scholarly journals ABOUT REGULARIZATION OF SEVERELY ILL-POSED PROBLEMS BY STANDARD TIKHONOV'S METHOD WITH THE BALANCING PRINCIPLE

2014 ◽  
Vol 19 (2) ◽  
pp. 199-215 ◽  
Author(s):  
Sergei G. Solodky ◽  
Ganna L. Myleiko
Keyword(s):  
Input Data ◽  
Regularization Parameter ◽  
Stable Solution ◽  
Order Of Accuracy ◽  
Tikhonov Method ◽  
Ill Posed ◽  
Balancing Principle

In the present paper for a stable solution of severely ill-posed problems with perturbed input data, the standard Tikhonov method is applied, and the regularization parameter is chosen according to balancing principle. We establish that the approach provides the order of accuracy on the class of problems under consideration.

scholarly journals COMPLEXITY ESTIMATES FOR SEVERELY ILL-POSED PROBLEMS UNDER A POSTERIORI SELECTION OF REGULARIZATION PARAMETER

2017 ◽  
Vol 22 (3) ◽  
pp. 283-299
Author(s):  
Sergii G. Solodky ◽  
Ganna L. Myleiko ◽  
Evgeniya V. Semenova
Keyword(s):  
Integral Equations ◽  
Regularization Parameter ◽  
Efficient Algorithms ◽  
Optimal Order ◽  
A Posteriori ◽  
Order Of Accuracy ◽  
Tikhonov Method ◽  
Discrete Information ◽  
Ill Posed ◽  
Selection Of

In the article the authors developed two efficient algorithms for solving severely ill-posed problems such as Fredholm’s integral equations. The standard Tikhonov method is applied as a regularization. To select a regularization parameter we employ two different a posteriori rules, namely, discrepancy and balancing principles. It is established that proposed strategies not only achieved optimal order of accuracy on the class of problems under consideration, but also they are economical in the sense of used discrete information.

scholarly journals Two-Step Newton-Tikhonov Method for Hammerstein-Type Equations: Finite-Dimensional Realization

2012 ◽  
Vol 2012 ◽  
pp. 1-22 ◽  
Author(s):  
Santhosh George ◽  
Monnanda Erappa Shobha
Keyword(s):  
Hilbert Space ◽  
Regularization Parameter ◽  
Real Hilbert Space ◽  
Initial Approximation ◽  
Numerical Illustration ◽  
Bounded Linear ◽  
Finite Dimensional ◽  
Tikhonov Method ◽  
Ill Posed ◽  
Balancing Principle

Finite-dimensional realization of a Two-Step Newton-Tikhonov method is considered for obtaining a stable approximate solution to nonlinear ill-posed Hammerstein-type operator equations KF(x)=f. Here F:D(F)⊆X→X is nonlinear monotone operator, K:X→Y is a bounded linear operator, X is a real Hilbert space, and Y is a Hilbert space. The error analysis for this method is done under two general source conditions, the first one involves the operator K and the second one involves the Fréchet derivative of F at an initial approximation x0 of the the solution x̂: balancing principle of Pereverzev and Schock (2005) is employed in choosing the regularization parameter and order optimal error bounds are established. Numerical illustration is given to confirm the reliability of our approach.

scholarly journals Determination of a space-dependent force function in the one-dimensional wave equation

2014 ◽  
Vol 12 (1) ◽  
Author(s):  
S. O. Hussein ◽  
D. Lesnic
Keyword(s):  
Input Data ◽  
Regularization Parameter ◽  
Stable Solution ◽  
Force Function ◽  
Tikhonov Regularization Method ◽  
Vibrating String ◽  
Ill Posed ◽  
Curve Method ◽  
The One

