A Riemannian steepest descent approach over the inhomogeneous symplectic group: Application to the averaging of linear optical systems

Simone Fiori

Applied Mathematics and Computation, 2016, vol. 283, issue C, 251-264

Abstract: The present manuscript describes a Riemannian-steepest-descent approach to compute the average out of a set of optical system transference matrices on the basis of a Lie-group averaging criterion function. The devised averaging algorithm is compared with the Harris’ exponential-mean-logarithm averaging rule, especially developed in computational ophthalmology to compute the average character of a set of biological optical systems. Results of numerical experiments show that the iterative algorithm based on gradient steepest descent implemented by exponential-map stepping converges to solutions that are in good agreement with those obtained by the application of Harris’ exponential-mean-logarithm averaging rule. Such results seem to confirm that Harris’ exponential-mean-logarithm averaging rule is numerically optimal in a Lie-group averaging sense.

Keywords: Averaging on Lie groups; Optimization on Riemannian Lie groups; Symplectic matrices; Hamiltonian matrices (search for similar items in EconPapers)
Date: 2016
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Persistent link: https://EconPapers.repec.org/RePEc:eee:apmaco:v:283:y:2016:i:c:p:251-264

DOI: 10.1016/j.amc.2016.02.018

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