A MODEL TRUST REGION MODIFICATION OF INEXACT NEWTON'S METHOD FOR NONLINEAR TWO POINT BOUNDARY VALUE PROBLEMS (QUASILINEARIZATION)
DEAN, EDWARD JEROME
Doctor of Philosophy
The method of quasilinearization for the solution of nonlinear two point boundary value problems is Newton's method for a nonlinear differential operator equation. Semilocal convergence results, with the attendant error estimates, are available from the Kantorovich Theorem. Since the linear boundary value problem to be solved at each iteration must be discretized, it is natural to consider quasilinearization in the framework of Inexact Newton methods. Conditions on the size of the relative residual of the linear differential equation, given by an approximate solution, can then be specified to guarantee rapid local convergence. If initial value techniques are used to solve the linear boundary value problem then it is possible to implement an integration step selection scheme so that the residual criteria is satisfied by the approximate solution. The result is a sequence of approximate solutions to the linear boundary value problems that converge to the true solution of the nonlinear boundary value problem. A model trust region approach to globalization can be extended to this infinite dimensional problem to allow convergence from an arbitrary initial point. The double dogleg implementation yields a globally convergent algorithm that is robust in solving difficult problems.