A Nonlinear Least Squares problem is an optimization problem for which the objective function to be minimized has the form
𝐹(𝑥) = [𝑚∑𝑖=1] ƒ𝑖2(𝑥) , 𝑥∈𝑅𝑛 , 𝑚 ≥ 𝑛
The structure of 𝐹(𝑥) allows the problem to be solved using Newton-type methods, in which the Hessian is efficiently approximated from the first derivatives, rather than explicitly calculated. In most of the work which has appeared in recent years, it has been assumed that the ƒ𝑖's are twice continuously differentiable, that there are no explicit constraints on the variables, and that the Jacobian is small enough to be held in core and factorized. The first part of this thesis presents an overview of some of the algorithms which have been used on this general problem.
Special types of nonlinear least squares problems are then examined, in particular
(i) problems for which there are general linear inequality constraints on 𝑥
(ii) problems in which the functions ƒ𝑖 may be nonsmooth (i.e. have discontinuous first partial derivatives)
(iii) problems for which the Jacobian matrix
𝐽(𝑥) = [ ∂ƒ𝑖 / ∂𝑥𝑗 ] 𝑖=1,..𝑚 ; 𝑗=1,..𝑛
is sparse; that is, most of the ƒ𝑖's are functions of only a small subset of the variables.
Algorithms for these three classes of problems are presented in Chapters 2, 3, and 4, together with convergence results and numerical experience.
In the final Chapter, we consider the geophysical problem of inverting seismic arrival time data, in order to accurately locate earthquake coordinates and to obtain density information about the earth's crust in the vicinity of the recording stations. This can be formulated as a sparse nonlinear least squares problem, and is solved using a modification of the algorithm of Chapter 4. Numerical results are obtained for two actual data sets, and are compared with the results obtained from a program currently in use at the U. S. Geological Survey.
The appendices contain listings of the FORTRAN programs which were used to implement the algorithms of Chapters 2, 3, and 4.