Abstract

In lens system design, Damped Least Squares method is a traditionally used local search method. In this paper we apply mutation operators to control damping factor in Damped Least Squares. We study the mutation behavior in this method when the algorithm confronts with local minima. The proposed method can go beyond local minima by taking mutation operators to control damping factor. The proposed method was successfully applied to design problems. The result indicates that the mutation operators provide an effective and rapid way to jump out of poor local minima.

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References

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  1. K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).
  2. M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. 2(6), 463–470 (1995).
    [CrossRef]
  3. D. Vasiljevic, “Classical and evolutionary algorithms in the optimization of optical systems,” (Wiley, 2001).
  4. V. Yakovlev and G. Tempea, “Optimization of chirped mirrors,” Appl. Opt. 41(30), 6514–6520 (2002).
    [CrossRef] [PubMed]
  5. C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
    [CrossRef]
  6. I. Ono, S. Kobayashi, and Y. Yoshida, “Optimal lens design by real-coded genetic algorithms using UNDX,” Comput. Methods Appl. Mech. Eng. 186(2-4), 483–497 (2000).
    [CrossRef]
  7. L. Li, Q. H. Wang, D. H. Li, and H. R. Peng, “Jump method for optical thin film design,” Opt. Express 17(19), 16920–16926 (2009).
    [CrossRef] [PubMed]
  8. L. Li, Q. H. Wang, X. Q. Xu, and D. H. Li, “Two-step method for lens system design,” Opt. Express 18(12), 13285–13300 (2010).
    [CrossRef] [PubMed]
  9. D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).
  10. M. van Turnhout and F. Bociort, “Chaotic behavior in an algorithm to escape from poor local minima in lens design,” Opt. Express 17(8), 6436–6450 (2009).
    [CrossRef] [PubMed]
  11. J. Meiron, “Damped Least-Squares method for automatic lens design,” J. Opt. Soc. Am. 55(9), 1105–1107 (1965).
    [CrossRef]
  12. H. E. Nusse and J. A. Yorke, “Basins of attraction,” Science 271(5254), 1376–1380 (1996).
    [CrossRef]
  13. H. P. Schwefel, “Numerical optimization of computer models,” (Wiley, 1981).
  14. J. M. Yang and C. Y. Kao, “A robust evolutionary algorithm for optical thin-film designs,” Evol. Comput. 2, 978–985 (2000).
  15. D. C. O’Shea, “The monochromatic quartet: a search for the global optimum,” Proc. SPIE 1354, 548–554 (1990).
    [CrossRef]

2010 (1)

2009 (2)

2008 (1)

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[CrossRef]

2002 (1)

2000 (2)

I. Ono, S. Kobayashi, and Y. Yoshida, “Optimal lens design by real-coded genetic algorithms using UNDX,” Comput. Methods Appl. Mech. Eng. 186(2-4), 483–497 (2000).
[CrossRef]

J. M. Yang and C. Y. Kao, “A robust evolutionary algorithm for optical thin-film designs,” Evol. Comput. 2, 978–985 (2000).

1996 (1)

H. E. Nusse and J. A. Yorke, “Basins of attraction,” Science 271(5254), 1376–1380 (1996).
[CrossRef]

1995 (1)

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. 2(6), 463–470 (1995).
[CrossRef]

1994 (1)

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).

1990 (1)

D. C. O’Shea, “The monochromatic quartet: a search for the global optimum,” Proc. SPIE 1354, 548–554 (1990).
[CrossRef]

1965 (1)

1944 (1)

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

Beaulieu, J.

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[CrossRef]

Bociort, F.

Gagné, C.

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[CrossRef]

Hiraga, K.

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. 2(6), 463–470 (1995).
[CrossRef]

Ishikawa, J.

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. 2(6), 463–470 (1995).
[CrossRef]

Isshiki, M.

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. 2(6), 463–470 (1995).
[CrossRef]

Kao, C. Y.

J. M. Yang and C. Y. Kao, “A robust evolutionary algorithm for optical thin-film designs,” Evol. Comput. 2, 978–985 (2000).

Kobayashi, S.

I. Ono, S. Kobayashi, and Y. Yoshida, “Optimal lens design by real-coded genetic algorithms using UNDX,” Comput. Methods Appl. Mech. Eng. 186(2-4), 483–497 (2000).
[CrossRef]

Levenberg, K.

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

Li, D. H.

Li, L.

Meiron, J.

Nakadate, S.

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. 2(6), 463–470 (1995).
[CrossRef]

Nusse, H. E.

H. E. Nusse and J. A. Yorke, “Basins of attraction,” Science 271(5254), 1376–1380 (1996).
[CrossRef]

O’Shea, D. C.

D. C. O’Shea, “The monochromatic quartet: a search for the global optimum,” Proc. SPIE 1354, 548–554 (1990).
[CrossRef]

Ono, H.

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. 2(6), 463–470 (1995).
[CrossRef]

Ono, I.

I. Ono, S. Kobayashi, and Y. Yoshida, “Optimal lens design by real-coded genetic algorithms using UNDX,” Comput. Methods Appl. Mech. Eng. 186(2-4), 483–497 (2000).
[CrossRef]

Parizeau, M.

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[CrossRef]

Peng, H. R.

Shafer, D.

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).

Tempea, G.

Thibault, S.

