-----------------------------------------------+
.. __: method-polynomial_chaos-expansion_order-dimension_preference.html
__ method-polynomial_chaos-expansion_order-basis_type.html
__ method-polynomial_chaos-expansion_order-collocation_points.html
__ method-polynomial_chaos-expansion_order-collocation_ratio.html
__ method-polynomial_chaos-expansion_order-expansion_samples.html
__ method-polynomial_chaos-expansion_order-import_build_points_file.html
**Description**
When the expansion_order for a a polynomial chaos expansion is
specified, the coefficients may be computed by integration based on
random samples or by regression using either random or sub-sampled
tensor product quadrature points.
Multidimensional integration by Latin hypercube sampling
(specified with ``expansion_samples``). In this case, the
expansion order *p* cannot be inferred from the numerical
integration specification and it is necessary to provide an
``expansion_order`` to specify *p* for a total-order expansion.
Linear regression (specified with either ``collocation_points`` or
``collocation_ratio``). A total-order expansion is used and must
be specified using ``expansion_order`` as described in the
previous option. To avoid requiring the
user to calculate *N* from *n* and *p*), the ``collocation_ratio``
allows for specification of a constant factor applied to
*N* (e.g., ``collocation_ratio`` = ``2``. produces samples = *2N*).
In addition, the default linear relationship with *N* can be
overridden using a real-valued exponent specified using
``ratio_order``. In this case, the number of samples becomes
:math:`cN^o` where :math:`c` is the ``collocation_ratio`` and :math:`o`
is the ``ratio_order``. The ``use_derivatives`` flag informs the
regression approach to include derivative matching equations
(limited to gradients at present) in the least squares solutions,
enabling the use of fewer collocation points for a given
expansion order and dimension (number of points required becomes
:math:`\frac{cN^o}{n+1}` ). When admissible, a constrained least
squares approach is employed in which response values are first
reproduced exactly and error in reproducing response derivatives
is minimized. Two collocation grid options are supported: the
default is Latin hypercube sampling ("point collocation"), and an
alternate approach of "probabilistic collocation" is also
available through inclusion of the ``tensor_grid`` keyword. In
this alternate case, the collocation grid is defined using a
subset of tensor-product quadrature points: the order of the
tensor-product grid is selected as one more than the expansion
order in each dimension (to avoid sampling at roots of the basis
polynomials) and then the tensor multi-index is uniformly sampled
to generate a non-repeated subset of tensor quadrature points.
..
HIDDEN: the order of the tensor-product grid is the minimum
required to meet or exceed the point requirement, and then this set
of points is filtered down to the requirement based on the subset
of points having highest product integration weight. In our
experience, the alternate probabilistic collocation approach using
a structured grid tends to suffer from ill-conditioning more
quickly than the default point collocation approach using an
unstructured grid.
If ``collocation_points`` or ``collocation_ratio`` is specified, the PCE
coefficients will be determined by regression.
If no regression
specification is provided, appropriate defaults are defined.
Specifically SVD-based least-squares will be used for solving
over-determined systems and under-determined systems will be solved
using LASSO. For the situation when the number of function values is
smaller than the number of terms in a PCE, but the total number of
samples including gradient values is greater than the number of terms,
the resulting over-determined system will be solved using equality
constrained least squares. Technical information on the various
methods listed below can be found in the Linear regression section of
the Theory Manual. Some of the regression methods (OMP, LASSO, and
LARS) are able to produce a set of possible PCE coefficient vectors
(see the Linear regression section in the Theory Manual). If cross
validation is inactive, then only one solution, consistent with the
``noise_tolerance``, will be returned. If cross validation is active,
Dakota will choose between possible coefficient vectors found
internally by the regression method across the set of expansion orders
(1,..., ``expansion_order``) and the set of specified noise tolerances and
return the one with the lowest cross validation error indicator.