polynomial

Polynomial surrogate model

Specification

• Alias: None

• Arguments: None

Child Keywords:

Required/Optional	Description of Group	Dakota Keyword	Dakota Keyword Description
Required (Choose One)	Polynomial Order	basis_order	Polynomial order
		linear	Use a linear polynomial or trend function
		quadratic	Use a quadratic polynomial or trend function
		cubic	Use a cubic polynomial
Optional		export_model	Exports surrogate model in user-specified format(s)
Optional		import_model	Import surrogate model from archive file

Description

Linear, quadratic, and cubic polynomial surrogate models are available in Dakota. The utility of the simple polynomial models stems from two sources:

• over a small portion of the parameter space, a low-order polynomial model is often an accurate approximation to the true data trends

• the least-squares procedure provides a surface fit that smooths out noise in the data.

Local surrogate-based optimization methods ( surrogate_based_local) are often successful when using polynomial models, particularly quadratic models. However, a polynomial surface fit may not be the best choice for modeling data trends globally over the entire parameter space, unless it is known a priori that the true data trends are close to linear, quadratic, or cubic. See [MM95] for more information on polynomial models.

Known Issue: When using discrete variables, there have been sometimes significant differences in surrogate behavior observed across computing platforms in some cases. The cause has not yet been fully diagnosed and is currently under investigation. In addition, guidance on appropriate construction and use of surrogates with discrete variables is under development. In the meantime, users should therefore be aware that there is a risk of inaccurate results when using surrogates with discrete variables.

Theory

The form of the linear polynomial model is

\[\hat{f}(\mathbf{x}) \approx c_{0}+\sum_{i=1}^{n}c_{i}x_{i}\]

the form of the quadratic polynomial model is:

\[\hat{f}(\mathbf{x}) \approx c_{0}+\sum_{i=1}^{n}c_{i}x_{i} +\sum_{i=1}^{n}\sum_{j \ge i}^{n}c_{ij}x_{i}x_{j}\]

and the form of the cubic polynomial model is:

\[\hat{f}(\mathbf{x}) \approx c_{0}+\sum_{i=1}^{n}c_{i}x_{i} +\sum_{i=1}^{n}\sum_{j \ge i}^{n}c_{ij}x_{i}x_{j} +\sum_{i=1}^{n}\sum_{j \ge i}^{n}\sum_{k \ge j}^{n} c_{ijk}x_{i}x_{j}x_{k}\]

In all of the polynomial models, \(\hat{f}(\mathbf{x})\) is the response of the polynomial model; the \(x_{i},x_{j},x_{k}\) terms are the components of the \(n\) -dimensional design parameter values; the \(c_{0}\) , \(c_{i}\) , \(c_{ij}\) , \(c_{ijk}\) terms are the polynomial coefficients, and \(n\) is the number of design parameters. The number of coefficients, \(n_{c}\) , depends on the order of polynomial model and the number of design parameters. For the linear polynomial:

\[n_{c_{linear}}=n+1\]

for the quadratic polynomial:

\[n_{c_{quad}}=\frac{(n+1)(n+2)}{2}\]

and for the cubic polynomial:

\[n_{c_{cubic}}=\frac{(n^{3}+6 n^{2}+11 n+6)}{6}\]

There must be at least \(n_{c}\) data samples in order to form a fully determined linear system and solve for the polynomial coefficients. In Dakota, a least-squares approach involving a singular value decomposition numerical method is applied to solve the linear system.