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[CrossRef]

van Turnhout, M.

Wang, Q. H.

Xu, X. Q.

Yakovlev, V.

Yang, J. M.

J. M. Yang and C. Y. Kao, “A robust evolutionary algorithm for optical thin-film designs,” Evol. Comput. 2, 978–985 (2000).

Yorke, J. A.

H. E. Nusse and J. A. Yorke, “Basins of attraction,” Science 271(5254), 1376–1380 (1996).
[CrossRef]

Yoshida, Y.

I. Ono, S. Kobayashi, and Y. Yoshida, “Optimal lens design by real-coded genetic algorithms using UNDX,” Comput. Methods Appl. Mech. Eng. 186(2-4), 483–497 (2000).
[CrossRef]

Appl. Opt. (1)

Appl. Soft Comput. (1)

C. Gagné, J. Beaulieu, M. Parizeau, and S. Thibault, “Human-competitive lens system design with evolution strategies,” Appl. Soft Comput. 8(4), 1439–1452 (2008).
[CrossRef]

Comput. Methods Appl. Mech. Eng. (1)

I. Ono, S. Kobayashi, and Y. Yoshida, “Optimal lens design by real-coded genetic algorithms using UNDX,” Comput. Methods Appl. Mech. Eng. 186(2-4), 483–497 (2000).
[CrossRef]

Comput. Phys. (1)

D. Shafer, “Global optimization in optical design,” Comput. Phys. 8, 188–195 (1994).

Evol. Comput. (1)

J. M. Yang and C. Y. Kao, “A robust evolutionary algorithm for optical thin-film designs,” Evol. Comput. 2, 978–985 (2000).

J. Opt. Soc. Am. (1)

Opt. Express (3)

Opt. Rev. (1)

M. Isshiki, H. Ono, K. Hiraga, J. Ishikawa, and S. Nakadate, “Lens design: global optimization with escape function,” Opt. Rev. 2(6), 463–470 (1995).
[CrossRef]

Proc. SPIE (1)

D. C. O’Shea, “The monochromatic quartet: a search for the global optimum,” Proc. SPIE 1354, 548–554 (1990).
[CrossRef]

Q. Appl. Math. (1)

K. Levenberg, “A method for the solution of certain non-linear problems in least squares,” Q. Appl. Math. 2, 164–168 (1944).

Science (1)

H. E. Nusse and J. A. Yorke, “Basins of attraction,” Science 271(5254), 1376–1380 (1996).
[CrossRef]

Other (2)

H. P. Schwefel, “Numerical optimization of computer models,” (Wiley, 1981).

D. Vasiljevic, “Classical and evolutionary algorithms in the optimization of optical systems,” (Wiley, 2001).

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Figures (9)

Fig. 1
Fig. 1

An example of doublet lens.

Fig. 2
Fig. 2

Optimization route (n = 3). (a) Route of 1200 iterations. (b) Route of 500-750 iterations. (c) Route of 750-950 iterations. (d) Route of 950-1200 iterations.

Fig. 3
Fig. 3

Optimization route (n = 13). (a) Route of 1200 iterations. (b) Route of 550-750 iterations. (c) Route of 750-950 iterations. (d) Route of 950-1050 iterations.

Fig. 4
Fig. 4

Optimization route (n = 23). (a) Route of 1200 iterations. (b) Route of 500-750 iterations. (c) Route of 620-750 iterations.

Fig. 5
Fig. 5

Optimization route when mutated P is near to –Pk . (a) Overview of optimization route. (b) Overview of optimization route with enlarged vertical axis.

Fig. 6
Fig. 6

Optimization route when mutated P is far from –Pk .

Fig. 7
Fig. 7

The design found with the proposed method.

Fig. 8
Fig. 8

The design found with Damped Least Squares method.

Fig. 9
Fig. 9

The design found with evolutionary algorithm.

Tables (7)

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Table 1 Physical constraints for the monochromatic quartet.

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Table 2 Configuration parameters of the initial rough design.

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Table 3 Configuration parameters of the design found with the proposed method.

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Table 4 Configuration parameters of the design found with Damped Least Squares method.

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Table 5 Comparison between the solution found with Damped Least Squares and the solution found with the proposed method.

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Table 6 Configuration parameters of the design found with evolutionary algorithm.

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Table 7 Comparison of the solutions found with proposed method and the solution found with evolutionary algorithm.

Equations (10)

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Δ X = ( A T A + P I ) 1 A T F 0 ,
X = X 0 + Δ X .
| A T A + P I | = 0.
| A T A P k I | = 0 ,
P = P k { ( 1 + exp [ τ N ( 0 , 1 ) + τ N i ( 0 , 1 ) ] ) } n [ 1 + Q N ( 0 , 1 ) ] n ,
τ = ( 2 N ) 1 ,
τ = ( 2 N ) 1 ,
P = P k { exp [ τ N ( 0 , 1 ) + τ N i ( 0 , 1 ) ] } n [ Q N ( 0 , 1 ) ] n .
P = P 0 { ( 1 + exp [ τ N ( 0 , 1 ) + τ N i ( 0 , 1 ) ] ) } n [ 1 + Q N ( 0 , 1 ) ] n ,
F = P d i s t + P i m g + P v i g n + 1 3 θ = { 0 , 10.5 , 15 } R M S ,

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