Abstract

The efficiency of the multicanonical procedure can be significantly improved by applying an additional bias to the numerically generated sample space. However, results obtained by biasing in different sampling regions cannot in general be accurately combined, since their relative normalization coefficient is not known precisely. We demonstrate that for overlapping biasing regions a simple iterative procedure can be employed to determine the required coefficients.

© 2007 Optical Society of America

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  1. P. Smith, M. Shafi, and H. Gao, "Quick simulation: a review of importance sampling techniques in communications systems," IEEE J. Sel. Areas Commun. 15, 597-613 (1997).
    [Crossref]
  2. J. Bucklew and R. Radeke, "On the Monte Carlo simulation of digital communication systems in Gaussian noise," IEEE Trans. Commun. 51, 267-274 (2003).
    [Crossref]
  3. B. Berg, "Introduction to multicanonical Monte Carlo simulations," Fields Inst. Commun. 26, 1-24 (2000).
  4. D. Yevick, A First Course in Computational Physics and Object-Oriented Programming with C++ (Cambridge U. Press, 2005).
  5. D. Yevick, "The accuracy of multicanonical system models," IEEE Photon. Technol. Lett. 15, 224-226 (2003).
    [Crossref]
  6. D. Yevick, "Multicanonical communication system modeling--application to PMD statistics," IEEE Photon. Technol. Lett. 14, 1512-1514 (2002).
    [Crossref]
  7. J. Gordon and H. Kogelnik, "PMD fundamentals: polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
    [Crossref] [PubMed]
  8. M. Karlsson, "Probability density functions of the differential group delay in optical fiber communication systems," J. Lightwave Technol. 19, 324-331 (2001).
    [Crossref]
  9. G. Foschini and C. Poole, "Statistical theory of polarization dispersion in single mode fibers," J. Lightwave Technol. 9, 1439-1456 (1991).
    [Crossref]
  10. M. Jeruchim, P. Hahn, K. Smyntek, and R. Ray, "An experimental investigation of conventional and efficient importance sampling," IEEE Trans. Commun. 37, 578-587 (1989).
    [Crossref]
  11. T. Lu, D. Yevick, L. Yan, B. Zhang, and A. E. Willner, "An experimental approach to multicanonical sampling," IEEE Photon. Technol. Lett. 16, 1978-1980 (2005).
    [Crossref]
  12. T. Lu, D. Yevick, B. Hamilton, D. Dumas, and M. Reimer, "An experimental realization of biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1583-1585 (2005).
  13. T. Lu and D. Yevick, "Biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1420-1422 (2005).
    [Crossref]
  14. D. Yevick and T. Lu, "Improved multicanonical algorithms," J. Opt. Soc. Am. A 23, 2912-2918 (2006).
    [Crossref]
  15. R. Wolfe, M. Jeruchim, and P. Hahn, "On optimum and suboptimum biasing procedures for importance sampling in communication simulation," IEEE Trans. Commun. 38, 639-647 (1990).
    [Crossref]
  16. N. Alves, B. Berg, and R. Villanova, "Ising-model Monte Carlo simulations: density of states and mass gap," Phys. Rev. B 41, 383-394 (1990).
    [Crossref]
  17. P. Virnau and M. Muller, "Calculation of free energy through successive umbrella sampling," J. Chem. Phys. 120, 10925-10930 (2004).
    [Crossref]
  18. G. Torrie and J. Valleau, "Monte Carlo free energy estimates using non-Boltzmann sampling: application to the sub-critical Lennard-Jones fluid," Chem. Phys. Lett. 28, 578-581 (1974).
    [Crossref]
  19. G. Torrie and J. Valleau, "Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling," J. Comput. Phys. 23, 187-199 (1977).
    [Crossref]
  20. C. Bennett, "Efficient estimation of free energy differences from Monte Carlo data," J. Comput. Phys. 22, 245-268 (1976).
    [Crossref]
  21. S. Kumar, D. Bouzids, R. Swendsen, P. Kollman, and J. Rosebberg, "The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method," J. Comput. Chem. 13, 1011-1021 (1992).
    [Crossref]
  22. M. Souaille and B. Roux, "Extension to the weighted histogram analysis method: combining umbrella sampling with free energy calculations," Comput. Phys. Commun. 135, 40-57 (2001).
    [Crossref]
  23. A. Ferrenberg, D. Landau, and R. Swendsen, "Statistical errors in histogram reweighting," Phys. Rev. E 51, 5092-5100 (1995).
    [Crossref]
  24. G. Smith and A. Bruce, "A study of the multi-canonical Monte Carlo method," J. Phys. A 28, 6623-6643 (1995).
    [Crossref]
  25. B. Berg, "Algorithmic aspects of multicanonical simulations," Nucl. Phys. B 63A-C, 982-984 (1998).
  26. B. Berg, "Multicanonical recursions," J. Stat. Phys. 82, 323-342 (1996).
    [Crossref]
  27. N. Mandayam and B. Aazhang, "Importance sampling for analysis of direct detection optical communication systems," IEEE Trans. Commun. 43, 229-239 (1995).
    [Crossref]
  28. E. Veach, "Robust Monte Carlo methods for light transport simulation," Ph.D. thesis (Stanford University, 1997).
  29. D. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization (Wiley, 1992).
    [Crossref]
  30. A. Acharya, "Free energy differences: representations, estimators and sampling strategies," Ph.D. thesis (University of Edinburgh, 2004).
  31. A. Ferrenberg and R. Swendsen, "Optimized Monte Carlo data analysis," Phys. Rev. Lett. 63, 1195-1198 (1989).
    [Crossref] [PubMed]
  32. T. Lu, W. Huang, D. Yevick, M. O'Sullivan, and M. Reimer, "Multicanonical comparison of polarization-mode dispersion compensator performance," J. Opt. Soc. Am. A 22, 2804-2809 (2005).
    [Crossref]
  33. A. Djupsjobacka, "On differential group-delay statistics for polarization-mode dispersion emulators," J. Lightwave Technol. 19, 285-290 (2001).
    [Crossref]
  34. B. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1068 (1992).
    [Crossref]
  35. M. Reimer and D. Yevick, "Least-squares analysis of the Mueller matrix," Opt. Lett. 31, 2399-2401 (2006).
    [Crossref] [PubMed]
  36. D. Sandel, V. Mirvoda, S. Bhandare, F. Wust, and R. Noe, "Some enabling techniques for polarization mode dispersion compensation," J. Lightwave Technol. 21, 1198-1210 (2003).
    [Crossref]

