Local multi-point model via two-point nonlinear approximation

Specification

Alias: none

Argument(s): none

Description

TANA stands for Two Point Adaptive Nonlinearity Approximation.

The TANA-3 method[91] is a multipoint approximation method based on the two point exponential approximation[24]. 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.

Theory

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.