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