<p class="p1">The determination of an unknown spacewice dependent force function acting on a vibrating string from over-specied Cauchy boundary data is investigated numerically using the boundary element method (BEM) combined with a regularized method of separating variables. This linear inverse problem is ill-posed since small errors in the input data cause large errors in the output force solution. Consequently, when the input data is contaminated with noise we use the Tikhonov regularization method in order to obtain a stable solution. The choice of the regularization parameter is based on the L-curve method. Numerical results show that the solution is accurate for exact data and stable for noisy data.</p>

scholarly journals Solving ill-posed magnetic inverse problem using a Parameterized Trust-Region Sub-problem

2013 ◽  
Vol 43 (2) ◽  
pp. 99-123 ◽  
Author(s):  
Maha Mohamed Abdelazeem
Keyword(s):  
Weighting Function ◽  
Regularization Parameter ◽  
Stable Solution ◽  
Trust Region ◽  
Real Data ◽  
Linear Constraints ◽  
Singular Value Decomposition Method ◽  
Ill Posed ◽  
Value Decomposition ◽  
Natural Decay

Abstract The aim of this paper is to find a plausible and stable solution for the inverse geophysical magnetic problem. Most of the inverse problems in geophysics are considered as ill-posed ones. This is not necessarily due to complex geological situations, but it may arise because of ill-conditioned kernel matrix. To deal with such ill-conditioned matrix, one may truncate the most ill part as in truncated singular value decomposition method (TSVD). In such a method, the question will be where to truncate? In this paper, for comparison, we first try the adaptive pruning algorithm for the discrete L-curve criterion to estimate the regularization parameter for TSVD method. Linear constraints have been added to the ill-conditioned matrix. The same problem is then solved using a global optimizing and regularizing technique based on Parameterized Trust Region Sub-problem (PTRS). The criteria of such technique are to choose a trusted region of the solutions and then to find the satisfying minimum to the objective function. The ambiguity is controlled mainly by proper choosing the trust region. To overcome the natural decay in kernel with depth, a specific depth weighting function is used. A Matlab-based inversion code is implemented and tested on two synthetic total magnetic fields contaminated with different levels of noise to simulate natural fields. The results of PTRS are compared with those of TSVD with adaptive pruning L-curve. Such a comparison proves the high stability of the PTRS method in dealing with potential field problems. The capability of such technique has been further tested by applying it to real data from Saudi Arabia and Italy.

Lavrentiev Regularization and Balancing Principle for Solving Ill-Posed Problems with Monotone Operators

2010 ◽  
Vol 10 (4) ◽  
pp. 444-454 ◽  
Author(s):  
E.V. Semenova
Keyword(s):  
Regularization Parameter ◽  
Monotone Operators ◽  
Smooth Solutions ◽  
Lavrentiev Regularization ◽  
Ill Posed ◽  
Balancing Principle ◽  
Selection Of

AbstractThe paper considers a method for solving nonlinear ill-posed problems with monotone operators. The approach combines the Lavrentiev method, the fixedpoint method, and the balancing principle for selection of the regularization parameter. The method’s optimality has been proved for some set of smooth solutions. A test example proves the efficiency of the proposed method.

A Meshless Regularization Method for a Two-Dimensional Two-Phase Linear Inverse Stefan Problem

2013 ◽  
Vol 5 (06) ◽  
pp. 825-845 ◽  
Author(s):  
B. Tomas Johansson ◽  
Daniel Lesnic ◽  
Thomas Reeve
Keyword(s):  
Stefan Problem ◽  
Regularization Method ◽  
Input Data ◽  
Stable Solution ◽  
Higher Dimensions ◽  
Fundamental Solutions ◽  
Method Of Fundamental Solutions ◽  
Two Dimensional ◽  
Two Phase ◽  
Ill Posed

AbstractIn this paper, a meshless regularization method of fundamental solutions is proposed for a two-dimensional, two-phase linear inverse Stefan problem. The numerical implementation and analysis are challenging since one needs to handle composite materials in higher dimensions. Furthermore, the inverse Stefan problem is ill-posed since small errors in the input data cause large errors in the desired output solution. Therefore, regularization is necessary in order to obtain a stable solution. Numerical results for several benchmark test examples are presented and discussed.

