Browsing by Author "George, Santhosh"

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A Study on Ill-Posed Equations and Iterative Methods
(National Institute of Technology Karnataka, Surathkal., 2024) R, Krishnendu; George, Santhosh; P, Jidesh
Many problems that arise in various fields of study can be modeled into equations that are well-posed/ill-posed (linear or nonlinear). Especially in science and engineer ing, most of the inverse problems are ill-posed. The first half of the thesis focuses on f inite dimensional realization of regularization methods for ill-posed problems. The second half deals with iterative methods for solving well-posed nonlinear equations. It is proved in the literature that the Fractional Tikhonov regularization method (FTR) reduces the over smoothing of the solution compared to the usual Tikhonov reg ularization method for ill-posed problems. In Chapter 2 of the thesis, the FTR method in the finite dimensional setting is studied. The regularization parameter is chosen using Raus and Gfrerer type discrepancy principle in this Chapter. The choice of regularization parameter and suitable source condition plays an im portant role in a regularization method. In Chapter 3, an efficient new parameter choice strategy is introduced. The advantage is that this parameter choice strategy computes the regularization parameter before computing the approximate solution and is depen dent on the given data of the problem. This new parameter choice also provide the optimal order. The proposed parameter choice strategy is depending on a new source condition. Higher order iterative methods are used to solve nonlinear equations. The conver gence order of these methods uses Taylor’s expansion and assumptions on the higher order Fréchet derivative of the operator. In Chapter 4 and Chapter 5, we have elimi nated the use of Taylor’s expansion and hence assumptions on the higher order Fréchet derivatives of the operator in the problem. Moreover, the desired convergence order of the iterative method is obtained without using assumptions on the higher order Fréchet derivatives and hence the applicability of these iterative methods are extended to prob lem which were not posible using earlier studies. These iterative methods are also applied to solve nonlinear ill-posed problems.
Fractional Regularization Methods For Ill-Posed Problems In Hilbert Scales
(National Institute of Technology Karnataka, Surathkal, 2022) Mekoth, Chitra; George, Santhosh; P, Jidesh
Regularization methods are widely used for solving ill-posed problems. In this thesis we consider the fractional regularization methods for solving equations of the form T (x) = y where T : X −→ Y is a linear operator between the Hilbert spaces X and Y . In practical applications, we mostly have only the noisy data yδ such that ‖y − yδ ‖ ≤ δ. Throughout our study, we work in the setting of Hilbert scales as it improves the order of convergence. We study the Fractional Tikhonov Regularization method in Hilbert scales and for selecting the regulariza- tion parameter the adaptive choice method introduced by Pereverzev and Schock (2005) is used. Also we introduce a new parameter choice strategy. While per- forming numerical calculations, it is always easier to work in a finite dimensional setting than in an infinite dimensional one. For this reason we study the finite dimensional realization of the fractional Tikhonov and the fractional Lavrentiev regularization methods in Hilbert scales. We also study the analogous of the dis- crepancy principle considered in George and Nair (1993) for Fractional Lavrentiev method.
Image Reconstruction Using PDE, Variational and Regularization Methods
(National Institute of Technology Karnataka, Surathkal, 2013) P, Jidesh; George, Santhosh
Image restoration and enhancement are two important requirements in the field of image processing. In this study three anisotropic non-linear diffusion filters are proposed for image restoration and enhancement and one filter for image inpainting. The orientation, type and extent of filtering are controlled by the decision mechanism based on the underlying image features. The first process is a conditionally anisotropic diffusion for deblurring and denoising images. This process is a time-dependent curvature based model and the steady state is attained at a faster rate, using the explicit time-marching scheme. The filter switches between isotropic and anisotropic behavior based on the local image features. Two other non-linear curvature based diffusion processes are devised, one for image enhancement and the other one for image inpainting. The diffusion process in these filters is driven by the Gauss curvature of the level curves of the image. Therefore, these methods are capable of preserving structures even with non-zero mean curvature values like curvy edges and corners. To be precise, the second process couples a hyperbolic shock filter together with a Gauss curvature driven diffusion term to enhance images. And the third one inpaints the intended domain based on the Gauss curvature. Finally, a fourth-order shock coupled diffusion filter is proposed for image enhancement. This is an anisotropic model that converges at a faster rate and preserves planar approximation while enhancing images. In this study a thorough theoretical and experimental analysis is carried out for each and every diffusion process introduced as a part of this thesis work. A variety of applications are presented for denoising and deblurring gray-level and color images. The required mathematical preliminaries are presented in the introduction of the thesis. We conclude the thesis highlighting some of the future enhancements that could be possibly taken forward for further research.
