surfpack
Use the Surfpack version of Gaussian Process surrogates
Specification
Alias: None
Arguments: None
Child Keywords:
Required/Optional 
Description of Group 
Dakota Keyword 
Dakota Keyword Description 

Optional 
Choose a trend function for a Gaussian process surrogate 

Optional 
Change the optimization method used to compute hyperparameters 

Optional 
Max number of likelihood function evaluations 

Optional (Choose One) 
Nugget 
Specify a nugget to handle illconditioning 

Have Surfpack compute a nugget to handle illconditioning 

Optional 
Specify the correlation lengths for the Gaussian process 

Optional 
Exports surrogate model in userspecified format(s) 

Optional 
Import surrogate model from archive file 
Description
This keyword specifies the use of the Gaussian process that is incorporated in our surface fitting library called Surfpack.
Several user options are available:
Optimization methods: Maximum Likelihood Estimation (MLE) is used to find the optimal values of the hyperparameters governing the trend and correlation functions. By default the global optimization method DIRECT is used for MLE, but other options for the optimization method are available. See
optimization_method
. The total number of evaluations of the likelihood function can be controlled using themax_trials
keyword followed by a positive integer. Note that the likelihood function does not require running the “truth” model, and is relatively inexpensive to compute.Trend Function: The GP models incorporate a parametric trend function whose purpose is to capture largescale variations. See
trend
.Correlation Lengths: Correlation lengths are usually optimized by Surfpack, however, the user can specify the lengths manually. See
correlation_lengths
.Illconditioning One of the major problems in determining the governing values for a Gaussian process or Kriging model is the fact that the correlation matrix can easily become illconditioned when there are too many input points close together. Since the predictions from the Gaussian process model involve inverting the correlation matrix, illconditioning can lead to poor predictive capability and should be avoided. Note that a sufficiently bad sample design could require correlation lengths to be so short that any interpolatory GP model would become inept at extrapolation and interpolation. The
surfpack
model handles illconditioning internally by default, but behavior can be modified usingGradient Enhanced Kriging (GEK). The
use_derivatives
keyword will cause the Surfpack GP to be constructed from a combination of function value and gradient information (if available). See notes in the Theory section.
Theory
Gradient Enhanced Kriging
Incorporating gradient information will only be beneficial if accurate and inexpensive derivative information is available, and the derivatives are not infinite or nearly so. Here “inexpensive” means that the cost of evaluating a function value plus gradient is comparable to the cost of evaluating only the function value, for example gradients computed by analytical, automatic differentiation, or continuous adjoint techniques. It is not cost effective to use derivatives computed by finite differences. In tests, GEK models built from finite difference derivatives were also significantly less accurate than those built from analytical derivatives. Note that GEK’s correlation matrix tends to have a significantly worse condition number than Kriging for the same sample design.
This issue was addressed by using a pivoted Cholesky factorization of Kriging’s correlation matrix (which is a small submatrix within GEK’s correlation matrix) to rank points by how much unique information they contain. This reordering is then applied to whole points (the function value at a point immediately followed by gradient information at the same point) in GEK’s correlation matrix. A standard nonpivoted Cholesky is then applied to the reordered GEK correlation matrix and a bisection search is used to find the last equation that meets the constraint on the (estimate of) condition number. The cost of performing pivoted Cholesky on Kriging’s correlation matrix is usually negligible compared to the cost of the nonpivoted Cholesky factorization of GEK’s correlation matrix. In tests, it also resulted in more accurate GEK models than when pivoted Cholesky or wholepointblock pivoted Cholesky was performed on GEK’s correlation matrix.