Abstract

In this article, we evaluate a quasi-Monte Carlo (QMC) method with various low-discrepancy sequences (LDS) in illumination optical systems which are adopted in some commercial products, and clarify the method’s effectiveness quantitatively. We assumed the evaluated systems were an illumination optical system with a perfectly diffusing surface, and we compared them against the theoretical irradiance distribution. The evaluation results indicate that the QMC method delivers higher asymptotic convergence rate than the MC method does, and there is little difference between each LDS. In evaluation of simple optical systems that can be boiled down to low-dimensional numerical integration problems, the QMC method was found to be extremely effective.

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References

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  1. H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods (Society for Industrial and Applied Mathematics, 1992).
  2. M. Drmota and F. Robert, Tichy, Sequences, Discrepancies and Applications (Springer, 1997).
  3. J. Dick and F. Pillichshammer, Digital Nets and Sequences (Cambridge University Press, 2010).
  4. S. Ninomiya and S. Tezuka, “Toward real-time pricing of complex financial derivatives,” Appl. Math. Finance 3, 1–20 (1996).
  5. A. Keller, “Instant radiosity,” in Proceedings of the 24th annual conference on Computer graphics and interactive techniques, G. S. Owen, T. Whitted and B. Mones-Hattal ed. (ACM Press, 1997).
  6. T. Kollig and A. Keller, “Efficient multidimensional sampling,” Comput. Graph. Forum 21, 557–563 (2002).
  7. I. M. Sobol', “On the distribution of points in a cube and the approximate evaluation of integrals,” USSR Comput. Math. Math. Phys. 7, 86–112 (1967).
  8. R. D. Richtmyer, “The evaluation of definite integrals, and a quasi-Monte-Carlo method based on the properties of algebraic numbers,” LA-1342, Los Alamos Scientific Laboratories, (1951).
  9. J. H. Halton, “On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals,” Numer. Math. 2, 84–90 (1960).
  10. J. M. Hammersley, “Monte Carlo methods for solving multivariable problems,” Ann. N. Y. Acad. Sci. 86, 844–874 (1960).
  11. H. Faure, “On the star-discrepancy of generalized Hammersley sequences in two dimensions,” Monatsh. Math. 101, 291–300 (1986).
  12. M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).

2002 (1)

T. Kollig and A. Keller, “Efficient multidimensional sampling,” Comput. Graph. Forum 21, 557–563 (2002).

1998 (1)

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).

1996 (1)

S. Ninomiya and S. Tezuka, “Toward real-time pricing of complex financial derivatives,” Appl. Math. Finance 3, 1–20 (1996).

1986 (1)

H. Faure, “On the star-discrepancy of generalized Hammersley sequences in two dimensions,” Monatsh. Math. 101, 291–300 (1986).

1967 (1)

I. M. Sobol', “On the distribution of points in a cube and the approximate evaluation of integrals,” USSR Comput. Math. Math. Phys. 7, 86–112 (1967).

1960 (2)

J. H. Halton, “On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals,” Numer. Math. 2, 84–90 (1960).

J. M. Hammersley, “Monte Carlo methods for solving multivariable problems,” Ann. N. Y. Acad. Sci. 86, 844–874 (1960).

Faure, H.

H. Faure, “On the star-discrepancy of generalized Hammersley sequences in two dimensions,” Monatsh. Math. 101, 291–300 (1986).

Halton, J. H.

J. H. Halton, “On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals,” Numer. Math. 2, 84–90 (1960).

Hammersley, J. M.

J. M. Hammersley, “Monte Carlo methods for solving multivariable problems,” Ann. N. Y. Acad. Sci. 86, 844–874 (1960).

Keller, A.

T. Kollig and A. Keller, “Efficient multidimensional sampling,” Comput. Graph. Forum 21, 557–563 (2002).

Kollig, T.

T. Kollig and A. Keller, “Efficient multidimensional sampling,” Comput. Graph. Forum 21, 557–563 (2002).

Matsumoto, M.

