Dakota Reference Manual
Version 6.4
LargeScale Engineering Optimization and Uncertainty Analysis

Use the Surfpack version of Gaussian Process surrogates
Alias: none
Argument(s): none
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 the max_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 using
Gradient 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.
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.