2006 (2)

2005 (4)

T. Lu, W. Huang, D. Yevick, M. O'Sullivan, and M. Reimer, "Multicanonical comparison of polarization-mode dispersion compensator performance," J. Opt. Soc. Am. A 22, 2804-2809 (2005).
[Crossref]

T. Lu, D. Yevick, L. Yan, B. Zhang, and A. E. Willner, "An experimental approach to multicanonical sampling," IEEE Photon. Technol. Lett. 16, 1978-1980 (2005).
[Crossref]

T. Lu, D. Yevick, B. Hamilton, D. Dumas, and M. Reimer, "An experimental realization of biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1583-1585 (2005).

T. Lu and D. Yevick, "Biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1420-1422 (2005).
[Crossref]

2004 (1)

P. Virnau and M. Muller, "Calculation of free energy through successive umbrella sampling," J. Chem. Phys. 120, 10925-10930 (2004).
[Crossref]

2003 (3)

J. Bucklew and R. Radeke, "On the Monte Carlo simulation of digital communication systems in Gaussian noise," IEEE Trans. Commun. 51, 267-274 (2003).
[Crossref]

D. Yevick, "The accuracy of multicanonical system models," IEEE Photon. Technol. Lett. 15, 224-226 (2003).
[Crossref]

D. Sandel, V. Mirvoda, S. Bhandare, F. Wust, and R. Noe, "Some enabling techniques for polarization mode dispersion compensation," J. Lightwave Technol. 21, 1198-1210 (2003).
[Crossref]

2002 (1)

