Blurb::
Use the Surfpack version of Gaussian Process surrogates
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:
<ol>
<li> 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 \ref model-surrogate-global-gaussian_process-surfpack-optimization_method.

  The total number of evaluations of the 
  likelihood function can 
  be controlled using the \c 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.

</li>
<li> Trend Function:

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

</li>
<li> Correlation Lengths:

  Correlation lengths are usually optimized by Surfpack, however, the
  user can specify the lengths manually.
  See \ref model-surrogate-global-gaussian_process-surfpack-correlation_lengths.
  
</li>
<li> 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 \c surfpack model handles ill-conditioning internally by default,
  but behavior can be modified using 

</li>
<li> Gradient Enhanced Kriging (GEK).

  The \c 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.

</li>
</ol>

Topics::	
Examples::
Theory::

<b> Gradient Enhanced Kriging </b>


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.

Faq::
See_Also::	
