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In the present paper, we consider the semilocal convergence problems of the two-step Newton method for solving nonlinear operator equation in Banach spaces. Under the assumption that the first derivative of the operator satisfies a generalized Lipschitz condition, a new semilocal convergence analysis for the two-step Newton method is presented. The Q-cubic convergence is obtained by an additional condition. This analysis also allows us to obtain three important spacial cases about the convergence results based on the premises of Kantorovich, Smale and Nesterov-Nemirovskii types. An application of our convergence results is to the approximation of minimal positive solution for a nonsymmetric algebraic Riccati equation arising from transport theory.
We analyze several Galerkin approximations of a Gaussian random field $mathcal{Z}colonmathcal{D}timesOmegatomathbb{R}$ indexed by a Euclidean domain $mathcal{D}subsetmathbb{R}^d$ whose covariance structure is determined by a negative fractional power
We present a local convergence analysis of inexact Newton-like methods for solving nonlinear equations under majorant conditions. This analysis provides an estimate of the convergence radius and a clear relationship between the majorant function, whi
Using deep neural networks to solve PDEs has attracted a lot of attentions recently. However, why the deep learning method works is falling far behind its empirical success. In this paper, we provide a rigorous numerical analysis on deep Ritz method
We propose and analyse a minimal-residual method in discrete dual norms for approximating the solution of the advection-reaction equation in a weak Banach-space setting. The weak formulation allows for the direct approximation of solutions in the Leb
This paper studies the problem of approximating a function $f$ in a Banach space $X$ from measurements $l_j(f)$, $j=1,dots,m$, where the $l_j$ are linear functionals from $X^*$. Most results study this problem for classical Banach spaces $X$ such as