D. Yevick, "Multicanonical communication system modeling--application to PMD statistics," IEEE Photon. Technol. Lett. 14, 1512-1514 (2002).
[Crossref]

2001 (3)

2000 (2)

J. Gordon and H. Kogelnik, "PMD fundamentals: polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[Crossref] [PubMed]

B. Berg, "Introduction to multicanonical Monte Carlo simulations," Fields Inst. Commun. 26, 1-24 (2000).

1998 (1)

B. Berg, "Algorithmic aspects of multicanonical simulations," Nucl. Phys. B 63A-C, 982-984 (1998).

1997 (1)

P. Smith, M. Shafi, and H. Gao, "Quick simulation: a review of importance sampling techniques in communications systems," IEEE J. Sel. Areas Commun. 15, 597-613 (1997).
[Crossref]

1996 (1)

B. Berg, "Multicanonical recursions," J. Stat. Phys. 82, 323-342 (1996).
[Crossref]

1995 (3)

N. Mandayam and B. Aazhang, "Importance sampling for analysis of direct detection optical communication systems," IEEE Trans. Commun. 43, 229-239 (1995).
[Crossref]

A. Ferrenberg, D. Landau, and R. Swendsen, "Statistical errors in histogram reweighting," Phys. Rev. E 51, 5092-5100 (1995).
[Crossref]

G. Smith and A. Bruce, "A study of the multi-canonical Monte Carlo method," J. Phys. A 28, 6623-6643 (1995).
[Crossref]

1992 (2)

B. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1068 (1992).
[Crossref]

S. Kumar, D. Bouzids, R. Swendsen, P. Kollman, and J. Rosebberg, "The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method," J. Comput. Chem. 13, 1011-1021 (1992).
[Crossref]

1991 (1)

G. Foschini and C. Poole, "Statistical theory of polarization dispersion in single mode fibers," J. Lightwave Technol. 9, 1439-1456 (1991).
[Crossref]

1990 (2)

R. Wolfe, M. Jeruchim, and P. Hahn, "On optimum and suboptimum biasing procedures for importance sampling in communication simulation," IEEE Trans. Commun. 38, 639-647 (1990).
[Crossref]

N. Alves, B. Berg, and R. Villanova, "Ising-model Monte Carlo simulations: density of states and mass gap," Phys. Rev. B 41, 383-394 (1990).
[Crossref]

1989 (2)

M. Jeruchim, P. Hahn, K. Smyntek, and R. Ray, "An experimental investigation of conventional and efficient importance sampling," IEEE Trans. Commun. 37, 578-587 (1989).
[Crossref]

A. Ferrenberg and R. Swendsen, "Optimized Monte Carlo data analysis," Phys. Rev. Lett. 63, 1195-1198 (1989).
[Crossref] [PubMed]

1977 (1)

G. Torrie and J. Valleau, "Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling," J. Comput. Phys. 23, 187-199 (1977).
[Crossref]

1976 (1)

C. Bennett, "Efficient estimation of free energy differences from Monte Carlo data," J. Comput. Phys. 22, 245-268 (1976).
[Crossref]

1974 (1)

G. Torrie and J. Valleau, "Monte Carlo free energy estimates using non-Boltzmann sampling: application to the sub-critical Lennard-Jones fluid," Chem. Phys. Lett. 28, 578-581 (1974).
[Crossref]

Aazhang, B.

N. Mandayam and B. Aazhang, "Importance sampling for analysis of direct detection optical communication systems," IEEE Trans. Commun. 43, 229-239 (1995).
[Crossref]

Acharya, A.

A. Acharya, "Free energy differences: representations, estimators and sampling strategies," Ph.D. thesis (University of Edinburgh, 2004).

Alves, N.

N. Alves, B. Berg, and R. Villanova, "Ising-model Monte Carlo simulations: density of states and mass gap," Phys. Rev. B 41, 383-394 (1990).
[Crossref]

Bennett, C.

C. Bennett, "Efficient estimation of free energy differences from Monte Carlo data," J. Comput. Phys. 22, 245-268 (1976).
[Crossref]

Berg, B.

