We present and analyze a new damping approach called backward step control for the globalization of the convergence of Newton-type methods for the numerical solution of nonlinear root-finding problems. We provide and discuss reasonable assumptions that imply convergence of backward step control on the basis of generalized Newton paths in conjunction with a backward analysis argument. In particu...

We investigate a semi-smooth Newton method for the numerical solution of optimal control problems subject to differential-algebraic equations (DAEs) and mixed control-state constraints. The necessary conditions are stated in terms of a local minimum principle. By use of the Fischer-Burmeister function the local minimum principle is transformed into an equivalent nonlinear and semi-smooth equati...

A Newton–Krylov method is an implementation of Newton’s method in which a Krylov subspace method is used to solve approximately the linear systems that characterize steps of Newton’s method. Newton–Krylov methods are often implemented in “matrix-free” form, in which the Jacobian-vector products required by the Krylov solver are approximated by finite differences. Here we consider using approxim...

Null space Newton algorithms are efficient in solving the nonlinear equations arising in hydraulic analysis of water distribution networks. In this article, we propose and evaluate an inexact Newton method that relies on partial updates of the network pipes’ frictional headloss computations to solve the linear systems more efficiently and with numerical reliability. The update set parameters ar...

We describe a Newton-based nonlinear solver for immiscible two-phase transport in the presence of significant viscous, buoyancy, and capillary forces. The evolution of CO2 plumes in heterogeneous saline aquifers, especially during the post-injection period, is an important example of this class of problem. The flux (fractional flow) function, which is strongly nonlinear function of saturation, ...

A Newton–Krylov method is an implementation of Newton’s method in which a Krylov subspace method is used to solve approximately the linear subproblems that determine Newton steps. To enhance robustness when good initial approximate solutions are not available, these methods are usually globalized, i.e., augmented with auxiliary procedures (globalizations) that improve the likelihood of converge...

This document gives an overview of the Next Steps in Signaling (NSIS) framework and protocol suite created by the NSIS Working Group during the period of 2001-2010. It also includes suggestions on how the industry can make use of the new protocols and how the community can exploit the extensibility of both the framework and existing protocols to address future signaling needs.

We study the numerical performance of a limited memory quasi Newton method for large scale optimization which we call the L BFGS method We compare its performance with that of the method developed by Buckley and LeNir which combines cyles of BFGS steps and conjugate direction steps Our numerical tests indicate that the L BFGS method is faster than the method of Buckley and LeNir and is better a...

A new method for quasi-Newton minimization outperforms BFGS by combining least-change updates of the Hessian with step sizes estimated from a Wishart model of uncertainty. The Hessian update is in the Broyden family but uses a negative parameter, outside the convex range, that is usually regarded as the safe zone for Broyden updates. Although full Newton steps based on this update tend to be to...

We study the numerical performance of a limited memory quasi-Newton method for large scale optimization, which we call the L-BFGS method. We compare its performance with that of the method developed by Buckley and LeNir (1985), which combines cyles of BFGS steps and conjugate direction steps. Our numerical tests indicate that the L-BFGS method is faster than the method of Buckley and LeNir, and...