We study a Newton-like method for the minimization of an objective function φ that is the sum of a smooth convex function and an `1 regularization term. This method, which is sometimes referred to in the literature as a proximal Newton method, computes a step by minimizing a piecewise quadratic model qk of the objective function φ. In order to make this approach efficient in practice, it is imp...

Nonlinear multigrid methods such as the Full Approximation Scheme (FAS) and Newton-multigrid (Newton-MG) are well established as fast solvers for nonlinear PDEs of elliptic and parabolic type. In this paper we consider Newton-MG and FAS iterations applied to second order differential operators with nonlinear diffusion coefficient. Under mild assumptions arising in practical applications, an app...

We propose a generalization of the Weierstrass iteration for over-constrained systems of equations and we prove that the proposed method is the Gauss-Newton iteration to find the nearest system which has at least k common roots and which is obtained via a perturbation of prescribed structure. In the univariate case we show the connection of our method to the optimization problem formulated by K...

We propose a semismooth active-set Newton algorithm for solving the nonlinear complementarity problems with degenerate solutions. This method introduces the active-set technique to identify the degenerate set. At each iteration, the search direction is obtained by two reduced linear systems. Instead of employing gradient steps as adjustments to guarantee the sufficient reduction of the merit fu...

We study the local convergence rates of several most widely used single-vector Newton-like methods for the solution of a degenerate eigenvalue of nonlinear algebraic eigenvalue problems of the form T (λ)v = 0. This problem has not been completely understood, since the Jacobian associated with Newton’s method is singular at the desired eigenpair, and the standard convergence theory is not applic...

| The use of the modiied Newton method for the solution of nonlinear magnetostatic problems arising from the vector potential Finite Element analysis is investigated. In particular the optimum choice of the relaxation factor is investigated. A new method is developed for determining the relaxation factor which minimizes the energy functional in the direction along the solution update at each no...

We describe a non-overlapping domain decomposition algorithm for nonlinear porous media flows discretized with the multiscale mortar mixed finite element method. There are two main ideas: (1) linearize the global system in both subdomain and interface variables simultaneously to yield a single Newton iteration; and (2) algebraically eliminate subdomain velocities (and optionally, subdomain pres...