B. Berg, "Introduction to multicanonical Monte Carlo simulations," Fields Inst. Commun. 26, 1-24 (2000).

B. Berg, "Algorithmic aspects of multicanonical simulations," Nucl. Phys. B 63A-C, 982-984 (1998).

B. Berg, "Multicanonical recursions," J. Stat. Phys. 82, 323-342 (1996).
[Crossref]

N. Alves, B. Berg, and R. Villanova, "Ising-model Monte Carlo simulations: density of states and mass gap," Phys. Rev. B 41, 383-394 (1990).
[Crossref]

Bhandare, S.

Bouzids, D.

S. Kumar, D. Bouzids, R. Swendsen, P. Kollman, and J. Rosebberg, "The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method," J. Comput. Chem. 13, 1011-1021 (1992).
[Crossref]

Bruce, A.

G. Smith and A. Bruce, "A study of the multi-canonical Monte Carlo method," J. Phys. A 28, 6623-6643 (1995).
[Crossref]

Bucklew, J.

J. Bucklew and R. Radeke, "On the Monte Carlo simulation of digital communication systems in Gaussian noise," IEEE Trans. Commun. 51, 267-274 (2003).
[Crossref]

Djupsjobacka, A.

Dumas, D.

T. Lu, D. Yevick, B. Hamilton, D. Dumas, and M. Reimer, "An experimental realization of biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1583-1585 (2005).

Ferrenberg, A.

A. Ferrenberg, D. Landau, and R. Swendsen, "Statistical errors in histogram reweighting," Phys. Rev. E 51, 5092-5100 (1995).
[Crossref]

A. Ferrenberg and R. Swendsen, "Optimized Monte Carlo data analysis," Phys. Rev. Lett. 63, 1195-1198 (1989).
[Crossref] [PubMed]

Foschini, G.

G. Foschini and C. Poole, "Statistical theory of polarization dispersion in single mode fibers," J. Lightwave Technol. 9, 1439-1456 (1991).
[Crossref]

Gao, H.

P. Smith, M. Shafi, and H. Gao, "Quick simulation: a review of importance sampling techniques in communications systems," IEEE J. Sel. Areas Commun. 15, 597-613 (1997).
[Crossref]

Gordon, J.

J. Gordon and H. Kogelnik, "PMD fundamentals: polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[Crossref] [PubMed]

Hahn, P.

R. Wolfe, M. Jeruchim, and P. Hahn, "On optimum and suboptimum biasing procedures for importance sampling in communication simulation," IEEE Trans. Commun. 38, 639-647 (1990).
[Crossref]

M. Jeruchim, P. Hahn, K. Smyntek, and R. Ray, "An experimental investigation of conventional and efficient importance sampling," IEEE Trans. Commun. 37, 578-587 (1989).
[Crossref]

Hamilton, B.

T. Lu, D. Yevick, B. Hamilton, D. Dumas, and M. Reimer, "An experimental realization of biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1583-1585 (2005).

Heffner, B.

B. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1068 (1992).
[Crossref]

Huang, W.

Jeruchim, M.

R. Wolfe, M. Jeruchim, and P. Hahn, "On optimum and suboptimum biasing procedures for importance sampling in communication simulation," IEEE Trans. Commun. 38, 639-647 (1990).
[Crossref]

M. Jeruchim, P. Hahn, K. Smyntek, and R. Ray, "An experimental investigation of conventional and efficient importance sampling," IEEE Trans. Commun. 37, 578-587 (1989).
[Crossref]

Karlsson, M.

Kogelnik, H.

J. Gordon and H. Kogelnik, "PMD fundamentals: polarization mode dispersion in optical fibers," Proc. Natl. Acad. Sci. U.S.A. 97, 4541-4550 (2000).
[Crossref] [PubMed]

Kollman, P.

S. Kumar, D. Bouzids, R. Swendsen, P. Kollman, and J. Rosebberg, "The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method," J. Comput. Chem. 13, 1011-1021 (1992).
[Crossref]

Kumar, S.

