Consider an under-determined system of nonlinear equations F (x) = 0, F : IR → IR, where F is continuously differentiable and m > n. This system appears in a variety of applications, including parameter–dependent systems, dynamical systems with periodic solutions, and nonlinear eigenvalue problems. Robust, efficient numerical methods are often required for the solution of this system. Newton’s ...

Least squares methods are effective for solving systems of partial differential equations. In the case of nonlinear systems the equations are usually linearized by a Newton iteration or successive substitution method, and then treated as a linear least squares problem. We show that it is often advantageous to form a sum of squared residuals first, and then compute a zero of the gradient with a ...

We compare the performance of several robust large-scale minimization algorithms for the unconstrained minimization of an ill-posed inverse problem. The parabolized Navier-Stokes equations model was used for adjoint parameter estimation. The methods compared consist of two versions of the nonlinear conjugate gradient method (CG), Quasi-Newton (BFGS), the limited memory Quasi-Newton (L-BFGS) [15...

The semismooth Newton method is a nonsmooth Newton-type method applied to a suitable reformulation of the complementarity problem as a nonlinear and nonsmooth system of equations. It is one of the standard methods for solving these kind of problems, and it can be implemented in an inexact way so that all linear systems of equations have to be solved only inexactly. However, from a practical poi...

We present a phase-based potential ordering that is an extension of the Cascade ordering introduced by Appleyard and Cheshire [1]. The ordering is valid for both two-phase and three-phase flow, and it can handle countercurrent flow due to gravity and/or capillarity. We show how this ordering can be used to reduce the nonlinear algebraic system that arise from the fully-implicit method (FIM) int...

We implemented and optimized seven finite-difference solvers for the full nonlinear Poisson-Boltzmann equation in biomolecular applications, including four relaxation methods, one conjugate gradient method, and two inexact Newton methods. The performance of the seven solvers was extensively evaluated with a large number of nucleic acids and proteins. Worth noting is the inexact Newton method in...