The a posteriori Fourier method for solving ill-posed problems

2012 ◽  
Vol 28 (9) ◽  
pp. 095002 ◽  
Author(s):  
Chu-Li Fu ◽  
Yuan-Xiang Zhang ◽  
Hao Cheng ◽  
Yun-Jie Ma
Keyword(s):  
Fourier Method ◽  
A Posteriori ◽  
Ill Posed

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.

scholarly journals Optimal order yielding discrepancy principle for simplified regularization in Hilbert scales: finite-dimensional realizations

2004 ◽  
Vol 2004 (37) ◽  
pp. 1973-1996 ◽  
Author(s):  
Santhosh George ◽  
M. Thamban Nair
Keyword(s):  
Wide Class ◽  
Noise Level ◽  
A Priori ◽  
Approximate Solutions ◽  
Optimal Order ◽  
Discrepancy Principle ◽  
A Posteriori ◽  
Operator Equations ◽  
Finite Dimensional ◽  
Ill Posed

Simplified regularization using finite-dimensional approximations in the setting of Hilbert scales has been considered for obtaining stable approximate solutions to ill-posed operator equations. The derived error estimates using an a priori and a posteriori choice of parameters in relation to the noise level are shown to be of optimal order with respect to certain natural assumptions on the ill posedness of the equation. The results are shown to be applicable to a wide class of spline approximations in the setting of Sobolev scales.

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 A LEPSKIJ-TYPE STOPPING-RULE FOR SIMPLIFIED ITERATIVELY REGULARIZED GAUSS-NEWTON METHOD

2017 ◽  
Author(s):  
Agah D. Garnadi
Keyword(s):  
Newton Method ◽  
Hilbert Spaces ◽  
Potential Problem ◽  
Initial Approximation ◽  
Stopping Rule ◽  
A Posteriori ◽  
Iterative Regularization ◽  
Ill Posed ◽  
Derivatives Of ◽  
Unknown Solution

Iterative regularization methods for nonlinear ill-posed equations of the form $ F(a)= y$, where $ F: D(F) \subset X \to Y$ is an operator between Hilbert spaces $ X $ and $ Y$, usually involve calculation of the Fr\'{e}chet derivatives of $ F$ at each iterate and at the unknown solution $ a^\sharp$. A modified form of the generalized Gauss-Newton method which requires the Fr\'{e}chet derivative of $F$ only at an initial approximation $ a_0$ of the solution $ a^\sharp$ as studied by Mahale and Nair \cite{MaNa:2k9}. This work studied an {\it a posteriori} stopping rule of Lepskij-type of the method. A numerical experiment from inverse source potential problem is demonstrated.

Sign in / Sign up

Export Citation Format

Share Document