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		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 One)	collocation_points	Number of collocation points used to estimate 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

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 \(cN^o\) where \(c\) is the collocation_ratio and \(o\) 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 \(\frac{cN^o}{n+1}\) ). 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.

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.