Iterative Methods and their Applications for Solving Non-Linear Ill-Posed Equations
(National Institute of Technology Karnataka, Surathkal., 2023) K., Muhammed Saeed; George, Santhosh; P., Jidesh
This thesis deals with iterative methods and their convergence for solving non-linear equations in Banach Spaces. As an application, it also deals with solving non-linear ill posed equations in a Hilbert space setting. Under various assumptions, local and semi local convergence analyses of some iterative schemes are studied. We have established the desired order of convergence using weaker assumptions than those available in the literature. We have also extended some of the methods efficiently. Computable radii of convergence and dynamics analysis using the basin of attractions are other highlights. The first contribution of the thesis is the convergence analysis of a fifth-order it erative method using conditions only on the first Fréchet derivative. This increased the applicability of the method. In our second work, we used the iterative method for solving the regularized equation corresponding to a non-linear ill-posed equation. We introduced a new source condition and parameter choice strategy for the desired results. Thirdly, using Lipschitz-type assumptions on first and second derivatives instead of Taylor series expansion, we established third-order convergence of an iterative Home ier method. We further extended this method to the fifth and sixth order. Lastly, we studied another iterative method introduced by Traub. We established third-order con vergence without using Taylor series expansion. We extended this method to the fifth and sixth order.
Iterative Regularization Theory for Nonlinear Ill-Posed Problems
(National Institute of Technology Karnataka, Surathkal, 2019) Sreedeep, C. D.; George, Santhosh
In science and engineering many practical problems can be formulated using mathematical modelling and can be classified as nonlinear ill-posed problems. Here we consider those ill-posed equations involving m-accretive operators in Banach spaces. Using a general H¨older type source condition we were able to obtain an optimal order error estimate. For nonlinear problems, obtaining a closed form solution is possible only in rare cases, so most of the methods considered for approximating the solution of nonlinear problems are iterative. Four different types of iterative schemes are being discussed in this thesis. Firstly, we consider a derivative and inverse free method and obtained second order convergence. Then, we produced an extended Newton-type iterative scheme that converges cubically to the solution which uses assumptions only on the first Fr´echet derivative of the operator. Afterwards, we studied Newton-Kantorovich regularization method and obtained second order convergence with weak assumptions. Finally, we examined Secant-type iteration and proved that the proposed iterative scheme has a convergence order at least 2.20557 using assumptions only on first Fr´echet derivative of the operator. Through out the work, for choosing the regularization parameter we have taken the adaptive parameter choice strategy given by Pereverzev and Schock (2005).
Mathematical Modeling for the Solutions of Equations and Systems of Equations with Applications Vol.1
(2018) Argyros, Ioannis K.; George, Santhosh; Thapa, Narayan
Mathematical Modeling for the Solutions of Equations and Systems of Equations with Applications Vol.2
(2018) Argyros, Ioannis K.; George, Santhosh; Thapa, Narayan
Mathematical Modeling for the Solutions of Equations and Systems of Equations with Applications Vol.3
(2019) Argyros, Ioannis K.; George, Santhosh
Mathematical Modeling for the Solutions of Equations and Systems of Equations with Applications Vol.4
(2020) Argyros, Ioannis K.; George, Santhosh
Newton Type Methods for Lavrentiev Regularization of Nonlinear Ill-Posed Operator Equations
(National Institute of Technology Karnataka, Surathkal, 2013) Pareth, Suresan; George, Santhosh
In this thesis we consider nonlinear ill-posed operator equations of the form F(x) = f; that arise from the study of nonlinear inverse problems, where F : X ! X is a nonlinear monotone operator defined on a real Hilbert space X: In applications, instead of f; usually only noisy data fδ are available. Then the problem of recovery of the exact solution ^ x from noisy equation F(x) = fδ is ill-posed, in the sense that a small perturbation in the data can cause large deviation in the solution. Thus the computation of a stable approximation for ^ x from the solution of F(x) = fδ; becomes an important issue in ill-posed problems, and the regularization techniques have to be taken into account. Approximation methods are an attractive choice since they are straightforward to implement, for getting the numerical solution of nonlinear ill-posed problems. Thus in the last few years more emphasis was put on the investigation of iterative regularization methods. We consider Newton type iterative regularization methods and their finite dimensional realizations, for obtaining approximation for ^ x in the Hilbert space and Hilbert scales settings. We use the adaptive scheme of Pereverzyev and Schock (2005), for choosing the regularization parameter.
