scholarly journals Error estimates for the iteratively regularized Newton–Landweber method in Banach spaces under approximate source conditions

2020 ◽  
Vol 0 (0) ◽  
Author(s):  
Pallavi Mahale ◽  
Sharad Kumar Dixit
Keyword(s):  
Identification Problem ◽  
Convergence Result ◽  
Stopping Rules ◽  
Duality Mapping ◽  
Optimal Error Estimate ◽  
Source Condition ◽  
Parameter Identification Problem ◽  
Ill Posed ◽  
Landweber Method ◽  
Inner Iteration

AbstractIn 2012, Jin Qinian considered an inexact Newton–Landweber iterative method for solving nonlinear ill-posed operator equations in the Banach space setting by making use of duality mapping. The method consists of two steps; the first one is an inner iteration which gives increments by using Landweber iteration, and the second one is an outer iteration which provides increments by using Newton iteration. He has proved a convergence result for the exact data case, and for the perturbed data case, a weak convergence result has been obtained under a Morozov type stopping rule. However, no error bound has been given. In 2013, Kaltenbacher and Tomba have considered the modified version of the Newton–Landweber iterations, in which the combination of the outer Newton loop with an iteratively regularized Landweber iteration has been used. The convergence rate result has been obtained under a Hölder type source condition. In this paper, we study the modified version of inexact Newton–Landweber iteration under the approximate source condition and will obtain an order-optimal error estimate under a suitable choice of stopping rules for the inner and outer iterations. We will also show that the results proved in this paper are more general as compared to the results proved by Kaltenbacher and Tomba in 2013. Also, we will give a numerical example of a parameter identification problem to support our method.

Simplified Iteratively Regularized Gauss–Newton Method in Banach Spaces Under a General Source Condition

2020 ◽  
Vol 20 (2) ◽  
pp. 321-341
Author(s):  
Pallavi Mahale ◽  
Sharad Kumar Dixit
Keyword(s):  
Newton Method ◽  
Regularization Parameter ◽  
A Priori ◽  
Identification Problem ◽  
Stopping Rule ◽  
Optimal Error Estimates ◽  
Source Condition ◽  
Parameter Identification Problem ◽  
Ill Posed ◽  
General Source

AbstractIn this paper, we consider a simplified iteratively regularized Gauss–Newton method in a Banach space setting under a general source condition. We will obtain order-optimal error estimates both for an a priori stopping rule and for a Morozov-type stopping rule together with a posteriori choice of the regularization parameter. An advantage of a general source condition is that it provides a unified setting for the error analysis which can be applied to the cases of both severely and mildly ill-posed problems. We will give a numerical example of a parameter identification problem to discuss the performance of the method.

scholarly journals Convergence Rate of the Modified Landweber Method for Solving Inverse Potential Problems

Mathematics ◽  
2020 ◽  
Vol 8 (4) ◽  
pp. 608
Author(s):  
Pornsarp Pornsawad ◽  
Parada Sungcharoen ◽  
Christine Böckmann
Keyword(s):  
Convergence Rate ◽  
Potential Problem ◽  
Regularization Parameter ◽  
Discrepancy Principle ◽  
Source Condition ◽  
Rate Analysis ◽  
Potential Problems ◽  
Ill Posed ◽  
Landweber Method ◽  
Convergence Rate Analysis

In this paper, we present the convergence rate analysis of the modified Landweber method under logarithmic source condition for nonlinear ill-posed problems. The regularization parameter is chosen according to the discrepancy principle. The reconstructions of the shape of an unknown domain for an inverse potential problem by using the modified Landweber method are exhibited.

scholarly journals Range-relaxed criteria for choosing the Lagrange multipliers in nonstationary iterated Tikhonov method

2018 ◽  
Vol 40 (1) ◽  
pp. 606-627 ◽  
Author(s):  
R Boiger ◽  
A Leitão ◽  
B F Svaiter
Keyword(s):  
Lagrange Multipliers ◽  
Approximate Solutions ◽  
Identification Problem ◽  
Linear Operators ◽  
Type Method ◽  
Geometrical Properties ◽  
Regularization Parameters ◽  
New Strategy ◽  
Parameter Identification Problem ◽  
Ill Posed

Abstract In this article we propose a novel nonstationary iterated Tikhonov (NIT)-type method for obtaining stable approximate solutions to ill-posed operator equations modeled by linear operators acting between Hilbert spaces. Geometrical properties of the problem are used to derive a new strategy for choosing the sequence of regularization parameters (Lagrange multipliers) for the NIT iteration. Convergence analysis for this new method is provided. Numerical experiments are presented for two distinct applications: (I) a two-dimensional elliptic parameter identification problem (inverse potential problem); and (II) an image-deblurring problem. The results obtained validate the efficiency of our method compared with standard implementations of the NIT method (where a geometrical choice is typically used for the sequence of Lagrange multipliers).

