Local multi-point model via two-point nonlinear approximation


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TANA stands for Two Point Adaptive Nonlinearity Approximation.

The TANA-3 method [XG98] is a multipoint approximation method based on the two point exponential approximation [FRB90]. This approach involves a Taylor series approximation in intermediate variables where the powers used for the intermediate variables are selected to match information at the current and previous expansion points.

Known Issue: When using discrete variables, there have been sometimes significant differences in surrogate behavior observed across computing platforms in some cases. The cause has not yet been fully diagnosed and is currently under investigation. In addition, guidance on appropriate construction and use of surrogates with discrete variables is under development. In the meantime, users should therefore be aware that there is a risk of inaccurate results when using surrogates with discrete variables.


The form of the TANA model is:

\[\hat{f}({\bf x}) \approx f({\bf x}_2) + \sum_{i=1}^n \frac{\partial f}{\partial x_i}({\bf x}_2) \frac{x_{i,2}^{1-p_i}}{p_i} (x_i^{p_i} - x_{i,2}^{p_i}) + \frac{1}{2} \epsilon({\bf x}) \sum_{i=1}^n (x_i^{p_i} - x_{i,2}^{p_i})^2\]

where \(n\) is the number of variables and:

\[p_i = 1 + \ln \left[ \frac{\frac{\partial f}{\partial x_i}({\bf x}_1)} {\frac{\partial f}{\partial x_i}({\bf x}_2)} \right] \left/ \ln \left[ \frac{x_{i,1}}{x_{i,2}} \right] \right. \epsilon({\bf x}) = \frac{H}{\sum_{i=1}^n (x_i^{p_i} - x_{i,1}^{p_i})^2 + \sum_{i=1}^n (x_i^{p_i} - x_{i,2}^{p_i})^2} H = 2 \left[ f({\bf x}_1) - f({\bf x}_2) - \sum_{i=1}^n \frac{\partial f}{\partial x_i}({\bf x}_2) \frac{x_{i,2}^{1-p_i}}{p_i} (x_{i,1}^{p_i} - x_{i,2}^{p_i}) \right]\]

and \({\bf x}_2\) and \({\bf x}_1\) are the current and previous expansion points. Prior to the availability of two expansion points, a first-order Taylor series is used.