Monotonieeigenschaften des Steffensen-Verfahrens.
Moving Dirichlet boundary conditions
This paper develops a framework to include Dirichlet boundary conditions on a subset of the boundary which depends on time. In this model, the boundary conditions are weakly enforced with the help of a Lagrange multiplier method. In order to avoid that the ansatz space of the Lagrange multiplier depends on time, a bi-Lipschitz transformation, which maps a fixed interval onto the Dirichlet boundary, is introduced. An inf-sup condition as well as existence results are presented for a class of second...
Multilevel Schwarz method for the minimization with constraints of non-quadratic functionals.
Multiple Zeros and Applications to Optimal Linear Functionals.
Neutron transport theory in the multigroup energetical approximation (Preliminary communication)
New convergence criteria for the Newton-Kantorovich method and some applications to nonlinear integral equations
New iterative approximation methods for a countable family of nonexpansive mappings in Banach spaces.
New iterative scheme for finite families of equilibrium, variational inequality, and fixed point problems in Banach spaces.
New perturbed iterations for a generalized class of strongly nonlinear operator inclusion problems in Banach spaces.
New sufficient convergence conditions for the secant method
We provide new sufficient conditions for the convergence of the secant method to a locally unique solution of a nonlinear equation in a Banach space. Our new idea uses “Lipschitz-type” and center-“Lipschitz-type” instead of just “Lipschitz-type” conditions on the divided difference of the operator involved. It turns out that this way our error bounds are more precise than the earlier ones and under our convergence hypotheses we can cover cases where the earlier conditions are violated.
New unifying convergence criteria for Newton-like methods
We present a local and a semilocal analysis for Newton-like methods in a Banach space. Our hypotheses on the operators involved are very general. It turns out that by choosing special cases for the "majorizing" functions we obtain all previous results in the literature, but not vice versa. Since our results give a deeper insight into the structure of the functions involved, we can obtain semilocal convergence under weaker conditions and in the case of local convergence a larger convergence radius....
Newton, Chebyshev, and Halley Basins of Attraction; A Complete Geometric Approach
Newton-Kantorovich method and its global convergence.
Newton's Method Under Mild Differentiability Conditions with Error Analysis.
Newton-type iterations as approximations of Chebyshev's method.
Newton-type iterative methods for nonlinear ill-posed Hammerstein-type equations
We use a combination of modified Newton method and Tikhonov regularization to obtain a stable approximate solution for nonlinear ill-posed Hammerstein-type operator equations KF(x) = y. It is assumed that the available data is with , K: Z → Y is a bounded linear operator and F: X → Z is a nonlinear operator where X,Y,Z are Hilbert spaces. Two cases of F are considered: where exists (F’(x₀) is the Fréchet derivative of F at an initial guess x₀) and where F is a monotone operator. The parameter...
Non linear schemes for the heat equation in 1D
Inspired by the growing use of non linear discretization techniques for the linear diffusion equation in industrial codes, we construct and analyze various explicit non linear finite volume schemes for the heat equation in dimension one. These schemes are inspired by the Le Potier’s trick [C. R. Acad. Sci. Paris, Ser. I 348 (2010) 691–695]. They preserve the maximum principle and admit a finite volume formulation. We provide a original functional setting for the analysis of convergence of such methods....
Nondiscrete induction and an inversion-free modification of Newton's method
Nondiscrete induction and double step secant method.
Nonlinear dynamics systems - bifurcations, continuation methods, periodic solutions