Abstract

Process variations can significantly degrade device performance and chip yield in silicon photonics. In order to reduce the design and production costs, it is highly desirable to predict the statistical behavior of a device before the final fabrication. Monte Carlo is the mainstream computational technique used to estimate the uncertainties caused by process variations. However, it is very often too expensive due to its slow convergence rate. Recently, stochastic spectral methods based on polynomial chaos expansions have emerged as a promising alternative, and they have shown significant speedup over Monte Carlo in many engineering problems. The existing literature mostly assumes that the random parameters are mutually independent. However, in practical applications such assumption may not be necessarily accurate. In this paper, we develop an efficient numerical technique based on stochastic collocation to simulate silicon photonics with correlated and non-Gaussian random parameters. The effectiveness of our proposed technique is demonstrated by the simulation results of a silicon-on-insulator based directional coupler example. Since the mathematic formulation in this paper is very generic, our proposed algorithm can be applied to a large class of photonic design cases as well as to many other engineering problems.

© 2015 Optical Society of America

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  1. A. Biberman and K. Bergman, “Optical interconnection networks for high-performance computing systems,” Reports on Progress in Physics 75, 046402 (2012).
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  3. W. A. Zortman, D. C. Trotter, and M. R. Watts, “Silicon photonics manufacturing,” Opt. Express 18, 23598–23607 (2010).
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  4. A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
    [Crossref]
  5. X. Chen, M. Mohamed, Z. Li, L. Shang, and A. R. Mickelson, “Process variation in silicon photonic devices,” Appl. Opt. 52, 7638–7647 (2013).
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  6. S. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in “2011 8th IEEE International Conference on Group IV Photonics (GFP),” (2011), pp. 71–73.
  7. X. Wang, W. Shi, H. Yun, S. Grist, N. A. Jaeger, and L. Chrostowski, “Narrow-band waveguide bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20, 15547–15558 (2012).
    [Crossref] [PubMed]
  8. L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Fard Talebi, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in “Optical Fiber Communication Conference,” (OSA, 2014), pp. Th2A-37.
  9. Z. Zhang, T. El-Moselhy, I. Elfadel, and L. Daniel, “Stochastic testing method for transistor-level uncertainty quantification based on generalized polynomial chaos,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 32, 1533–1545 (2013).
    [Crossref]
  10. Z. Zhang, T. El-Moselhy, P. Maffezzoni, I. Elfadel, and L. Daniel, “Efficient uncertainty quantification for the periodic steady state of forced and autonomous circuits,” IEEE Trans. Circuits Syst. II: Exp. Briefs 60, 687–691 (2013).
    [Crossref]
  11. Z. Zhang, I. Elfadel, and L. Daniel, “Uncertainty quantification for integrated circuits: Stochastic spectral methods,” in “IEEE/ACM International Conference on Computer-Aided Design,” (2013), pp. 803–810.
  12. K. Strunz and Q. Su, “Stochastic formulation of spice-type electronic circuit simulation with polynomial chaos,” ACM Trans. Model Comput. Simul. 18, 15 (2008).
    [Crossref]
  13. N. Agarwal and N. Aluru, “Stochastic analysis of electrostatic MEMS subjected to parameter variations,” J. Microelectromech. Syst. 18, 1454–1468 (2009).
    [Crossref]
  14. Z. Zhang, X. Yang, G. Marucci, P. Maffezzoni, I. M. Elfadel, G. Karnidadakis, and L. Danie, “Stochastic testing simulator for integrated circuits and MEMS: Hierarchical and sparse techniques,” in “IEEE Custom Integrated Circuits Conference,” (2014).
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    [Crossref]
  17. D. Xiu and G. E. Karniadakis, “The Wiener–Askey polynomial chaos for stochastic differential equations,” SIAM J. Sci. Comput. 24, 619–644 (2002).
    [Crossref]
  18. D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in “Signal Processing in Photonic Communications,” (OSA, 2013), pp. JT3A–8.
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    [Crossref]
  21. F. Nobile, R. Tempone, and C. G. Webster, “A sparse grid stochastic collocation method for partial differential equations with random input data,” SIAM J. Numer. Anal. 46, 2309–2345 (2008).
    [Crossref]
  22. T. El-Moselhy and L. Daniel, “Stochastic integral equation solver for efficient variation-aware interconnect extraction,” in “Proceedings of Design Automation Conference,” (ACM, 2008), pp. 415–420.
  23. T. El-Moselhy and L. Daniel, “Variation-aware stochastic extraction with large parameter dimensionality: Review and comparison of state of the art intrusive and non-intrusive techniques,” in “International Symposium on Quality Electronic Design (ISQED),” (IEEE, 2011), pp. 1–10.
  24. D. VandeGinste, D. De Zutter, D. Deschrijver, T. Dhaene, P. Manfredi, and F. Canavero, “Stochastic modeling-based variability analysis of on-chip interconnects,” IEEE Trans. Compon. Packag. Manuf. Technol. 2, 1182–1192 (2012).
    [Crossref]
  25. I. S. Stievano, P. Manfredi, and F. G. Canavero, “Carbon nanotube interconnects: Process variation via polynomial chaos,” IEEE Trans. Electromagn. Compat. 54, 140–148 (2012).
    [Crossref]
  26. C. Soize and R. Ghanem, “Physical systems with random uncertainties: chaos representations with arbitrary probability measure,” SIAM J. Sci. Comput. 26, 395–410 (2004).
    [Crossref]
  27. T.-W. Weng, Z. Zhang, Z. Su, and L. Daniel, “Fast stochastic simulation of silicon waveguide with non-gaussian correlated process variations,” in “Asia Communications and Photonics Conference,” (OSA, 2014), p. AF3B.7.
  28. A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. R. Stat. Soc. Series B 39, 1–38 (1977).
  29. G. H. Golub and J. H. Welsch, “Calculation of Gauss quadrature rules,” Math. Comput. 23, 221–230 (1969).
    [Crossref]
  30. Q. Liu and D. A. Pierce, “A note on gausshermite quadrature,” Biometrika 81, 624–629 (1994).
  31. H. Wilf, Mathematics for the Physical Sciences, Problem 9 (Wiley, 1962).

