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
Choosing Regularization Parameters in Iterative Methods for Ill-Posed Problems
SIAM Journal on Matrix Analysis and Applications
Introduction to Numerical Continuation Methods
Introduction to Numerical Continuation Methods
Control Perspectives on Numerical Algorithms And Matrix Problems (Advances in Design and Control) (Advances in Design and Control 10)
Regularization Parameter Selection in Discrete Ill-Posed Problems-The Use of the U-Curve
International Journal of Applied Mathematics and Computer Science
Original Article: Comparingparameter choice methods for regularization of ill-posed problems
Mathematics and Computers in Simulation
The Chaotic Nature of Faster Gradient Descent Methods
Journal of Scientific Computing
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This paper revisits a class of recently proposed so-called invariant manifold methods for zero finding of ill-posed problems, showing that they can be profitably viewed as homotopy methods, in which the homotopy parameter is interpreted as a learning parameter. Moreover, it is shown that the choice of this learning parameter can be made in a natural manner from a control Liapunov function approach CLF. From this viewpoint, maintaining manifold invariance is equivalent to ensuring that the CLF satisfies a certain ordinary differential equation, involving the learning parameter, that allows an estimate of rate of convergence. In order to illustrate this approach, algorithms recently proposed using the invariant manifold approach, are rederived, via CLFs, in a unified manner. Adaptive regularization parameters for solving linear algebraic ill-posed problems were also proposed. This paper also shows that the discretizations of the ODEs to solve the zero finding problem, as well as the different adaptive choices of the regularization parameter, yield iterative methods for linear systems, which are also derived using the Liapunov optimizing control LOC method.