Dakota Reference Manual
Version 6.2
LargeScale Engineering Optimization and Uncertainty Analysis

Polynomial surrogate model
Alias: none
Argument(s): none
Required/Optional  Description of Group  Dakota Keyword  Dakota Keyword Description  

Required (Choose One)  polynomial order (Group 1)  linear  Use a linear polynomial or trend function  
quadratic  Use a quadratic polynomial or trend function  
cubic  Use a cubic polynomial  
Optional  export_model_file  Export surrogate to Surfpack model file 
Linear, quadratic, and cubic polynomial surrogate models are available in Dakota. The utility of the simple polynomial models stems from two sources:
Local surrogatebased 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.
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 leastsquares approach involving a singular value decomposition numerical method is applied to solve the linear system.