2013 (4)

Z. Zhang, T. El-Moselhy, I. Elfadel, and L. Daniel, “Stochastic testing method for transistor-level uncertainty quantification based on generalized polynomial chaos,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 32, 1533–1545 (2013).
[Crossref]

Z. Zhang, T. El-Moselhy, P. Maffezzoni, I. Elfadel, and L. Daniel, “Efficient uncertainty quantification for the periodic steady state of forced and autonomous circuits,” IEEE Trans. Circuits Syst. II: Exp. Briefs 60, 687–691 (2013).
[Crossref]

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref] [PubMed]

X. Chen, M. Mohamed, Z. Li, L. Shang, and A. R. Mickelson, “Process variation in silicon photonic devices,” Appl. Opt. 52, 7638–7647 (2013).
[Crossref] [PubMed]

2012 (4)

A. Biberman and K. Bergman, “Optical interconnection networks for high-performance computing systems,” Reports on Progress in Physics 75, 046402 (2012).
[Crossref] [PubMed]

X. Wang, W. Shi, H. Yun, S. Grist, N. A. Jaeger, and L. Chrostowski, “Narrow-band waveguide bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20, 15547–15558 (2012).
[Crossref] [PubMed]

D. VandeGinste, D. De Zutter, D. Deschrijver, T. Dhaene, P. Manfredi, and F. Canavero, “Stochastic modeling-based variability analysis of on-chip interconnects,” IEEE Trans. Compon. Packag. Manuf. Technol. 2, 1182–1192 (2012).
[Crossref]

I. S. Stievano, P. Manfredi, and F. G. Canavero, “Carbon nanotube interconnects: Process variation via polynomial chaos,” IEEE Trans. Electromagn. Compat. 54, 140–148 (2012).
[Crossref]

2011 (1)

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

2010 (2)

W. A. Zortman, D. C. Trotter, and M. R. Watts, “Silicon photonics manufacturing,” Opt. Express 18, 23598–23607 (2010).
[Crossref] [PubMed]

A. Singhee and R. A. Rutenbar, “Why quasi-Monte Carlo is better than Monte Carlo or latin hypercube sampling for statistical circuit analysis,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 29, 1763–1776 (2010).
[Crossref]

2009 (1)

N. Agarwal and N. Aluru, “Stochastic analysis of electrostatic MEMS subjected to parameter variations,” J. Microelectromech. Syst. 18, 1454–1468 (2009).
[Crossref]

2008 (2)

F. Nobile, R. Tempone, and C. G. Webster, “A sparse grid stochastic collocation method for partial differential equations with random input data,” SIAM J. Numer. Anal. 46, 2309–2345 (2008).
[Crossref]

K. Strunz and Q. Su, “Stochastic formulation of spice-type electronic circuit simulation with polynomial chaos,” ACM Trans. Model Comput. Simul. 18, 15 (2008).
[Crossref]

2005 (1)

D. Xiu and J. S. Hesthaven, “High-order collocation methods for differential equations with random inputs,” SIAM J. Sci. Comput. 27, 1118–1139 (2005).
[Crossref]

2004 (1)

C. Soize and R. Ghanem, “Physical systems with random uncertainties: chaos representations with arbitrary probability measure,” SIAM J. Sci. Comput. 26, 395–410 (2004).
[Crossref]

2002 (1)

D. Xiu and G. E. Karniadakis, “The Wiener–Askey polynomial chaos for stochastic differential equations,” SIAM J. Sci. Comput. 24, 619–644 (2002).
[Crossref]

1994 (1)

Q. Liu and D. A. Pierce, “A note on gausshermite quadrature,” Biometrika 81, 624–629 (1994).

1977 (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. R. Stat. Soc. Series B 39, 1–38 (1977).

