We introduce a class of stabilizing Newton-Kaczmarz methods for nonlinear ill-posed problems and analyze their convergence and regularization behaviour. As usual for iterative methods for solving nonlinear ill-posed problems, conditions on the nonlinearity (or the derivatives) have to be imposed in order to obtain convergence. As we shall discuss in general and in some specific examples, the no...

where 5 > 0, a is a constant, and F\(p) is analytic and bounded in the half plane Re(j>) > c. We assume that this condition is satisfied. Whether this condition is also sufficient for the convergence of the w-point quadrature formula to the true value of f(t) in (1), when n tends to infinity, has not been determined. The author makes use here of the fact that the convergence occurs whenever F(p...

Abstract: This paper addresses the numerical solution of optimal control problems for systems described by ordinary differential equations with control constraints. The state equation is discretized by a general explicit Runge-Kutta scheme and the controls are approximated by functions that are piecewise polynomial, but not necessarily continuous. We then propose an approximate gradient project...

Since the solution of a second-kind Volterra integral equation with weakly singular kernel has, in general, unbounded derivatives at the left endpoint of the interval of integration, its numerical solution by polynomial spline collocation on uniform meshes will lead to poor convergence rates. In this paper we investigate the convergence rates with respect to graded meshes, and we discuss the pr...

We present a new algorithm for numerically computing quadrature formulas for arbi-trary domains which exactly integrate a given polynomial space. An effective method for constructingquadrature formulas has been to numerically solve a nonlinear set of equations for the quadraturepoints and their associated weights. Symmetry conditions are often used to reduce the number ofequatio...

We examine the convergence of product quadrature formulas of interpolatory type, based on the zeros of certain generalized Jacobi polynomials, for the discretization of integrals of the type / K(x,y)f(x)dx, -l<y<\, • i where the kernel K(x, y) is weakly singular and the function f{x) has singularities only at the endpoints ± 1 . In particular, when K(x , y) = log \x — y\, K(x ,y) = \x — y\", v ...

In this paper we present an efficient branch-following procedure that can be used not only to compute branches of periodic solutions of periodically forced dynamical systems but also to determine the stability of the periodic solutions. The procedure combines Broyden’s method with a subspace iteration method to determine the dominant eigenvalues. The method has connections with the hybrid Newto...

In this paper, we propose new variants of Newton’s method based on quadrature formula and power mean for solving nonlinear unconstrained optimization problems. It is proved that the order of convergence of the proposed family is three. Numerical comparisons are made to show the performance of the presented methods. Furthermore, numerical experiments demonstrate that the logarithmic mean Newton’...

The importance of singular and hypersingular integral transforms, coming from their many applications, justifies some interest in their numerical approximation. The literature about the numerical evaluation of such integrals on bounded intervals is wide and quite satisfactory; instead only few papers deal with the numerical evaluation of such integral transforms on half-infinite intervals or on...