Abstract:

Quasi-Newton methods are a class of numerical techniques aimed at solving nonlinear root-finding and optimization problems. The secant method recursively improves the approximation of a functions root by using successive secant lines. As a simple, one-dimensional case of Quasi-Newton methods, the secant method shares the same core idea of approximating derivatives but is limited to root-finding in one dimension. In contrast, Quasi-Newton methods generalize this approach to tackle more complex optimization problems in higher-dimensional spaces. These methods are also employed to accelerate the convergence of coupled problems without requiring any knowledge of the individual participant solvers. The Quasi-Newton methods operate in an infinitive space by default while processing the data, which can result in issues when Quasi-Newton is post-processing the physically bounded values such as concentrations within the range [0, 1]. This may lead to values exceeding the specified range, potentially causing participant solvers to break down or generate nonphysical outcomes. To address this concern, a cut-off step can be incorporated to confine the bounded values, that means the part exceeding the range after the original Quasi-Newton acceleration process is cut off. According to numerical experiments, this cut-off step could prevent the solver from breaking down, but it might harm the convergence property of the original problem. Additionally, it easily leads the simulation to be trapped in "fake convergence", i.e. the data is accelerated to somewhere outside the bound and is cut off to the same bound value consecutively in two iterations, giving the false impression of convergence. An alternative method involves applying transformations before and after the cut-off step. Numerous experiments suggest that this approach needs in average fewer iterations to converge, compared to the pure cut-off method and in general, it faces a lower likelihood of fake convergence. However, the performance of both methods varies with different testing functions, gradients, starting points and other parameters.

This thesis aims to numerically analyze and visualize the impact of the cut-off and transformation methods on the convergence property of general functions for one-dimensional and two-dimensional cases. The evaluation is conducted using the outcomes from various parameter settings, including different testing functions, starting points and normalization values.