scholarly journals NEW RULE FOR CHOICE OF THE REGULARIZATION PARAMETER IN (ITERATED) TIKHONOV METHOD

2009 ◽  
Vol 14 (2) ◽  
pp. 187-198 ◽  
Author(s):  
Toomas Raus ◽  
Uno Hämarik
Keyword(s):  
Hilbert Spaces ◽  
Noise Level ◽  
Regularization Parameter ◽  
Optimal Error Estimates ◽  
A Posteriori ◽  
Optimal Error ◽  
Numerical Examples ◽  
Tikhonov Method ◽  
Mild Assumption ◽  
Ill Posed

We propose a new a posteriori rule for choosing the regularization parameter α in (iterated) Tikhonov method for solving linear ill‐posed problems in Hilbert spaces. We assume that data are noisy but noise level δ is given. We prove that (iterated) Tikhonov approximation with proposed choice of α converges to the solution as δ → 0 and has order optimal error estimates. Under certain mild assumption the quasioptimality of proposed rule is also proved. Numerical examples show the advantage of the new rule over the monotone error rule, especially in case of rough δ.

scholarly journals Inverse Free Iterative Methods for Nonlinear Ill-Posed Operator Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8 ◽  
Author(s):  
Ioannis K. Argyros ◽  
Santhosh George ◽  
P. Jidesh
Keyword(s):  
Approximate Solution ◽  
Regularization Parameter ◽  
Optimal Order ◽  
Operator Equations ◽  
Source Condition ◽  
Numerical Examples ◽  
Ill Posed ◽  
Balancing Principle ◽  
General Source ◽  
New Iterative Method

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 equationF(x)=y. 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.

scholarly journals A modified Tikhonov regularization method based on Hermite expansion for solving the Cauchy problem of the Laplace equation

2020 ◽  
Vol 18 (1) ◽  
pp. 1685-1697
Author(s):  
Zhenyu Zhao ◽  
Lei You ◽  
Zehong Meng
Keyword(s):  
Cauchy Problem ◽  
Laplace Equation ◽  
Tikhonov Regularization ◽  
Regularization Method ◽  
Regularization Parameter ◽  
Solution Process ◽  
Tikhonov Regularization Method ◽  
Hermite Expansion ◽  
The Cauchy Problem ◽  
Ill Posed

Abstract In this paper, a Cauchy problem for the Laplace equation is considered. We develop a modified Tikhonov regularization method based on Hermite expansion to deal with the ill posed-ness of the problem. The regularization parameter is determined by a discrepancy principle. For various smoothness conditions, the solution process of the method is uniform and the convergence rate can be obtained self-adaptively. Numerical tests are also carried out to verify the effectiveness of the method.

scholarly journals Regularization of backward time-fractional parabolic equations by Sobolev-type equations

2020 ◽  
Vol 28 (5) ◽  
pp. 659-676
Author(s):  
Dinh Nho Hào ◽  
Nguyen Van Duc ◽  
Nguyen Van Thang ◽  
Nguyen Trung Thành
Keyword(s):  
Error Estimates ◽  
Parabolic Equations ◽  
Regularization Parameter ◽  
A Priori ◽  
Sobolev Type ◽  
A Posteriori ◽  
Parameter Choice ◽  
Numerical Tests ◽  
Ill Posed ◽  
Parameter Choice Rules

AbstractThe problem of determining the initial condition from noisy final observations in time-fractional parabolic equations is considered. This problem is well known to be ill-posed, and it is regularized by backward Sobolev-type equations. Error estimates of Hölder type are obtained with a priori and a posteriori regularization parameter choice rules. The proposed regularization method results in a stable noniterative numerical scheme. The theoretical error estimates are confirmed by numerical tests for one- and two-dimensional equations.

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