Investigation of Line Search Globalization and Scaling Aspects of Newton's Method in Two Industrial Implementations

Abstract: Simulating complex physical systems often requires solving systems of nonlinear algebraic equations. One of the most frequently used numerical methods to solve systems of nonlinear equations is Newton's method with its advantage of quadratic local convergence. However, Newton's method does not guarantee global convergence. This raises the need for combining Newton's method with a globalization strategy. One more problem that affects Newton's method convergence is caused by large differences in the scales of the iteration variables as well as the residuals. Although the Newton iteration is affine invariant, the termination criteria and norm calculations are not. This in turn affects the convergence. In this thesis, we address topics of Newton's method globalization using line search and the scaling of both variables and residuals from theoretical and implementation perspective.