Abstract

A modified simplex optimization method is developed for the design of illumination systems. The simplex method is a judicious choice for illumination optimization because of its robustness and convergence properties. To optimize the simplex method, its four parameters are adjusted dependent on the dimensionality of the space to converge with fewer iterations. This work is presented for the end game, when the optimizer is converging on a local optimum rather than searching for it. Up to a 37% reduction in the number of computations is realized. An example using a compound parabolic concentrator is compared between the standard and the modified simplex methods, providing over 22% improvement in the end game.

© 2005 Optical Society of America

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References

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  1. W. J. Cassarly and M. J. Hayford, Proc. SPIE 4832, 258 (2002).
    [CrossRef]
  2. N. E. Shatz and J. C. Bortz, Proc. SPIE 2538, 136 (1995).
    [CrossRef]
  3. W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, England, 1992), pp. 395–455.
  4. J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
    [CrossRef]
  5. R. J. Koshel, Proc. SPIE 4832, 270 (2002).
    [CrossRef]
  6. W. T. Welford and R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989).

2002 (2)

W. J. Cassarly and M. J. Hayford, Proc. SPIE 4832, 258 (2002).
[CrossRef]

R. J. Koshel, Proc. SPIE 4832, 270 (2002).
[CrossRef]

1995 (1)

N. E. Shatz and J. C. Bortz, Proc. SPIE 2538, 136 (1995).
[CrossRef]

1965 (1)

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

Bortz, J. C.

N. E. Shatz and J. C. Bortz, Proc. SPIE 2538, 136 (1995).
[CrossRef]

Cassarly, W. J.

W. J. Cassarly and M. J. Hayford, Proc. SPIE 4832, 258 (2002).
[CrossRef]

Flannery, B. P.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, England, 1992), pp. 395–455.

Hayford, M. J.

W. J. Cassarly and M. J. Hayford, Proc. SPIE 4832, 258 (2002).
[CrossRef]

Koshel, R. J.

R. J. Koshel, Proc. SPIE 4832, 270 (2002).
[CrossRef]

Mead, R.

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

Nelder, J. A.

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

Press, W. H.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, England, 1992), pp. 395–455.

Shatz, N. E.

N. E. Shatz and J. C. Bortz, Proc. SPIE 2538, 136 (1995).
[CrossRef]

Teukolsky, S. A.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, England, 1992), pp. 395–455.

Vetterling, W. T.

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, England, 1992), pp. 395–455.

Welford, W. T.

W. T. Welford and R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989).

Winston, R.

W. T. Welford and R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989).

Comput. J. (1)

J. A. Nelder and R. Mead, Comput. J. 7, 308 (1965).
[CrossRef]

Proc. SPIE (3)

R. J. Koshel, Proc. SPIE 4832, 270 (2002).
[CrossRef]

W. J. Cassarly and M. J. Hayford, Proc. SPIE 4832, 258 (2002).
[CrossRef]

N. E. Shatz and J. C. Bortz, Proc. SPIE 2538, 136 (1995).
[CrossRef]

Other (2)

W. H. Press, B. P. Flannery, S. A. Teukolsky, and W. T. Vetterling, Numerical Recipes in C (Cambridge U. Press, Cambridge, England, 1992), pp. 395–455.

W. T. Welford and R. Winston, High Collection Nonimaging Optics (Academic, San Diego, Calif., 1989).

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Figures (4)

Fig. 1
Fig. 1

Depiction of the reflection step in a two-dimensional parabolic well. The reflection is along the centroid line of the original simplex, not including high-function point P 1 in the calculation of this centroid. After reflection, the function value for the new point ( P R ) is calculated and compared with the previous value and the other points within the simplex.

Fig. 2
Fig. 2

Flow chart of the two embedded optimization routines: The first optimizes the values of the actions for the simplex method (see Table 1), whereas the second optimizes the simplex in an N-dimensional parabolic well. For the bottom row steps a no causes the routine to return to the upper row step (dashed arrows), and a yes causes the routine to progress to the next step.

Fig. 3
Fig. 3

FOM for the factor optimization is plotted versus the dimensionality of the parabolic well for the two cases, with and without independent contraction factors. The FOM is governed by Eq. (1), such that a value of less than 1 indicates fewer iterations of the modified simplex routine in comparison with the standard routine.

Fig. 4
Fig. 4

Dependencies of the α (reflection), γ (expansion), and β (contraction) factors plotted as a function of dimensionality of the parabolic well. Note that in general α increases, γ decreases, and β increases as N increases. The dashed lines denote the standard values for these factors.

Tables (2)

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Table 1 Description of the Simplex Steps

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Table 2 Results for the CPC Optimization Using the Standard and Modified Simplex Routines

Equations (1)

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FOM fact = 1 n p i = 1 n p FOM i , par , altered FOM i , par , std ,

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