A Simplified Gauss–Newton Iterative Scheme with an a Posteriori Parameter Choice Rule for Solving Nonlinear Ill-Posed Problems

2015 ◽  
Vol 2 (1) ◽  
pp. 97-112 ◽  
Author(s):  
D. Pradeep ◽  
M. P. Rajan
Keyword(s):  
Iterative Scheme ◽  
Choice Rule ◽  
A Posteriori ◽  
Parameter Choice ◽  
A Posteriori Parameter Choice ◽  
Ill Posed

An a posteriori truncation regularization method for an ill-posed problem for the heat equation with an integral boundary condition

2019 ◽  
Vol 26 (1) ◽  
pp. 35-45
Author(s):  
Mohamed Denche ◽  
Abdelali Benchikha
Keyword(s):  
Boundary Condition ◽  
Heat Equation ◽  
Regularization Method ◽  
Initial Conditions ◽  
Integral Boundary Condition ◽  
A Posteriori ◽  
Parameter Choice ◽  
Integral Boundary ◽  
A Posteriori Parameter Choice ◽  
Ill Posed

Abstract The aim of this paper is to investigate the problem of control by the initial conditions of the heat equation with an integral boundary condition. Using the truncation method with an a posteriori parameter choice rule, we give the error estimate between the exact and the regularized solutions. A numerical implementation shows the efficiency of the proposed method.

Pointwise Computation in an Ill-Posed Spherical Pseudo-Differential Equation

2015 ◽  
Vol 15 (2) ◽  
pp. 213-219 ◽  
Author(s):  
Sergei V. Pereverzyev ◽  
Pavlo Tkachenko
Keyword(s):  
Differential Equation ◽  
Least Squares ◽  
Hilbert Spaces ◽  
Least Squares Method ◽  
A Posteriori ◽  
Parameter Choice ◽  
Pseudo Differential Equation ◽  
A Posteriori Parameter Choice ◽  
Ill Posed ◽  
Numer Anal

AbstractIn the present paper, we consider the approximation of the solution of an ill-posed spherical pseudo-differential equation at a given point. While the methods for approximating the whole solution are well-studied in Hilbert spaces, such as the space of square-summable functions, the computation of values of the solution at given points is much less studied. This can be explained, in particular, by the fact that for square-summable functions the functional of pointwise evaluation is, in general, not well defined. To overcome this limitation we adjust the regularized least-squares method of An, Chen, Sloan and Womersley [Siam J. Numer. Anal. 50 (2012), no. 3, 1513–1534] by using a special a posteriori parameter choice rule. We also illustrate our theoretical findings by numerical results for the reconstruction of the solution at a given point.

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