Affiliations: Department of Electrical Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil
Correspondence:
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Corresponding author: Fernando Pazos, Department of Electrical Engineering, Federal University of Rio de Janeiro, Rio de Janeiro, Brazil. E-mail: quini@ort.org.br
Note: [1] This work was supported by grants PNPD/CNPq for the first author and grants BPP/CNPq and CNE/FAPERJ for the second author.
Abstract: 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.
Keywords: Liapunov Function Approach, Liapunov Optimizing Control, Zero finding