S. Kumar, D. Bouzids, R. Swendsen, P. Kollman, and J. Rosebberg, "The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method," J. Comput. Chem. 13, 1011-1021 (1992).
[Crossref]

Landau, D.

A. Ferrenberg, D. Landau, and R. Swendsen, "Statistical errors in histogram reweighting," Phys. Rev. E 51, 5092-5100 (1995).
[Crossref]

Lu, T.

D. Yevick and T. Lu, "Improved multicanonical algorithms," J. Opt. Soc. Am. A 23, 2912-2918 (2006).
[Crossref]

T. Lu, W. Huang, D. Yevick, M. O'Sullivan, and M. Reimer, "Multicanonical comparison of polarization-mode dispersion compensator performance," J. Opt. Soc. Am. A 22, 2804-2809 (2005).
[Crossref]

T. Lu and D. Yevick, "Biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1420-1422 (2005).
[Crossref]

T. Lu, D. Yevick, B. Hamilton, D. Dumas, and M. Reimer, "An experimental realization of biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1583-1585 (2005).

T. Lu, D. Yevick, L. Yan, B. Zhang, and A. E. Willner, "An experimental approach to multicanonical sampling," IEEE Photon. Technol. Lett. 16, 1978-1980 (2005).
[Crossref]

Mandayam, N.

N. Mandayam and B. Aazhang, "Importance sampling for analysis of direct detection optical communication systems," IEEE Trans. Commun. 43, 229-239 (1995).
[Crossref]

Mirvoda, V.

Muller, M.

P. Virnau and M. Muller, "Calculation of free energy through successive umbrella sampling," J. Chem. Phys. 120, 10925-10930 (2004).
[Crossref]

Noe, R.

O'Sullivan, M.

Poole, C.

G. Foschini and C. Poole, "Statistical theory of polarization dispersion in single mode fibers," J. Lightwave Technol. 9, 1439-1456 (1991).
[Crossref]

Radeke, R.

J. Bucklew and R. Radeke, "On the Monte Carlo simulation of digital communication systems in Gaussian noise," IEEE Trans. Commun. 51, 267-274 (2003).
[Crossref]

Ray, R.

M. Jeruchim, P. Hahn, K. Smyntek, and R. Ray, "An experimental investigation of conventional and efficient importance sampling," IEEE Trans. Commun. 37, 578-587 (1989).
[Crossref]

Reimer, M.

Rosebberg, J.

S. Kumar, D. Bouzids, R. Swendsen, P. Kollman, and J. Rosebberg, "The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method," J. Comput. Chem. 13, 1011-1021 (1992).
[Crossref]

Roux, B.

M. Souaille and B. Roux, "Extension to the weighted histogram analysis method: combining umbrella sampling with free energy calculations," Comput. Phys. Commun. 135, 40-57 (2001).
[Crossref]

Sandel, D.

Scott, D.

D. Scott, Multivariate Density Estimation: Theory, Practice, and Visualization (Wiley, 1992).
[Crossref]

Shafi, M.

P. Smith, M. Shafi, and H. Gao, "Quick simulation: a review of importance sampling techniques in communications systems," IEEE J. Sel. Areas Commun. 15, 597-613 (1997).
[Crossref]

Smith, G.

G. Smith and A. Bruce, "A study of the multi-canonical Monte Carlo method," J. Phys. A 28, 6623-6643 (1995).
[Crossref]

Smith, P.

P. Smith, M. Shafi, and H. Gao, "Quick simulation: a review of importance sampling techniques in communications systems," IEEE J. Sel. Areas Commun. 15, 597-613 (1997).
[Crossref]

Smyntek, K.

M. Jeruchim, P. Hahn, K. Smyntek, and R. Ray, "An experimental investigation of conventional and efficient importance sampling," IEEE Trans. Commun. 37, 578-587 (1989).
[Crossref]

Souaille, M.

M. Souaille and B. Roux, "Extension to the weighted histogram analysis method: combining umbrella sampling with free energy calculations," Comput. Phys. Commun. 135, 40-57 (2001).
[Crossref]

Swendsen, R.

