Derivative Free Regularization Method for Nonlinear Ill-Posed Equations in Hilbert Scales

Abstract

In this paper, we deal with nonlinear ill-posed operator equations involving a monotone operator in the setting of Hilbert scales. Our convergence analysis of the proposed derivative-free method is based on the simple property of the norm of a self-adjoint operator. Using a general Hölder-type source condition, we obtain an optimal order error estimate. Also we consider the adaptive parameter choice strategy proposed by Pereverzev and Schock (2005) for choosing the regularization parameter. Finally, we applied the proposed method to the parameter identification problem in an elliptic PDE in the setting of Hilbert scales and compare the results with the corresponding method in Hilbert space.

Extended Efficient Multistep Solvers for Solving Equations in Banach Spaces

“…Finally, we conclude that our research contributes to the development of more efficient and robust numerical methods for solving nonlinear equations in a variety of scientific and engineering applications. Similar methodology is applicable to other methods [2,[8][9][10][11][12][13][14]. Thus, this can be the chosen course of action for upcoming projects.…”

“…The accurate computation of the radius of convergence is essential for optimizing the performance and reliability of iterative solvers in practical applications. Further, the computable upper error bounds on ∥y (j) n − x * ∥ and results on the uniqueness of x * were not considered in a previous study [7] or in similar methods using Taylor series to determine the order of convergence [2,[8][9][10][11][12][13][14].…”

Extended Higher Order Iterative Method for Nonlinear Equations and its Convergence Analysis in Banach Spaces

“…In numerical analysis, higher order iterative methods have acquired foremost significance for solving nonlinear equations that arise in numerous branches of science and technology [1,2]. Various researchers have developed a plethora of iterative methods [3][4][5][6][7][8][9][10][11][12][13] for solving nonlinear equations given in the form ( ) 0, f x = where f : D ⊂ R → R is a continuously differentiable nonlinear function and D is an open interval. Most widely used iterative method is quadratically convergent Newton's method given by 1 ( ) , 0,1, 2, .…”

Two Point Iterative Schemes for Nondifferentiable Equations in Banach Space

“…Numerous studies exist in the local as well as the semi-local convergence analysis of iterative schemes [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. Recently, there has been a surge in the development of schemes for solving equations or systems of equations involving nondifferentiable operators.…”

“…A finer and more flexible local and semi-local convergence analysis for the scheme (1.2) is developed involving an invertible operator P , which if chosen appropriately leads to weaker convergence conditions, better uniqueness of the solution and a larger radius of convergence than if P is chosen to be as in earlier studies F (x * ) or F (x 0 ) or [x 0 , x −1 ; F ]. This idea can be extended to multistep and multipoint schemes [1][2][3][4][5][6][7][8][9][10][11][12][13][14][15][16]. This is the direction of our future research.…”