Santhosh George

Local convergence for multi-point-parametric Chebyshev-Halley-type methods of high convergence order

We present a local convergence analysis for general multi-point-Chebyshev-Halley-type methods (MMCHTM) of high convergence order in order to approximate a solution of an equation in a Banach space setting. MMCHTM includes earlier methods given by others as special cases. The convergence ball for a class of MMCHTM methods is obtained under weaker hypotheses than before. Numerical examples are also presented in this study.

An analysis of Lavrentiev regularization method and Newton type process for nonlinear ill-posed problems

Abstract In this paper we consider the Lavrentiev regularization method and a modified Newton method for obtaining stable approximate solution to nonlinear ill-posed operator equations F ( x ) = y where F : D ( F ) ⊆ X ⟶ X is a nonlinear monotone operator or F ′ ( x 0 ) is nonnegative selfadjoint operator defined on a real Hilbert space X . We assume that only a noisy data y δ ∈ X with ‖ y - y δ ‖ ⩽ δ are available. Further we assume that Frechet derivative F ′ of F satisfies center-type Lipschitz condition. A priori choice of regularization parameter is presented. We proved that under a general source condition on x 0 - x ˆ , the error ‖ x ˆ - x n , α δ ‖ between the regularized approximation x n , α δ ( x 0 , α δ ≔ x 0 ) and the solution x ˆ is of optimal order. In the concluding section the algorithm is applied to numerical solution of the inverse gravimetry problem.

Shock coupled fourth-order diffusion for image enhancement

In this paper a shock coupled fourth-order diffusion filter is proposed for image enhancement. This filter converges at a faster rate while preserving and enhancing edges, ramps and textures present in the images. The proposed filter diffuses with varying magnitudes in the directions normal to the level-curve and along it. The magnitude of the directional diffusion is controlled by a diffusion function, meant to provide a good response in the direction along the level-curves, than across them. The proposed filter can still preserve the planar approximation of the image, thereby avoiding the discrepancy caused due to the staircase effect, as in the second-order counterparts. The anisotropic property of the filter is thoroughly studied, analyzed and demonstrated with perspective and quantitative results. The performance of the proposed filter is compared with the state-of-the-art methods for image enhancement. The quantitative and perspective measures provided endorse the capability of the method to enhance various kinds of images.

A time-dependent switching anisotropic diffusion model for denoising and deblurring images

A conditionally anisotropic diffusion based deblurring and denoising filter is introduced in this paper. This is a time-dependent curvature based model and the steady state can be attained at a faster rate, using the explicit time-marching scheme. The filter switches between isotropic and anisotropic diffusion depending on the local image features. The switching of the filter is controlled by a binary function, which returns either zero or one, based on the underlying local image gradient features. The parameters in the proposed filter can be fine-tuned to get the desired output image. The filter is applied to various kinds of input test images and the response is analyzed. The filter is found to be effective in the reconstruction of partially textured, textured, constant-intensity and color images, as is evident from the results provided.

Expanding the applicability of Lavrentiev regularization methods for ill-posed problems

In this paper, we are concerned with the problem of approximating a solution of an ill-posed problem in a Hilbert space setting using the Lavrentiev regularization method and, in particular, expanding the applicability of this method by weakening the popular Lipschitz-type hypotheses considered in earlier studies such as (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009). Numerical examples are given to show that our convergence criteria are weaker and our error analysis tighter under less computational cost than the corresponding works given in (Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 26:35-48, 2005; Bakushinskii and Smirnova in Nonlinear Anal. 64:1255-1261, 2006; Bakushinskii and Smirnova in Numer. Funct. Anal. Optim. 28:13-25, 2007; Jin in Math. Comput. 69:1603-1623, 2000; Mahale and Nair in ANZIAM J. 51:191-217, 2009).MSC:65F22, 65J15, 65J22, 65M30, 47A52.

Unified convergence domains of Newton-like methods for solving operator equations

We present a unified semilocal convergence analysis in order to approximate a locally unique zero of an operator equation in a Banach space setting. Using our new idea of restricted convergence domains we generate smaller Lipschitz constants than in earlier studies leading to the following advantages: weaker sufficient convergence criteria, tighter error estimates on the distances involved and an at least as precise information on the location of the zero. Hence, the applicability of these methods is extended. These advantages are obtained under the same cost on the parameters involved. Numerical examples where the old sufficient convergence criteria cannot apply to solve equations but the new criteria can apply are also provided in this study.

An application of Newton-type iterative method for the approximate implementation of Lavrentiev regularization

Abstract. Motivated by the two-step directional Newton method considered by Argyros and Hilout (2010) for approximating a zero of a differentiable function F defined on a convex set of a Hilbert space H, we consider a two-step Newton–Lavrentiev method (TSNLM) for obtaining an approximate solution to the nonlinear ill-posed operator equation , where is a nonlinear monotone operator defined on a real Hilbert space X. It is assumed that and that the only available data are with . We prove that the TSNLM converges cubically to a solution of the equation (such solution is an approximation of ) where x0 is the initial guess. Under a general source condition on , we derive order optimal error bounds by choosing the regularization parameter α according to the balancing principle considered by Perverzev and Schock (2005). The computational results provided endorse the reliability and effectiveness of our method.

Local convergence for deformed Chebyshev-type method in Banach space under weak conditions

We present a local convergence analysis for deformed Chebyshev methods in order to approximate a solution of a nonlinear equation in a Banach space setting. Our methods include the Chebyshev and other high-order methods under hypotheses up to the first Fréchet derivative in contrast to earlier studies using hypotheses up to the second or third Fréchet derivative. The convergence ball and error estimates are given for these methods. Numerical examples are also provided in this study.

Iterative regularization methods for nonlinear ill-posed operator equations with m-accretive mappings in banach spaces

Abstract In this paper, a modified Newton type iterative method is considered for approximately solving ill-posed nonlinear operator equations involving m -accretive mappings in Banach space. Convergence rate of the method is obtained based on an a priori choice of the regularization parameter. Our analysis is not based on the sequential continuity of the normalized duality mapping.

Inverse Free Iterative Methods for Nonlinear Ill-Posed Operator Equations

We present a new iterative method which does not involve inversion of the operators for obtaining an approximate solution for the nonlinear ill-posed operator equation . The proposed method is a modified form of Tikhonov gradient (TIGRA) method considered by Ramlau (2003). The regularization parameter is chosen according to the balancing principle considered by Pereverzev and Schock (2005). The error estimate is derived under a general source condition and is of optimal order. Some numerical examples involving integral equations are also given in this paper.