Nguyen S. Hoang

Dynamical Systems Gradient Method for Solving Nonlinear Equations with Monotone Operators

A version of the Dynamical Systems Gradient Method for solving ill-posed nonlinear monotone operator equations is studied in this paper. A discrepancy principle is proposed and justified. A numerical experiment was carried out with the new stopping rule. Numerical experiments show that the proposed stopping rule is efficient. Equations with monotone operators are of interest in many applications.

Solving ill-conditioned linear algebraic systems by the dynamical systems method

An iterative scheme for the dynamical systems method (DSM) is given, such that one does not have to solve the Cauchy problem occuring in the application of the DSM for solving ill-conditioned Linear Algebraic Systems (LAS). The novelty of the algorithm is that the algorithm does not have to find the regularization parameter α by solving a nonlinear equation. Numerical experiments show that DSM competes favorably with the variational regularization.

DYNAMICAL SYSTEMS METHOD FOR SOLVING NONLINEAR EQUATIONS WITH MONOTONE OPERATORS

A version of the Dynamical Systems Method (DSM) for solving ill-posed nonlinear equations with monotone operators in a Hilbert space is studied in this paper. An a posteriori stopping rule, based on a discrepancy-type principle is proposed and justified mathematically. The results of two numerical experiments are presented. They show that the proposed version of DSM is numerically efficient. The numerical experiments consist of solving nonlinear integral equations.

Dynamical systems : method and applications : theoretical developments and numerical examples

PART I 1 Introduction 3 2 Ill-posed problems 11 3 DSM for well-posed problems 57 4 DSM and linear ill-posed problems 71 5 Some inequalities 93 6 DSM for monotone operators 133 7 DSM for general nonlinear operator equations 145 8 DSM for operators satisfying a spectral assumption 155 9 DSM in Banach spaces 161 10 DSM and Newton-type methods without inversion of the derivative 169 11 DSM and unbounded operators 177 12 DSM and nonsmooth operators 181 13 DSM as a theoretical tool 195 14 DSM and iterative methods 201 15 Numerical problems arising in applications 213 PART II 16 Solving linear operator equations by a Newton-type DSM 255 17 DSM of gradient type for solving linear operator equations 269 18 DSM for solving linear equations with finite-rank operators 281 19 A discrepancy principle for equations with monotone continuous operators 295 20 DSM of Newton-type for solving operator equations with minimal smoothness assumptions 307 21 DSM of gradient type 347 22 DSM of simple iteration type 373 23 DSM for solving nonlinear operator equations in Banach spaces 409 PART III 24 Solving linear operator equations by the DSM 423 25 Stable solutions of Hammerstein-type integral equations 441 26 Inversion of the Laplace transform from the real axis using an adaptive iterative method 455

A NONLINEAR INEQUALITY

A quadratic inequality is formulated in the paper. An estimate on the rate of decay of solutions to this inequality is obtained. This inequality is of interest in a study of dynamical systems and nonlinear evolution equations.

On the equivalence of the continuous Adams–Bashforth method and Nordsieck’s technique for changing the step size

Abstract Recent research has raised the question of whether Nordsieck’s technique for changing the step size in the Adams–Bashforth method is equivalent to the explicit continuous Adams–Bashforth method. This work provides a complete proof that the two approaches are indeed equivalent.

Dynamical Systems Method for solving nonlinear equations with locally Holder continuous monotone operators

A version of the Dynamical Systems Method (DSM) for solving ill-posed nonlinear equations with monotone and locally Hoelder continuous operators is studied in this paper. A discrepancy principle is proposed and justified under natural and weak assumptions. The only smoothness assumption on F is the local Hoelder continuity of order α>1/2.