AUTHORS: J. Arroyo, O. J. Garay, J. Mencia
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ABSTRACT: Different gradient descent methods have been introduced in [1] to study a quite general family of variational problems under affine and isoperimetric constraints. Thus, in [1] gradient descent sequences are created by using linear, quadratic or cubic approximations to the gradient descent trajectories, and methods are numerically implemented in a computational platform (which we call XEL-platform). In this work, performance of the above quadratic gradient descent versions are analyzed under the influence of transversality constraints. We see that, within this context, transversality conditions can be dealt as isoperimetric constraints and, then, the XEL-platform can be used to localize minimizers in the spaces of curves determined by the prescribed constraints. Efficiency of the approach is analyzed by considering two very well known classical problems, the brachystochrone and closed planar elastica. In the first case, the effect of introducing an isoperimetric constraint is also considered and the estimated errors are shown to be numerically insignificant. In the second case, minimizers are well known (circles and eight-figure curves) and we see how the XEL-platform takes very distant curves (from the energy point of view) within the same homotopy class to the minimizer included in that class. It is also capable to detect local minimizers which may appear during the gradient descent from the initial curve towards the minimum.
KEYWORDS: Gradient descent, energy minimizers, isoperimetric conditions, tranversality
REFERENCES:
[1] J. Arroyo, O.J. Garay, J.J. Menc´ıa and A. Pampano, A gradient-descent method for La- ´ grangian densities depending on multiple derivatives, Preprint, 2018.
[2] J. Arroyo, O.J. Garay and A. Pampano, Bound- ´ ary value problems for Euler-Bernoulli planar elastica. A solution construction procedure, Submitted.
[3] H. Brezis, ´ Analyse fonctionnelle, Mason Editeur, Paris, 1993.
[4] W. Dunham, Journey Through Genius, Penguin Books, New York, 1991.
[5] L. Euler, Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes, sive solutio problematis isoperimetrici lattisimo sensu accepti, Bousquet, Lausannae et Genevae E 65A. O.O. Ser. I vol 24 1744.
[6] I.M. Gelfand and S.V. Fomin, Calculus of Variations, Prentice-Hall Inc., Englewood Cliffs, NJ, 1963.
[7] R. Levien, The elastica: a mathematical history. Technical Report No. UCB/EECS-2008-103, Univ. of Berkeley, http://www.eecs.berkeley.edu/Pubs/TechRpts/ 2008/EECS-2008-103.html.
[8] J. Langer and D. A. Singer, The total squared curvature of closed curves, J. Differ. Geom. 20 (1984) 1-22.
[9] A. Linner, Gradients, preferred met- ´ rics and asymmetries. Preprint, 2001, http://www.math.niu.edu/ alinner/
[10] A.E. Love, A treatise on the Mathematical theory of Elasticity, Dover Publications, New York, 1944.
[11] R.S. Palais, Morse theory on Hilbert manifolds, Topology 2 (1963) 299-340.
[12] G. Smyrlis and V. Zisis, Local convergence of the steepest descent method in Hilbert spaces, J. Math. Anal. Appl. 300 (2004) 436-453
