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

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  • Fiori, Simone

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.

Suggested Citation

  • Fiori, Simone, 2016. "A Riemannian steepest descent approach over the inhomogeneous symplectic group: Application to the averaging of linear optical systems," Applied Mathematics and Computation, Elsevier, vol. 283(C), pages 251-264.
    • Handle: RePEc:eee:apmaco:v:283:y:2016:i:c:p:251-264
    DOI: 10.1016/j.amc.2016.02.018
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    References listed on IDEAS

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    1. David G. Luenberger, 1972. "The Gradient Projection Method Along Geodesics," Management Science, INFORMS, vol. 18(11), pages 620-631, July.
    2. Celledoni, Elena & Fiori, Simone, 2008. "Descent methods for optimization on homogeneous manifolds," Mathematics and Computers in Simulation (MATCOM), Elsevier, vol. 79(4), pages 1298-1323.
    3. R.-B. Wu & R. Chakrabarti & H. Rabitz, 2010. "Critical Landscape Topology for Optimization on the Symplectic Group," Journal of Optimization Theory and Applications, Springer, vol. 145(2), pages 387-406, May.
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