Other Gradient-Based Methods

Beside these optimization methods some others have been proven to be well suited for device simulation purposes and have therefore also qualified for optimization in general.

• The ``method of steepest descent'' [251,252] is a iterative optimization method and uses the negative gradient of the function as search direction and combine that with a line search algorithm. However, when the condition number of the system matrix is large, the convergence speed is drastically reduced.
• The ``conjugate gradient'' (CG) algorithm [219,253] is used to optimize for instance the matrix function $\Phi({\mathbf{{x}}}) = {{\mathbf{{x}}}^T} {\bf {b}} - {{{\frac{1}{2}}}} {{\mathbf{{x}}}^T} {A} {{\mathbf{{x}}}}$ , which is equivalent to solving $A{\mathbf{{x}}}= {\mathbf{{b}}}$ , where $A$ is a symmetric, positive definite system matrix. Other CG variants are the``Biconjugate gradient'' method [254,255] and ``conjugate gradient squared'' method [256] were introduced to deal with not symmetric or even with non-positive definite matrices $A$ .