Applying Variable Projection Methods to Nonlinear Least Squares Problems

This thesis focuses on solving separable nonlinear least squares (SNLLS) problems and explores how the so-called Variable Projection (VarPro) method can be used to solve this particular type of problem. First, there is a brief discussion on curve fitting methods and SNLLS models. Then, an overview of the VarPro algorithm is discussed, along with the optimization concepts that facilitate the method's success. We examine how to derive the Jacobian for the nonlinear solvers and consider different ways to approximate it numerically. This leads into a section focusing on a variety of numerical experiments that illustrate the effectiveness of the VarPro method. The tests demonstrate how different initial guesses, noise levels, and Jacobian approximations affect the accuracy and efficiency of the computations. The thesis also briefly talks through some of the many applications of VarPro across a wide spectrum of topics, which include numerical analysis, biomedical imaging, spectroscopy, and chemistry.