A. Ferrenberg, D. Landau, and R. Swendsen, "Statistical errors in histogram reweighting," Phys. Rev. E 51, 5092-5100 (1995).
[Crossref]

S. Kumar, D. Bouzids, R. Swendsen, P. Kollman, and J. Rosebberg, "The weighted histogram analysis method for free-energy calculations on biomolecules. I. The method," J. Comput. Chem. 13, 1011-1021 (1992).
[Crossref]

A. Ferrenberg and R. Swendsen, "Optimized Monte Carlo data analysis," Phys. Rev. Lett. 63, 1195-1198 (1989).
[Crossref] [PubMed]

Torrie, G.

G. Torrie and J. Valleau, "Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling," J. Comput. Phys. 23, 187-199 (1977).
[Crossref]

G. Torrie and J. Valleau, "Monte Carlo free energy estimates using non-Boltzmann sampling: application to the sub-critical Lennard-Jones fluid," Chem. Phys. Lett. 28, 578-581 (1974).
[Crossref]

Valleau, J.

G. Torrie and J. Valleau, "Nonphysical sampling distributions in Monte Carlo free-energy estimation: umbrella sampling," J. Comput. Phys. 23, 187-199 (1977).
[Crossref]

G. Torrie and J. Valleau, "Monte Carlo free energy estimates using non-Boltzmann sampling: application to the sub-critical Lennard-Jones fluid," Chem. Phys. Lett. 28, 578-581 (1974).
[Crossref]

Veach, E.

E. Veach, "Robust Monte Carlo methods for light transport simulation," Ph.D. thesis (Stanford University, 1997).

Villanova, R.

N. Alves, B. Berg, and R. Villanova, "Ising-model Monte Carlo simulations: density of states and mass gap," Phys. Rev. B 41, 383-394 (1990).
[Crossref]

Virnau, P.

P. Virnau and M. Muller, "Calculation of free energy through successive umbrella sampling," J. Chem. Phys. 120, 10925-10930 (2004).
[Crossref]

Willner, A. E.

T. Lu, D. Yevick, L. Yan, B. Zhang, and A. E. Willner, "An experimental approach to multicanonical sampling," IEEE Photon. Technol. Lett. 16, 1978-1980 (2005).
[Crossref]

Wolfe, R.

R. Wolfe, M. Jeruchim, and P. Hahn, "On optimum and suboptimum biasing procedures for importance sampling in communication simulation," IEEE Trans. Commun. 38, 639-647 (1990).
[Crossref]

Wust, F.

Yan, L.

T. Lu, D. Yevick, L. Yan, B. Zhang, and A. E. Willner, "An experimental approach to multicanonical sampling," IEEE Photon. Technol. Lett. 16, 1978-1980 (2005).
[Crossref]

Yevick, D.

M. Reimer and D. Yevick, "Least-squares analysis of the Mueller matrix," Opt. Lett. 31, 2399-2401 (2006).
[Crossref] [PubMed]

D. Yevick and T. Lu, "Improved multicanonical algorithms," J. Opt. Soc. Am. A 23, 2912-2918 (2006).
[Crossref]

T. Lu, W. Huang, D. Yevick, M. O'Sullivan, and M. Reimer, "Multicanonical comparison of polarization-mode dispersion compensator performance," J. Opt. Soc. Am. A 22, 2804-2809 (2005).
[Crossref]

T. Lu, D. Yevick, L. Yan, B. Zhang, and A. E. Willner, "An experimental approach to multicanonical sampling," IEEE Photon. Technol. Lett. 16, 1978-1980 (2005).
[Crossref]

T. Lu and D. Yevick, "Biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1420-1422 (2005).
[Crossref]

T. Lu, D. Yevick, B. Hamilton, D. Dumas, and M. Reimer, "An experimental realization of biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1583-1585 (2005).

D. Yevick, "The accuracy of multicanonical system models," IEEE Photon. Technol. Lett. 15, 224-226 (2003).
[Crossref]

D. Yevick, "Multicanonical communication system modeling--application to PMD statistics," IEEE Photon. Technol. Lett. 14, 1512-1514 (2002).
[Crossref]

D. Yevick, A First Course in Computational Physics and Object-Oriented Programming with C++ (Cambridge U. Press, 2005).

Zhang, B.

