expansion_order
The (initial) order of a polynomial expansion
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 
Specify the type of basis truncation to be used for a Polynomial Chaos Expansion. 

Required (Choose One) 
Required (Choose One) 
Number of collocation points used to estimate expansion coefficients 

Set the number of points used to build a PCE via regression to be proportional to the number of terms in the expansion. 

Number of simulation samples used to estimate the expected value of a set of PCE coefficients 

Optional 
File containing points you wish to use to build a surrogate 
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 subsampled 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 anexpansion_order
to specify p for a totalorder expansion. Linear regression (specified with either
collocation_points
or collocation_ratio
). A totalorder expansion is used and must be specified usingexpansion_order
as described in the previous option. To avoid requiring the user to calculate N from n and p), thecollocation_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 realvalued exponent specified usingratio_order
. In this case, the number of samples becomes \(cN^o\) where \(c\) is thecollocation_ratio
and \(o\) is theratio_order
. Theuse_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 \(\frac{cN^o}{n+1}\) ). 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 thetensor_grid
keyword. In this alternate case, the collocation grid is defined using a subset of tensorproduct quadrature points: the order of the tensorproduct 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 multiindex is uniformly sampled to generate a nonrepeated subset of tensor quadrature points.
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 SVDbased leastsquares will be used for solving
overdetermined systems and underdetermined 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 overdetermined 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.