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		trend	Choose a trend function for a Gaussian process surrogate
Optional		optimization_method	Change the optimization method used to compute hyperparameters
Optional		max_trials	Max number of likelihood function evaluations
Optional (Choose One)	Nugget	nugget	Specify a nugget to handle ill-conditioning
		find_nugget	Have Surfpack compute a nugget to handle ill-conditioning
Optional		correlation_lengths	Specify the correlation lengths for the Gaussian process
Optional		export_model	Exports surrogate model in user-specified format(s)
Optional		import_model	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:

1. Optimization methods: Maximum Likelihood Estimation (MLE) is used to find the optimal values of the hyper-parameters 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 model-surrogate-global-gaussian_process-surfpack-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.

2. Trend Function: The GP models incorporate a parametric trend function whose purpose is to capture large-scale variations. See model-surrogate-global-gaussian_process-surfpack-trend.

3. Correlation Lengths: Correlation lengths are usually optimized by Surfpack, however, the user can specify the lengths manually. See model-surrogate-global-gaussian_process-surfpack-correlation_lengths.

4. Ill-conditioning 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 ill-conditioned when there are too many input points close together. Since the predictions from the Gaussian process model involve inverting the correlation matrix, ill-conditioning 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 ill-conditioning internally by default, but behavior can be modified using

5. 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.

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 sub-matrix 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 non-pivoted 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 non-pivoted Cholesky factorization of GEK’s correlation matrix. In tests, it also resulted in more accurate GEK models than when pivoted Cholesky or whole-point-block pivoted Cholesky was performed on GEK’s correlation matrix.