1Blurb::
2Local multi-point model via two-point nonlinear approximation
3Description::
4<b> TANA </b> stands for Two Point Adaptive Nonlinearity Approximation.
5
6The TANA-3 method \cite Xu98 is a multipoint approximation method
7based on the two point exponential approximation \cite Fad90. This
8approach involves a Taylor series approximation in intermediate
9variables where the powers used for the intermediate variables are
10selected to match information at the current and previous expansion
11points.
12
13<em>Known Issue: When using discrete variables, there have been
14sometimes significant differences in surrogate behavior observed
15across computing platforms in some cases.  The cause has not yet been
16fully diagnosed and is currently under investigation.  In addition,
17guidance on appropriate construction and use of surrogates with
18discrete variables is under development.  In the meantime, users
19should therefore be aware that there is a risk of inaccurate results
20when using surrogates with discrete variables.</em>
21
22
23Topics::
24Examples::
25Theory::
26The form of the TANA model is:
27
28\f[ \hat{f}({\bf x}) \approx f({\bf x}_2) + \sum_{i=1}^n
29\frac{\partial f}{\partial x_i}({\bf x}_2) \frac{x_{i,2}^{1-p_i}}{p_i}
30(x_i^{p_i} - x_{i,2}^{p_i}) + \frac{1}{2} \epsilon({\bf x}) \sum_{i=1}^n
31(x_i^{p_i} - x_{i,2}^{p_i})^2 \f]
32
33where \f$n\f$ is the number of variables and:
34
35\f[ p_i = 1 + \ln \left[ \frac{\frac{\partial f}{\partial x_i}({\bf x}_1)}
36{\frac{\partial f}{\partial x_i}({\bf x}_2)} \right] \left/
37\ln \left[ \frac{x_{i,1}}{x_{i,2}} \right] \right.
38\epsilon({\bf x}) = \frac{H}{\sum_{i=1}^n (x_i^{p_i} - x_{i,1}^{p_i})^2 +
39\sum_{i=1}^n (x_i^{p_i} - x_{i,2}^{p_i})^2}
40H = 2 \left[ f({\bf x}_1) - f({\bf x}_2) - \sum_{i=1}^n
41\frac{\partial f}{\partial x_i}({\bf x}_2) \frac{x_{i,2}^{1-p_i}}{p_i}
42(x_{i,1}^{p_i} - x_{i,2}^{p_i}) \right] \f]
43
44and \f${\bf x}_2\f$ and \f${\bf x}_1\f$ are the current and previous expansion
45points. Prior to the availability of two expansion points, a
46first-order Taylor series is used.
47
48Faq::
49See_Also::
50