Spherical Continuation Algorithm with Spheres of Variable Radius to Trace Homotopy Curves

Torres-Muñoz, Delia; Hernandez-Martinez, L.; Vázquez-Leal, H.

doi:10.1007/s40819-015-0067-1

Cited by 9 publications

(7 citation statements)

References 26 publications

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“…ence, the proposed rational expression degree [enumer- [3,4] (i) Multiple solutions are obtained from system (9) resulting that, using Newton-Raphson, we can find global or local minimums. is is an open issue for CCLM that could be solved using homotopy continuation methods [28,29,39] due to their capability of finding multiple solutions. (ii) e nature of rational approximations (as (10)) does not produce a uniform convergence as the approximation order increases, for instance, Pade pproximations.…”

Section: Numerical Simulation and Discussionmentioning

confidence: 99%

“…Furthermore, we can expect that, as the CC point x 1 gets closer to zero, y ′ (0) will improve its accuracy; nevertheless, (9) becomes hard to solve using NR. Hence, it requires further research to explore other numerical tools, like homotopy continuation methods (HCM) [28][29][30], to take full advantage of the CCLM capabilities. An interesting benefit of HCM is the possibility to find multiple solutions, opening a future research for selecting the most suitable solution.…”

Section: E Omas-fermi Singular Equation For the Neutralmentioning

confidence: 99%

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Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

Vázquez-Leal

2020

Discrete Dynamics in Nature and Society

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This work presents the novel continuum-cancellation Leal-method (CCLM) for the approximation of nonlinear differential equations. CCLM obtains accurate approximate analytical solutions resorting to a process that involves the continuum cancellation (CC) of the residual error of multiple selected points; such CC process occurs during the successive derivatives of the differential equation resulting in an accuracy increase of the inner region of the CC-points and, thus, extends the domain of convergence and accuracy. Users of CCLM can propose their own trial functions to construct the approximation as long as they are continuous in the CC-points, that is, it can be polynomials, exponentials, and rational polynomials, among others. In addition, we show how the process to obtain the approximations is straightforward and simple to achieve and capable to generate compact, and easy, computable expressions. A convergence control is proposed with the aim to establish a solid scheme to obtain optimal CCLM approximations. Furthermore, we present the application of CCLM in several examples: Thomas–Fermi singular equation for the neutral atom, magnetohydrodynamic flow of blood in a porous channel singular boundary-valued problem, and a system of initial condition differential equations to model the dynamics of cocaine consumption in Spain. We present a computational convergence study for the proposed approximations resulting in a tendency of the RMS error to zero as the approximation order increases for all case studies. In addition, a computation time analysis (using Fortran) for the proposed approximations presents average times from 3.5 nanoseconds to 7 nanoseconds for all the case studies. Thence, CCML approximations can be used for intensive computing simulations.

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Section: Numerical Simulation and Discussionmentioning

confidence: 99%

Section: E Omas-fermi Singular Equation For the Neutralmentioning

confidence: 99%

Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

Vázquez-Leal

2020

Discrete Dynamics in Nature and Society

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“…An appropriate path tracking procedure that supports HPPM should be considered. One of them is the Hyperspherical Algorithm [34][35][36] which, as its name implies, is based on the generation of an n-dimension sphere with radius r and its center lies on the homotopic curve in O i . The contour of this sphere touches at least two points of the same curve, O i+1 and O i−1 , as can be seen in Figure 2.…”

Section: Homotopy Continuation Methods As a Collision-free Path Plann...mentioning

confidence: 99%

FPGA Implementation of Homotopic Path Planning Method with Automatic Assignment of Repulsion Parameter

et al. 2020

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In recent times, autonomous robots have become more relevant, aiming not only to be an extension of mobility and human performance but also allowing them to independently solve specific problems such as finding free-collision paths within some defined environments. In order to achieve this, several techniques have been developed, like action-reaction algorithms, sampling-based algorithms, and deterministic algorithms such as the Homotopy Path Planning Method (HPPM). This work presents, for the first time, a complete deterministic collision-free path planning scheme implemented in FPGA, which is mounted on a Scribbler 2 robot from Parallax. Then, an automatic algorithm of the repulsion parameter for the HPPM method is presented, using as a reference the minimum distance between the center of each obstacle with respect to the homotopic ideal path; furthermore, an algorithm is proposed for discriminating dead-end routes and collision risk trajectories, which allows us to obtain a feasible free-collision path that takes into account the robot dimensions. Besides, comparative performance tests have been carried out against other path-finding methods from the low degrees of freedom (low DoF) and sampling-based planners. Our proposal exhibits path calculation times which are 5 to 10 times faster on FPGA implementation, compared to the other methods and 10 to 100 times faster on PC implementation also compared to the rest. Similar results are obtained with regards to memory consumption, namely 20 to 200 times lower on FPGA implementation and 10 to 100 times lower on PC implementation.

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“…The homotopy continuation method (HCM) is a technique recommended to solve nonlinear equation systems with multiple solutions [29,30]. Fig.…”

Section: Homotopy Continuation Methods Generalitiesmentioning

confidence: 99%

Exploring a homotopy approach for the design of nanometer digital circuits tolerant to process variations

Rodríguez

Garcia-Gervacio

Leal

2018

IEICE Electron. Express

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It is known that process parameter variation degrades the performance of nanometer integrated circuits. Process variations reduce the maximum clock frequency operation of the chips. Diverse strategies have been proposed in the literature to overcome this issue, especially optimization algorithms (gate-sizing algorithms). However, convergence related problems have limited their use. In this work, a homotopy approach for the design of nanometer digital circuits tolerant to process variations is proposed. Two optimization strategies are developed in this work, the first based on the homotopy continuation method (HCM) and the second based on a modification of HCM, called in this work reboot homotopy continuation method (RHCM). Three logic paths were implemented to validate the algorithms. The optimization results obtained with the proposed strategies are compared with a Lagrange-Multipliers-based framework. Results obtained from HCM method are equivalent to the obtained with Lagrange Multipliers. On the other side, results obtained from the RHCM method are more accurate than the obtained with HCM and Lagrange Multipliers. Furthermore, the area used to implement the logic paths is lower when RHCM is applied. Moreover, the number of Newton-Rhapson iterations required to find the solutions are lower when RHCM is used; consequently, time computing is also lower.

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Spherical Continuation Algorithm with Spheres of Variable Radius to Trace Homotopy Curves

Cited by 9 publications

References 26 publications

Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

Exploring the Novel Continuum-Cancellation Leal-Method for the Approximate Solution of Nonlinear Differential Equations

FPGA Implementation of Homotopic Path Planning Method with Automatic Assignment of Repulsion Parameter

Exploring a homotopy approach for the design of nanometer digital circuits tolerant to process variations

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