**Title: Inexact Newton Methods, Newton-Krylov Methods, and Extensions for Large-Scale Underdetermined Systems
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**Author: Homer F. Walker, Professor Mathematical Sciences Department Worcester Polytechnic Institute and Department of Energy Program Manager Applied Mathematics Research **

**Abstract: **

Inexact Newton methods provide a natural framework for implementing Newton's method when an iterative linear algebra method is used to solve approximately the linear subproblems that characterize Newton steps. When the linear algebra method is a Krylov subspace method, the resulting inexact Newton method is known as a Newton-Krylov method. This talk will begin with a brief introduction to inexact Newton and Newton-Krylov methods, outlining their local and global convergence properties, efficient implementation, and performance on several benchmark CFD problems. Following this, I will discuss extensions of these methods suitable for large-scale nonlinear systems that are underdetermined, i.e., have more unknowns than equations, and report on numerical experiments with a Newton-GMRES implementation applied to solving parameter-dependent systems and to determining periodic solutions of time-dependent problems. This work is joint with John Shadid and Roger Pawlowski, Sandia National Laboratories, and with Joseph Simonis, the Boeing Company.