4.2.2.3 LEVENBERG-MARQUARDT Algorithm

The LEVENBERG-MARQUARDT algorithm [242] is an efficient method to solve non-linear least squares problems [243]. Thus, it is well suited for complex inverse modeling tasks especially for TCAD applications where the aim of the LEVENBERG-MARQUARDT algorithm is to optimize (minimize) a twice differentiable function

$$f({\mathbf{{x}}}):\mathbb{R}^n\rightarrow\mathbb{R}.$$ (4.22)

If the original objective function is vector valued, an additional norm has to applied to map the vector to a scalar-valued quantity. The second derivative of the function $f$ is determined by its HESSian^4.6 matrix. Because the optimization tasks for TCAD problems cannot be described analytically, the derivatives have to be calculated for each single point. Since there is no guarantee that the HESSian $H(f,{\mathbf{{x}}})$ is positive definite for non-quadratic forms, the search algorithm might search in the wrong direction. Therefore, a correction term can be introduced to cover this problem by [242]

$$H^k({\mathbf{{x}}}^k) {=}H(f,{\mathbf{{x}}}^k) + \nu^k \tilde{I}.$$ (4.23)

If $H^k({\mathbf{{x}}}^k)$ is still not positive definite, the factor $\nu^k$ is increased by a certain user-defined factor. Since $H^k({\mathbf{{x}}}^k)$ is now per definitionem positive definite, the next point ${\mathbf{{x}}}^{k+1}$ can be calculated by

$${\mathbf{{x}}}^{k+1} {=}{\mathbf{{x}}}^k - H^k({\mathbf{{x}}}^k) \cdot \nabla f({\mathbf{{x}}}^k).$$ (4.24)

However, if there is no improvement in the last minimization step ( $f({\mathbf{{x}}}^{k+1}) > f({\mathbf{{x}}}^k)$ ), the factor $\nu^k$ has to be modified again and the previously described steps have to be recalculated.

This method is a more robust method than the GAUSS-NEWTON method [244] and provides in general an optimum on less iterations. Nevertheless, if the initial guess of ${\mathbf{{x}}}$ is too close to the optimal value, the convergence might be slower than that of the GAUSS-NEWTON method.