The tensor Golub–Kahan–Tikhonov method applied to the solution of ill‐posed problems with a t‐product structure

2021 ◽  
Author(s):  
Lothar Reichel ◽  
Ugochukwu O. Ugwu
Keyword(s):  
Product Structure ◽  
Tikhonov Method ◽  
Ill Posed

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.

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.

scholarly journals Identification of Source Term for the Time-Fractional Diffusion-Wave Equation by Fractional Tikhonov Method

Mathematics ◽  
2019 ◽  
Vol 7 (10) ◽  
pp. 934
Author(s):  
Le Dinh Long ◽  
Nguyen Hoang Luc ◽  
Yong Zhou ◽  
and Can Nguyen
Keyword(s):  
Source Term ◽  
A Priori ◽  
Fractional Diffusion ◽  
Behavior Changes ◽  
Diffusion Wave ◽  
Tikhonov Method ◽  
A Posteriori Parameter Choice ◽  
Ill Posed ◽  
Well Posed ◽  
Parameter Choice Rules

In this article, we consider an inverse problem to determine an unknown source term in a space-time-fractional diffusion equation. The inverse problems are often ill-posed. By an example, we show that this problem is NOT well-posed in the Hadamard sense, i.e., this problem does not satisfy the last condition-the solution’s behavior changes continuously with the input data. It leads to having a regularization model for this problem. We use the Tikhonov method to solve the problem. In the theoretical results, we also propose a priori and a posteriori parameter choice rules and analyze them.

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.

A mixed Newton-Tikhonov method for nonlinear ill-posed problems

2009 ◽  
Vol 30 (6) ◽  
pp. 741-752 ◽  
Author(s):  
Chuan-gang Kang ◽  
Guo-qiang He
Keyword(s):  
Tikhonov Method ◽  
Ill Posed

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.

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