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

the form of the quadratic polynomial model is:

and the form of the cubic polynomial model is:

In all of the polynomial models, is the response of the polynomial model; the terms are the components of the -dimensional design parameter values; the , , , terms are the polynomial coefficients, and is the number of design parameters. The number of coefficients, , depends on the order of polynomial model and the number of design parameters. For the linear polynomial:

for the quadratic polynomial:

and for the cubic polynomial:

There must be at least 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.