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

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.