In this paper, we investigate the semismooth Newton and quasi-Newton methods for the minimization problem in the weighted `−regularization of nonlinear inverse problems. We propose the conditions for obtaining the convergence of two methods. The semismooth Newton method is proven to locally converge with superlinear rate and the semismooth quasi-Newton method is proven to locally converge at le...

More than a decade agao, Newton’s method has been proposed for constructing the convex best interpolant. Its local quadratic convergence has only been established recently by recasting it as the generalized Newton method for semismooth equations. It still remains mysterious that the Newton method coupled with line search strategies works practically well in global sense. Similar to the classica...

We first recall some properties of infinite tridiagonal matrices considered as matrix transformations in sequence spaces of the forms sξ , sξ , s (c) ξ , or lp(ξ). Then, we give some results on the finite section method for approximating a solution of an infinite linear system. Finally, using a quasi-Newton method, we construct a sequence that converges fast to a solution of an infinite linear ...

A Multigrid Newton–Krylov Method for Multimaterial Equilibrium Radiation Diffusion1 William J. Rider,∗ Dana A. Knoll,∗ and Gordon L. Olson† ∗Hydrodynamic Methods Group, Applied Theoretical and Computational Physics Division, Los Alamos National Laboratory, MS D413, Los Alamos, New Mexico 87545; and †Transport Methods Group, Applied Theoretical and Computational Physics Division, Los Alamos Nati...

In this paper, we present a new family of two-step Newton-type iterative methods with memory for solving nonlinear equations. In order to obtain a Newton-type method with memory, we first present an optimal two-parameter fourth-order Newton-type method without memory. Then, based on the two-parameter method without memory, we present a new two-parameter Newton-type method with memory. Using two...