Numerical Analysis in Nonlinear Least Squares Methods and Applications

The approximate greatest descent (AGD) method and a two-phase AGD method (AGDN) are proposed as new methods for a nonlinear least squares problem. Numerical experiments show that these methods outperform existing methods including the Levenberg-Marquardt method. However, the AGDN method outperforms the AGD method with a faster convergence. If the AGDN method fails due to singularity of the Hessian matrix, the AGD method should be used

Continuous Methods for Elliptic Inverse Problems

Numerous mathematical models in applied and industrial mathematics take the form of a partial differential equation involving certain variable coefficients. These coefficients are known and they often describe some physical properties of the model. The direct problem in this context is to solve the partial differential equation. By contrast, an inverse problem asks for the identification of the variable coefficient when a certain measurement of a solution of the partial differential equation is available. A commonly used approach to inverse problems is to solve an optimization problem whose solution is an approximation of the sought coefficient. Such optimization problems are typically solved by discrete iterative schemes. It turns out that most known iterative schemes have their continuous counterparts given in terms of dynamical systems. However, such differential equations are usually solved by specific differential equation solvers. The primary objective of this thesis is to test the feasibility of differential equations based solvers for solving elliptic inverse problems. We will use differential equation solvers such as Euler\u27s Method, Trapezoidal Method, Runge-Kutta Method and Adams-Bashforth Method. In addition, these solvers will also be compared to built-in MATLAB ODE solvers. The performance and accuracy of these methods to solve inverse problems will be thoroughly discussed