On The Implementation of Regularization Methods for Nonlinear Ill-Posed Operator Equations
(National Institute of Technology Karnataka, Surathkal, 2016) V. S, Shubha; P, Jidesh; George, Santhosh
In this thesis we consider nonlinear ill-posed operator equations of the form F(x) = y; where F : X ! Y is a nonlinear operator between Hilbert spaces X and Y: Many problems from computational sciences and other disciplines can be brought to the form F(x) = y: In practical applications, usually noisy data yδ are available instead of y: The problem of recovery of the exact solution x^ from noisy equation F(x) = yδ is ill posed, in the sense that a small perturbation in the data can cause large deviation in the solution and the solutions of these equations are usually unknown in the closed form. Thus the computation of a stable approximation for x^ from the solution of F(x) = yδ; becomes an important issue in the ill-posed problems, and most methods for solving these equations are iterative. We consider iterative regularization methods and their finite dimensional realization, for obtaining an approximation for x^ in the Hilbert space. The choice of regularization parameter plays an important role in the convergence of regularization methods. We use the adaptive scheme of Pereverzev and Schock (2005), for choosing the regularization parameter. The error bounds obtained are of optimal order with respect to a general source condition.
Regularization Methods for Nonlinear Ill-Posed Hammerstein Type Operator Equations
(National Institute of Technology Karnataka, Surathkal, 2014) M. E, Shobha; George, Santhosh
This thesis is devoted for obtaining a stable approximate solution for nonlinear ill-posed Hammerstein type operator equations KF (x) = f. Here K : X → Y is a bounded linear operator, F : X → X is a non-linear operator, X and Y are Hilbert spaces. It is assumed throughout that the available data is fδ with kf − fδk ≤ δ. Many problems from computational sciences and other disciplines can be brought in a form similar to equation KF (x) = y using mathematical modelling (Engl et al. (1990), Scherzer, Engl and Anderssen (1993), Scherzer (1989)). The solutions of these equations can rarely be found in closed form. That is why most solution methods for these equations are iterative. The study about convergence matter of iterative procedures is usually based on two types: semi-local and local convergence analysis. The semi-local convergence matter is, based on the information around an initial point, to give conditions ensuring the convergence of the iterative procedure; while the local one is, based on the information around a solution, to find estimates of the radii of convergence balls. We aim at approximately solving the non-linear ill-posed Hammerstein type operator equations KF (x) = f using a combination of Tikhonov regularization with Newton-type Method in Hilbert spaces and in Hilbert Scales. Also we consider a combination of Tikhonov regularization with Dynamical System Method in Hilbert spaces. Precisely in the methods discussed in this thesis we considered two cases of the operator F : in the first case it is assumed that F ′(.)−1 exist (F ′(.) denotes the Fre´chet derivative of F ) and in the second case it is assumed that F ′(.)−1 does not exist but F is a monotone operator. The choice of regularization parameter plays an important role in the convergence of regularization method. We use the adaptive scheme suggested by Pereverzev and Schock (2005) for the selection of regularization parameter. The error bounds obtained are of optimal order with respect to a general source condition. Algorithms to implement the method is suggested and the computational results provided endorse the reliability and effectiveness of our methods.
Steepest Descent Type Methods for Nonlinear Ill-Posed Operator Equations
(National Institute of Technology Karnataka, Surathkal, 2018) M, Sabari; George, Santhosh
In this thesis, we consider steepest descent method and minimal error method for approximating a solution of the nonlinear ill-posed operator equation F(x) = y, where F : D(F) ⊆ X → Y is nonlinear Fr´echet differentiable operator between the Hilbert spaces X and Y. In practical application, we have only noisy data yδ with ∥y − yδ∥ ≤ δ. To our knowledge, convergence rate result for the steepest descent method and minimal error method with noisy data are not known. We provide error estimate for these methods with noisy data. We modified these methods with less computational cost. Error estimate for steepest descent method and minimal error method is not known under H¨older-type source condition. We provide an error estimate for these methods under H¨older-type source condition and also with noisy data. We also studied the regularized version of steepest descent method and regularization parameter in this regularized version is selected through the adaptive scheme of Pereverzev and Schock (2005).
Weighted Regularization Methods for Ill-Posed Problems
(National Institute of Technology Karnataka, Surathkal, 2020) Kanagaraj, K.; George, Santhosh
This thesis is devoted for obtaining a stable approximate solution for ill-posed operator equation F x = y: In the second Chapter we consider a non-linear illposed equation F x = y; where F is monotone operator defined on a Hilbert space. Our analysis in Chapter 2 is in the setting of a Hilbert scale. In the rest of the thesis, we studied weighted or fractional regularization method for linear ill-posed equation. Precisely, in Chapter 3 we studied fractional Tikhonov regularization method and in Chapters 4 and 5 we studied fractional Lavrentiv regularization method for the linear ill-posed equation A x = y; where A is a positive self-adjoint operator. Numerical examples are provided to show the reliability and effectiveness of our methods.