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).

Ninomiya, S.

S. Ninomiya and S. Tezuka, “Toward real-time pricing of complex financial derivatives,” Appl. Math. Finance 3, 1–20 (1996).

Nishimura, T.

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).

Sobol', I. M.

I. M. Sobol', “On the distribution of points in a cube and the approximate evaluation of integrals,” USSR Comput. Math. Math. Phys. 7, 86–112 (1967).

Tezuka, S.

S. Ninomiya and S. Tezuka, “Toward real-time pricing of complex financial derivatives,” Appl. Math. Finance 3, 1–20 (1996).

ACM Trans. Model. Comput. Simul. (1)

M. Matsumoto and T. Nishimura, “Mersenne twister: a 623-dimensionally equidistributed uniform pseudo-random number generator,” ACM Trans. Model. Comput. Simul. 8, 3–30 (1998).

Ann. N. Y. Acad. Sci. (1)

J. M. Hammersley, “Monte Carlo methods for solving multivariable problems,” Ann. N. Y. Acad. Sci. 86, 844–874 (1960).

Appl. Math. Finance (1)

S. Ninomiya and S. Tezuka, “Toward real-time pricing of complex financial derivatives,” Appl. Math. Finance 3, 1–20 (1996).

Comput. Graph. Forum (1)

T. Kollig and A. Keller, “Efficient multidimensional sampling,” Comput. Graph. Forum 21, 557–563 (2002).

Monatsh. Math. (1)

H. Faure, “On the star-discrepancy of generalized Hammersley sequences in two dimensions,” Monatsh. Math. 101, 291–300 (1986).

Numer. Math. (1)

J. H. Halton, “On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals,” Numer. Math. 2, 84–90 (1960).

USSR Comput. Math. Math. Phys. (1)

I. M. Sobol', “On the distribution of points in a cube and the approximate evaluation of integrals,” USSR Comput. Math. Math. Phys. 7, 86–112 (1967).

Other (5)

R. D. Richtmyer, “The evaluation of definite integrals, and a quasi-Monte-Carlo method based on the properties of algebraic numbers,” LA-1342, Los Alamos Scientific Laboratories, (1951).

A. Keller, “Instant radiosity,” in Proceedings of the 24th annual conference on Computer graphics and interactive techniques, G. S. Owen, T. Whitted and B. Mones-Hattal ed. (ACM Press, 1997).

H. Niederreiter, Random Number Generation and Quasi-Monte Carlo Methods (Society for Industrial and Applied Mathematics, 1992).

M. Drmota and F. Robert, Tichy, Sequences, Discrepancies and Applications (Springer, 1997).

J. Dick and F. Pillichshammer, Digital Nets and Sequences (Cambridge University Press, 2010).

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Figures (4)

Fig. 1
Fig. 1

Distribution of two dimensional LDSs and pseudorandom numbers. (a) MT. (b) Richtmyer. (c) Halton. (d) Faure. (e) Sobol’.

Fig. 2
Fig. 2

Optical system in the simulation.

Fig. 3
Fig. 3

Radiant intensity distributions obtained through simulations. (a) MT. (b) Richtmyer. (c) Halton. (d) Faure. (e) Sobol’.

Fig. 4
Fig. 4

Root-mean-square error evaluation results.

Tables (1)

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Table 1 Asymptotic Convergence Rate

Equations (7)

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D N (k) = sup t [0,1] k | A( [0,t);N ) N λ k ( [0,t) ) |.
D N (k) C k ( logN ) N k .
I θ = I n cosθ,
E θ = I θ cosθ d 2 = I n cos 2 θ d 2 ,
G(θ)= 1 2 ( 1cos2θ ),
θ= G 1 (x)= 1 2 arccos( 12x ),
RMSE= E E ^ 2 N = i j [ E(i,j) E ^ (i,j) ] 2 N .

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