Dakota Reference Manual  Version 6.4
Large-Scale Engineering Optimization and Uncertainty Analysis
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polynomial


Polynomial surrogate model

Specification

Alias: none

Argument(s): none

Required/Optional Description of Group Dakota Keyword Dakota Keyword Description
Required
(Choose One)
polynomial order (Group 1) 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-selected format

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[63] for more information on polynomial models.

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.