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The traditional Newton method for solving nonlinear operator equations in Banach spaces is discussed within the context of the continuous Newton method. This setting makes it possible to interpret the Newton method as a discrete dynamical…
A nonlinear partial differential equation is a nonlinear relationship between an unknown function and how it changes due to two or more input variables. A numerical method reduces such an equation to arithmetic for quick visualization, but…
Nonlinear systems with model uncertainty are often described by stochastic differential equations. Some techniques from random dynamical systems are discussed. They are relevant to better understanding of solution processes of stochastic…
We design and analyse a new numerical method to solve ODE system based on the structural method. We compute approximations of solutions together with its derivatives up to order $K$ by solving an entire block corresponding to $R$ time…
In this work, a class of non-linear weakly singular fractional integro-differential equations is considered, and we first prove existence, uniqueness, and smoothness properties of the solution under certain assumptions on the given data. We…
In this paper we present a subgradient method with non-monotone line search for the minimization of convex functions with simple convex constraints. Different from the standard subgradient method with prefixed step sizes, the new method…
Very high dimensional nonlinear systems arise in many engineering problems due to semi-discretization of the governing partial differential equations, e.g. through finite element methods. The complexity of these systems present…
When a system of first order linear ordinary differential equations has eigenvalues of large magnitude, its solutions exhibit complicated behaviour, such as high-frequency oscillations, rapid growth or rapid decay. The cost of representing…
In this paper, we propose a globally convergent method for solving constrained nonlinear systems. The method combines an efficient Newton conditional gradient method with a derivative-free and nonmonotone linesearch strategy. The global…
Nonlinear inverse problems pervade engineering and science, yet noisy, non-differentiable, or expensive residual evaluations routinely defeat Jacobian-based solvers. Derivative-free alternatives either demand smoothness, require large…
To extract the approximate solutions in the case of nonlinear fractional order differential equations with the homogeneous and nonhomogeneous boundary conditions, the weighted residual method is embedded here. We exploit three methods such…
Data-driven modeling and machine learning are widely used to model the behavior of dynamic systems. One application is the residual evaluation of technical systems where model predictions are compared with measurement data to create…
A nonlinear algebraic equation system of 5 variables is numerically solved, which is derived from the application of the Fourier transform to a differential equation system that allows modeling the behavior of the temperatures and the…
Sparse solution problems play an important role in both signal processing and image restoration. In this paper, we propose a stochastic column-block nonlinear Bregman method for efficiently computing sparse solutions to nonlinear systems.…
In this paper we introduce a new fix point iteration scheme for solving nonlinear electromagnetic scattering problems. The method is based on a spectral formulation of Maxwell's equations called the Bidirectional Pulse Propagation…
Plotting the residuals is a recommended procedure to diagnose deviations from linear model assumptions, such as non-linearity, heteroscedasticity, and non-normality. The presence of structure in residual plots can be tested using the lineup…
This paper deals with speeding up the convergence of a class of two-step iterative methods for solving linear systems of equations. To implement the acceleration technique, the residual norm associated with computed approximations for each…
A key issue in dimension reduction of dissipative dynamical systems with spectral gaps is the identification of slow invariant manifolds. We present theoretical and numerical results for a variational approach to the problem of computing…
A zero-finding technique for solving nonlinear equations more efficiently than they usually are with traditional iterative methods in which the order of convergence is improved is presented. The key idea in deriving this procedure is to…
The spectral gradient method is known to be a powerful low-cost tool for solving large-scale optimization problems. In this paper, our goal is to exploit its advantages in the stochastic optimization framework, especially in the case of…