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