On the choice of Lagrange multipliers in the iterated Tikhonov method for linear ill-posed equations in Banach spaces

2019 ◽  
Vol 28 (6) ◽  
pp. 796-826 ◽  
Author(s):  
M. P. Machado ◽  
F. Margotti ◽  
A. Leitão
Keyword(s):  
Banach Spaces ◽  
Lagrange Multipliers ◽  
Tikhonov Method ◽  
Ill Posed

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).

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 Range-relaxed criteria for choosing the Lagrange multipliers in the Levenberg–Marquardt method

2020 ◽  
Author(s):  
A Leitão ◽  
F Margotti ◽  
B F Svaiter
Keyword(s):  
Hilbert Spaces ◽  
Lagrange Multipliers ◽  
A Priori ◽  
Impedance Tomography ◽  
Nonlinear Operators ◽  
Marquardt Method ◽  
Exact Data ◽  
Levenberg Marquardt ◽  
Ill Posed ◽  
Novel Strategy

Abstract In this article we propose a novel strategy for choosing the Lagrange multipliers in the Levenberg–Marquardt method for solving ill-posed problems modeled by nonlinear operators acting between Hilbert spaces. Convergence analysis results are established for the proposed method, including monotonicity of iteration error, geometrical decay of the residual, convergence for exact data, stability and semi-convergence for noisy data. Numerical experiments are presented for an elliptic parameter identification two-dimensional electrical impedance tomography problem. The performance of our strategy is compared with standard implementations of the Levenberg–Marquardt method (using a priori choice of the multipliers).

Levenberg–Marquardt method for ill-posed inverse problems with possibly non-smooth forward mappings between Banach spaces

2021 ◽  
Author(s):  
Huu Nhu Vu
Keyword(s):  
Inverse Problems ◽  
Heat Source ◽  
Banach Spaces ◽  
Elliptic Boundary ◽  
Uniformly Convex ◽  
Bounded Operators ◽  
Marquardt Method ◽  
Levenberg Marquardt ◽  
Ill Posed ◽  
Forward Mapping

Abstract In this paper, we consider a Levenberg–Marquardt method with general regularization terms that are uniformly convex on bounded sets to solve the ill-posed inverse problems in Banach spaces, where the forward mapping might not Gˆateaux differentiable and the image space is unnecessarily reflexive. The method therefore extends the one proposed by Jin and Yang in (Numer. Math. 133:655–684, 2016) for smooth inverse problem setting with globally uniformly convex regularization terms. We prove a novel convergence analysis of the proposed method under some standing assumptions, in particular, the generalized tangential cone condition and a compactness assumption. All these assumptions are fulfilled when investigating the identification of the heat source for semilinear elliptic boundary-value problems with a Robin boundary condition, a heat source acting on the boundary, and a possibly non-smooth nonlinearity. Therein, the Clarke subdifferential of the non-smooth nonlinearity is employed to construct the family of bounded operators that is a replacement for the nonexisting Gˆateaux derivative of the forward mapping. The efficiency of the proposed method is illustrated with a numerical example.

scholarly journals The generalized condition numbers of bounded linear operators in Banach spaces

2004 ◽  
Vol 76 (2) ◽  
pp. 281-290 ◽  
Author(s):  
Guoliang Chen ◽  
Yimin Wei ◽  
Yifeng Xue
Keyword(s):  
Linear Operator ◽  
Banach Spaces ◽  
Perturbation Analysis ◽  
Bounded Linear Operator ◽  
Linear Operators ◽  
Condition Numbers ◽  
Bounded Linear ◽  
Infinite Dimensional ◽  
Ill Posed ◽  
Linear Operator Equations

AbstractFor any bounded linear operator A in a Banach space, two generalized condition numbers, k(A) and k(A) are defined in this paper. These condition numbers may be applied to the perturbation analysis for the solution of ill-posed differential equations and bounded linear operator equations in infinite dimensional Banach spaces. Different expressions for the two generalized condition numbers are discussed in this paper and applied to the perturbation analysis of the operator equation.

An Inverse Diffraction Problem: Shape Reconstruction

2015 ◽  
Vol 5 (4) ◽  
pp. 342-360 ◽  
Author(s):  
Yanfeng Kong ◽  
Zhenping Li ◽  
Xiangtuan Xiong
Keyword(s):  
Error Estimates ◽  
Diffraction Problem ◽  
Shape Reconstruction ◽  
Numerical Examples ◽  
Tikhonov Method ◽  
Ill Posed ◽  
Tikhonov Regularisation ◽  
Slow Evolution ◽  
Inverse Diffraction

AbstractAn inverse diffraction problem is considered. Both classical Tikhonov regularisation and a slow-evolution-from-the-continuation-boundary (SECB) method are used to solve the ill-posed problem. Regularisation error estimates for the two methods are compared, and the SECB method is seen to be an improvement on the classical Tikhonov method. Two numerical examples demonstrate their feasibility and efficiency.

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