T. Lu, D. Yevick, L. Yan, B. Zhang, and A. E. Willner, "An experimental approach to multicanonical sampling," IEEE Photon. Technol. Lett. 16, 1978-1980 (2005).
[Crossref]

Chem. Phys. Lett. (1)

G. Torrie and J. Valleau, "Monte Carlo free energy estimates using non-Boltzmann sampling: application to the sub-critical Lennard-Jones fluid," Chem. Phys. Lett. 28, 578-581 (1974).
[Crossref]

Comput. Phys. Commun. (1)

M. Souaille and B. Roux, "Extension to the weighted histogram analysis method: combining umbrella sampling with free energy calculations," Comput. Phys. Commun. 135, 40-57 (2001).
[Crossref]

Fields Inst. Commun. (1)

B. Berg, "Introduction to multicanonical Monte Carlo simulations," Fields Inst. Commun. 26, 1-24 (2000).

IEEE J. Sel. Areas Commun. (1)

P. Smith, M. Shafi, and H. Gao, "Quick simulation: a review of importance sampling techniques in communications systems," IEEE J. Sel. Areas Commun. 15, 597-613 (1997).
[Crossref]

IEEE Photon. Technol. Lett. (6)

D. Yevick, "The accuracy of multicanonical system models," IEEE Photon. Technol. Lett. 15, 224-226 (2003).
[Crossref]

D. Yevick, "Multicanonical communication system modeling--application to PMD statistics," IEEE Photon. Technol. Lett. 14, 1512-1514 (2002).
[Crossref]

T. Lu, D. Yevick, L. Yan, B. Zhang, and A. E. Willner, "An experimental approach to multicanonical sampling," IEEE Photon. Technol. Lett. 16, 1978-1980 (2005).
[Crossref]

T. Lu, D. Yevick, B. Hamilton, D. Dumas, and M. Reimer, "An experimental realization of biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1583-1585 (2005).

T. Lu and D. Yevick, "Biased multicanonical sampling," IEEE Photon. Technol. Lett. 17, 1420-1422 (2005).
[Crossref]

B. Heffner, "Automated measurement of polarization mode dispersion using Jones matrix eigenanalysis," IEEE Photon. Technol. Lett. 4, 1066-1068 (1992).
[Crossref]

IEEE Trans. Commun. (4)

N. Mandayam and B. Aazhang, "Importance sampling for analysis of direct detection optical communication systems," IEEE Trans. Commun. 43, 229-239 (1995).
[Crossref]

R. Wolfe, M. Jeruchim, and P. Hahn, "On optimum and suboptimum biasing procedures for importance sampling in communication simulation," IEEE Trans. Commun. 38, 639-647 (1990).
[Crossref]

M. Jeruchim, P. Hahn, K. Smyntek, and R. Ray, "An experimental investigation of conventional and efficient importance sampling," IEEE Trans. Commun. 37, 578-587 (1989).
[Crossref]

J. Bucklew and R. Radeke, "On the Monte Carlo simulation of digital communication systems in Gaussian noise," IEEE Trans. Commun. 51, 267-274 (2003).
[Crossref]

J. Chem. Phys. (1)

P. Virnau and M. Muller, "Calculation of free energy through successive umbrella sampling," J. Chem. Phys. 120, 10925-10930 (2004).
[Crossref]

J. Comput. Chem. (1)

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Figures (6)

Fig. 1
Fig. 1

Maximum relative error, Eq. (17), for the initial estimate of the normalization constants of Eq. (14) (solid curve) and for the iterated results A n ( 0 ) = 1 , n = 1 , , N I for these constants (dotted curve). Results are illustrated for a 100-segment fiber emulator with τ avg = 25 ps and ten biased multicanonical iterations with 2 × 10 4 points per iteration with γ = 0.3 .

