Journal of Approximation Theory
Journal of Optimization Theory and Applications
The use of the L-curve in the regularization of discrete ill-posed problems
SIAM Journal on Scientific Computing
Gradient Method with Retards and Generalizations
SIAM Journal on Numerical Analysis
Modified Two-Point Stepsize Gradient Methods for Unconstrained Optimization
Computational Optimization and Applications
Choosing Regularization Parameters in Iterative Methods for Ill-Posed Problems
SIAM Journal on Matrix Analysis and Applications
The Barzilai and Borwein Gradient Method for the Large Scale Unconstrained Minimization Problem
SIAM Journal on Optimization
Relaxed Steepest Descent and Cauchy-Barzilai-Borwein Method
Computational Optimization and Applications
Control Perspectives on Numerical Algorithms And Matrix Problems (Advances in Design and Control) (Advances in Design and Control 10)
New model function methods for determining regularization parameters in linear inverse problems
Applied Numerical Mathematics
An efficient dynamical systems method for solving singularly perturbed integral equations with noise
Computers & Mathematics with Applications
A doubly optimized solution of linear equations system expressed in an affine Krylov subspace
Journal of Computational and Applied Mathematics
A globally optimal tri-vector method to solve an ill-posed linear system
Journal of Computational and Applied Mathematics
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The Tikhonov method is a famous technique for regularizing ill-posed linear problems, wherein a regularization parameter needs to be determined. This article, based on an invariant-manifold method, presents an adaptive Tikhonov method to solve ill-posed linear algebraic problems. The new method consists in building a numerical minimizing vector sequence that remains on an invariant manifold, and then the Tikhonov parameter can be optimally computed at each iteration by minimizing a proper merit function. In the optimal vector method (OVM) three concepts of optimal vector, slow manifold and Hopf bifurcation are introduced. Numerical illustrations on well known ill-posed linear problems point out the computational efficiency and accuracy of the present OVM as compared with classical ones.