Optimal convergence rates for inexact Newton regularization with CG as inner iteration

2020 ◽  
Vol 28 (1) ◽  
pp. 145-153 ◽  
Author(s):  
Andreas Neubauer
Keyword(s):  
Parameter Identification ◽  
Conjugate Gradient ◽  
Gradient Method ◽  
Regularization Method ◽  
Convergence Rates ◽  
Identification Problem ◽  
Discrepancy Principle ◽  
Parameter Identification Problem ◽  
Optimal Convergence Rates ◽  
Inner Iteration

AbstractIn this paper we prove order optimality of an inexact Newton regularization method, where the linearized equations are solved approximately using the conjugate gradient method. The outer and inner iterations are stopped via the discrepancy principle. We show that the conditions needed for convergence rates are satisfied for a certain parameter identification problem.

Error estimates for the simplified iteratively regularized Gauss–Newton method in Banach spaces under a Morozov-type stopping rule

2018 ◽  
Vol 26 (3) ◽  
pp. 311-333 ◽  
Author(s):  
Pallavi Mahale ◽  
Sharad Kumar Dixit
Keyword(s):  
Banach Spaces ◽  
Error Estimates ◽  
Newton Method ◽  
A Priori ◽  
Identification Problem ◽  
Stopping Rule ◽  
Optimal Error Estimates ◽  
Optimal Error ◽  
Parameter Identification Problem ◽  
Ill Posed

AbstractJin Qinian and Min Zhong [10] considered an iteratively regularized Gauss–Newton method in Banach spaces to find a stable approximate solution of the nonlinear ill-posed operator equation. They have considered a Morozov-type stopping rule (Rule 1) as one of the criterion to stop the iterations and studied the convergence analysis of the method. However, no error estimates have been obtained for this case. In this paper, we consider a modified variant of the method, namely, the simplified Gauss–Newton method under both an a priori as well as a Morozov-type stopping rule. In both cases, we obtain order optimal error estimates under Hölder-type approximate source conditions. An example of a parameter identification problem for which the method can be implemented is discussed in the paper.

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

2019 ◽  
Vol 19 (4) ◽  
pp. 765-778
Author(s):  
Santhosh George ◽  
K. Kanagaraj
Keyword(s):  
Regularization Parameter ◽  
Adjoint Operator ◽  
Identification Problem ◽  
Optimal Order ◽  
Order Error ◽  
Adaptive Parameter ◽  
Derivative Free ◽  
Parameter Identification Problem ◽  
Ill Posed ◽  
Parameter Choice Strategy

AbstractIn 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.

Computational Removing of Electrical Resistance Measurement Error Caused by Thermic Effects

1995 ◽  
Author(s):  
Nicolae A. Damean
Keyword(s):  
Measurement Error ◽  
Electrical Resistance ◽  
Identification Problem ◽  
Resistance Measurement ◽  
Resistance Temperature Detector ◽  
Temperature Detector ◽  
Parameter Identification Problem ◽  
Ill Posed ◽  
Principal Subject ◽  
Well Posed

Abstract Thermic behaviour of electrical resistor passed by a constant DC is analysed both theoretical and numerical. This problem, a ill-posed one is changed in a well-posed one solving a parameter identification problem. The last problem is the principal subject of this paper. Numerical simulations presented here correspond to resistance temperature detector Pt-100. They lead to improving of resistance measurement error (and of temperature certainly for this particular case) through computational removing of error caused by thermic effects.

Simplified Iterated Lavrentiev Regularization for Nonlinear Ill-Posed Monotone Operator Equations

2017 ◽  
Vol 17 (2) ◽  
pp. 269-285 ◽  
Author(s):  
Pallavi Mahale
Keyword(s):  
Exact Solution ◽  
Monotone Operator ◽  
Initial Approximation ◽  
Fréchet Derivative ◽  
Optimal Error Estimate ◽  
Source Condition ◽  
Frechet Derivative ◽  
Lavrentiev Regularization ◽  
Ill Posed ◽  
General Source

AbstractMahale and Nair [12] considered an iterated form of Lavrentiev regularization for obtaining stable approximate solutions for ill-posed nonlinear equations of the form ${F(x)=y}$, where ${F:D(F)\subseteq X\to X}$ is a nonlinear monotone operator and X is a Hilbert space. They considered an a posteriori strategy to find a stopping index which not only led to the convergence of the method, but also gave an order optimal error estimate under a general source condition. However, the iterations defined in [12] require calculation of Fréchet derivatives at each iteration. In this paper, we consider a simplified version of the iterated Lavrentiev regularization which will involve calculation of the Fréchet derivative only at the point ${x_{0}}$, i.e., at the initial approximation of the exact solution ${x^{\dagger}}$. Moreover, the general source condition and stopping rule which we use in this paper involve calculation of the Fréchet derivative at the point ${x_{0}}$, instead at the unknown exact solution ${x^{\dagger}}$ as in [12].

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