1Blurb::
2Use the Surfpack version of Gaussian Process surrogates
3Description::
4This keyword specifies the use of the Gaussian process that is
5incorporated in our surface fitting library called Surfpack.
6
7Several user options are available:
8<ol>
9<li> Optimization methods:
10
11  Maximum Likelihood Estimation
12  (MLE) is used to find the optimal values of the hyper-parameters
13  governing the trend and correlation functions.
14  By default
15  the global optimization method DIRECT is used for MLE, but
16  other options for the optimization method are available.
17  See \ref model-surrogate-global-gaussian_process-surfpack-optimization_method.
18
19  The total number of evaluations of the
20  likelihood function can
21  be controlled using the \c max_trials keyword followed by a
22  positive integer. Note that the likelihood function does not require
23  running the "truth" model, and is relatively inexpensive to compute.
24
25</li>
26<li> Trend Function:
27
28  The GP models incorporate a parametric trend
29  function whose purpose is to capture large-scale variations.
30  See \ref model-surrogate-global-gaussian_process-surfpack-trend.
31
32</li>
33<li> Correlation Lengths:
34
35  Correlation lengths are usually optimized by Surfpack, however, the
36  user can specify the lengths manually.
37  See \ref model-surrogate-global-gaussian_process-surfpack-correlation_lengths.
38
39</li>
40<li> Ill-conditioning
41
42  One of the major problems in determining
43  the governing values for a Gaussian process or Kriging model is
44  the fact that the correlation matrix can easily become
45  ill-conditioned when there are too many input points close together.
46  Since the predictions from the Gaussian process model involve
47  inverting the correlation matrix, ill-conditioning can lead to poor
48  predictive capability and should be avoided.
49
50  Note that a
51  sufficiently bad sample design could require correlation lengths to
52  be so short that any interpolatory GP model would become
53  inept at extrapolation and interpolation.
54
55  The \c surfpack model handles ill-conditioning internally by default,
56  but behavior can be modified using
57
58</li>
59<li> Gradient Enhanced Kriging (GEK).
60
61  The \c use_derivatives keyword will cause the Surfpack GP to be
62  constructed from a combination of function value and gradient
63  information (if available).
64
65  See notes in the Theory section.
66
67</li>
68</ol>
69
70Topics::
71Examples::
72Theory::
73
74<b> Gradient Enhanced Kriging </b>
75
76
77Incorporating gradient information will only be
78beneficial if accurate and inexpensive derivative information is
79available, and the derivatives are not infinite or nearly so. Here
80"inexpensive" means that the cost of evaluating a function value
81plus gradient is comparable to the cost of evaluating only the
82function value, for example gradients computed by analytical,
83automatic differentiation, or continuous adjoint techniques. It is
84not cost effective to use derivatives computed by finite differences.
85In tests, GEK models built from finite difference derivatives were
86also significantly less accurate than those built from analytical
87derivatives. Note that GEK's correlation matrix tends to have a
88significantly worse condition number than Kriging for the same
89sample design.
90
91This issue was addressed by using a pivoted Cholesky
92factorization of Kriging's correlation matrix (which is a small
93sub-matrix within GEK's correlation matrix) to rank points by how
94much unique information they contain. This reordering is then
95applied to whole points (the function value at a point immediately
96followed by gradient information at the same point) in GEK's
97correlation matrix. A standard non-pivoted Cholesky is then
98applied to the reordered GEK correlation matrix and a bisection
99search is used to find the last equation that meets the constraint on
100the (estimate of) condition number. The cost of performing pivoted
101Cholesky on Kriging's correlation matrix is usually negligible
102compared to the cost of the non-pivoted Cholesky factorization of
103GEK's correlation matrix. In tests, it also resulted in more
104accurate GEK models than when pivoted Cholesky or
105whole-point-block pivoted Cholesky was performed on GEK's
106correlation matrix.
107
108Faq::
109See_Also::
110