Stability of a Regularized Newton Method with Two Potentials
Boushra Abbas,
Ramez Koudsieh
Issue:
Volume 7, Issue 1, February 2021
Pages:
1-11
Received:
14 February 2020
Accepted:
30 November 2020
Published:
22 January 2021
Abstract: In a Hilbert space setting, we introduce dynamical systems, which are linked to Newton and Levenberg-Marquardt methods. They are intended to solve, by splitting methods, inclusions governed by structured monotone operators M = A + B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. Based on the Minty representation of A as a Lipschitz manifold, we show that these dynamics can be formulated as differential systems, which are relevant to the Cauchy-Lipschitz theorem, and involve separately B and the resolvents of A. In the convex subdifferential case, by using Lyapunov asymptotic analysis, we prove a descent minimizing property, and weak convergence to equilibria of the trajectories. Time discretization of these dynamics gives algorithms combining Newton’s method and forward-backward methods for solving structured monotone inclusions. The Levenberg-Marquardt regularization term acts in an open loop way. As a byproduct of our study, we can take the regularization coefficient of bounded variation. These stability results are directly related to the study of numerical algorithms that combine forward-backward and Newton’s methods.
Abstract: In a Hilbert space setting, we introduce dynamical systems, which are linked to Newton and Levenberg-Marquardt methods. They are intended to solve, by splitting methods, inclusions governed by structured monotone operators M = A + B, where A is a general maximal monotone operator, and B is monotone and locally Lipschitz continuous. Based on the Min...
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Establishments of Empiric Intervening Formulas Methods, Procedures and Mathematical Elementary Algorithms: Medical Applications
Luis Alberto Escalona-Fernández
Issue:
Volume 7, Issue 1, February 2021
Pages:
12-16
Received:
11 December 2020
Accepted:
18 December 2020
Published:
9 February 2021
Abstract: Background: updating methods, procedures and mathematical elementary algorithms, constitute a tool of work for the reliability of the results of any study, and in medical applications. Aims: confirming the behavior of the experimental intervening data the modelation, through the linear regression equations, according to the methods, procedures and mathematical elementary algorithms, the ones that they are not necessary in knowledge and abilities of differential calculus. Method: they use the theoretic methods: Analysis synthesis, induction deduction and abstraction concretion. Process of understanding, explanation and interpretation. Methods, procedures and mathematical algorithms, as well as statistical methods and information-technology professional programs are applicable. Results: empiric formulas follow on from a mathematical model, which confirms the behavior of the experimental data by means of the simulation, a medical problem gets worked out, the correlation coefficient is checked for sampling, a confidence interval of the linear correlation coefficient estimates itself, helped in information-technology professional programs. Conclusions: they indicate methods, procedures and the mathematical elementary algorithms to construct a model, which confirm the behavior of the experimental data, and theoreticians for simulation, as from nature and the logical order of the study, helped for information-technology professional programs.
Abstract: Background: updating methods, procedures and mathematical elementary algorithms, constitute a tool of work for the reliability of the results of any study, and in medical applications. Aims: confirming the behavior of the experimental intervening data the modelation, through the linear regression equations, according to the methods, procedures and ...
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