Fig. 2
Fig. 2

Pdf, Eq. (10), obtained from the calculation of Fig. 1 after joining the results of ten different biased calculations after 1 (crosses), 2000 (circles), and 20,000 (dotted curve) iterations of Eq. (12). The solid curve is the exact result of [8]. The normalization constants are here initialized to A n ( 0 ) = 1 , n = 1 , , N I .

Fig. 3
Fig. 3

Top, biased pdf estimate p ̂ 2 before (circles) and after (dashed curve) combination with the initial Monte Carlo estimate p ̂ 1 (solid curve) according to Eq. (10) with γ = 0.3 . Bottom, analogous results for the second biasing iteration in which p ̂ 1 , p ̂ 2 , and p ̂ 3 (crosses) are combined according to Eq. (10) (solid curve). The vertical lines indicate the biasing region.

Fig. 4
Fig. 4

Pdf of the DGD of a 100-section fiber emulator calculated with fifty 5 × 10 4 -sample (dashed–dotted curve) and fifteen 1.2 × 10 4 -sample (dashed curve) iterations of the biased multicanonical method, with the standard multicanonical procedure for fifteen 1.67 × 10 5 -sample iterations (circles) and with the analytic result (solid curve).

Fig. 5
Fig. 5

Biased multicanonical experimental setup.

Fig. 6
Fig. 6

Experimentally determined pdf of the DGD of an eight-section fiber emulator for 45,000 samples measured with the piecewise biased multicanonical method (circles) and the standard Monte Carlo procedure (crosses). The solid curve indicates the corresponding numerical results for three 5 × 10 5 -sample iterations in the standard multicanonical method.

Equations (18)

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F ( E ) = { e ( E E n L ) 2 ( 2 σ 2 ) E < E n L e ( E E n R ) 2 ( 2 σ 2 ) E > E n R 1 E n L E E n R } ,
p ( E k ) = i = 1 m 1 H i ( E k ) p ̂ i ( E k ) i = 1 m 1 H i ( E k ) .
p ̂ ( E k ) = i = 1 m 1 A i w i ( E k ) p ̂ i ( E k ) ,
i = 1 m 1 w i ( E k ) = 1
p i ( E ) = { p ( E ) A i E R i 0 otherwise } ,
MSE { p ̂ ( E k ) } E { [ p ̂ ( E k ) p ( E k ) ] 2 } = i = 1 m 1 A i 2 w i 2 ( E k ) Var { p ̂ i ( E k ) } + [ E { p ̂ ( E k ) } p ( E k ) ] 2 ,
f ̂ ( E k ) = MSE { p ̂ ( E k ) } λ { i = 1 m 1 w i ( E k ) 1 } ,
w i ( E k ) = I i ( k ) [ A i 2 Var { p ̂ i ( E k ) } ] j = 1 m 1 I j ( k ) [ A j 2 Var { p ̂ j ( E k ) } ] .
Var { p ̂ i ( E k ) } = 1 n i Δ V g i ( E k ) p i ( E k ) .
p ̂ ( E k ) i = 1 m 1 n i p ̂ i ( E k ) j = 1 m 1 I j ( k ) n j A j .
A n = R n p ( E ) d E { k n E k n R n } p ̂ ( E k n ) Δ V .
A n { k n E k n R n } i = 1 m 1 n i p ̂ i ( E k n ) j = 1 m 1 I j ( k n ) n j A j .
k = 1 N w k ls [ p ̂ n 1 ( E k ) x n p ̂ n ( E k ) ] 2 ,
A n ( 0 ) A n 1 ( 0 ) = k = 1 N w k ls p ̂ n 1 ( E k ) p ̂ n ( E k ) k = 1 N w k ls [ p ̂ n ( E k ) ] 2 .
Δ E = 2 E max E 0 1 + 2 γ ( N I 1 ) ,
E n L = E 0 + Δ E [ ( n 1 ) γ 1 2 ] ,
E n R = E 0 + Δ E [ ( n 1 ) γ + 1 2 ] ,
ϵ = max n [ A n ( i ) A n ( i 1 ) A n ( i 1 ) ] × 100 % ,

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