1969 (1)

G. H. Golub and J. H. Welsch, “Calculation of Gauss quadrature rules,” Math. Comput. 23, 221–230 (1969).
[Crossref]

Absil, P.

S. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in “2011 8th IEEE International Conference on Group IV Photonics (GFP),” (2011), pp. 71–73.

Agarwal, N.

N. Agarwal and N. Aluru, “Stochastic analysis of electrostatic MEMS subjected to parameter variations,” J. Microelectromech. Syst. 18, 1454–1468 (2009).
[Crossref]

Aluru, N.

N. Agarwal and N. Aluru, “Stochastic analysis of electrostatic MEMS subjected to parameter variations,” J. Microelectromech. Syst. 18, 1454–1468 (2009).
[Crossref]

Bergman, K.

A. Biberman and K. Bergman, “Optical interconnection networks for high-performance computing systems,” Reports on Progress in Physics 75, 046402 (2012).
[Crossref] [PubMed]

Biberman, A.

A. Biberman and K. Bergman, “Optical interconnection networks for high-performance computing systems,” Reports on Progress in Physics 75, 046402 (2012).
[Crossref] [PubMed]

Bogaerts, W.

S. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in “2011 8th IEEE International Conference on Group IV Photonics (GFP),” (2011), pp. 71–73.

Canavero, F.

D. VandeGinste, D. De Zutter, D. Deschrijver, T. Dhaene, P. Manfredi, and F. Canavero, “Stochastic modeling-based variability analysis of on-chip interconnects,” IEEE Trans. Compon. Packag. Manuf. Technol. 2, 1182–1192 (2012).
[Crossref]

Canavero, F. G.

I. S. Stievano, P. Manfredi, and F. G. Canavero, “Carbon nanotube interconnects: Process variation via polynomial chaos,” IEEE Trans. Electromagn. Compat. 54, 140–148 (2012).
[Crossref]

Cassano, D.

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in “Signal Processing in Photonic Communications,” (OSA, 2013), pp. JT3A–8.

Chen, X.

Chrostowski, L.

X. Wang, W. Shi, H. Yun, S. Grist, N. A. Jaeger, and L. Chrostowski, “Narrow-band waveguide bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20, 15547–15558 (2012).
[Crossref] [PubMed]

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Fard Talebi, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in “Optical Fiber Communication Conference,” (OSA, 2014), pp. Th2A-37.

Cunningham, J.

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

Danie, L.

Z. Zhang, X. Yang, G. Marucci, P. Maffezzoni, I. M. Elfadel, G. Karnidadakis, and L. Danie, “Stochastic testing simulator for integrated circuits and MEMS: Hierarchical and sparse techniques,” in “IEEE Custom Integrated Circuits Conference,” (2014).

Daniel, L.

Z. Zhang, T. El-Moselhy, I. Elfadel, and L. Daniel, “Stochastic testing method for transistor-level uncertainty quantification based on generalized polynomial chaos,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 32, 1533–1545 (2013).
[Crossref]

Z. Zhang, T. El-Moselhy, P. Maffezzoni, I. Elfadel, and L. Daniel, “Efficient uncertainty quantification for the periodic steady state of forced and autonomous circuits,” IEEE Trans. Circuits Syst. II: Exp. Briefs 60, 687–691 (2013).
[Crossref]

Z. Zhang, I. Elfadel, and L. Daniel, “Uncertainty quantification for integrated circuits: Stochastic spectral methods,” in “IEEE/ACM International Conference on Computer-Aided Design,” (2013), pp. 803–810.

T. El-Moselhy and L. Daniel, “Stochastic integral equation solver for efficient variation-aware interconnect extraction,” in “Proceedings of Design Automation Conference,” (ACM, 2008), pp. 415–420.

T. El-Moselhy and L. Daniel, “Variation-aware stochastic extraction with large parameter dimensionality: Review and comparison of state of the art intrusive and non-intrusive techniques,” in “International Symposium on Quality Electronic Design (ISQED),” (IEEE, 2011), pp. 1–10.

T.-W. Weng, Z. Zhang, Z. Su, and L. Daniel, “Fast stochastic simulation of silicon waveguide with non-gaussian correlated process variations,” in “Asia Communications and Photonics Conference,” (OSA, 2014), p. AF3B.7.

De Zutter, D.

D. VandeGinste, D. De Zutter, D. Deschrijver, T. Dhaene, P. Manfredi, and F. Canavero, “Stochastic modeling-based variability analysis of on-chip interconnects,” IEEE Trans. Compon. Packag. Manuf. Technol. 2, 1182–1192 (2012).
[Crossref]

Dempster, A. P.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. R. Stat. Soc. Series B 39, 1–38 (1977).

Deschrijver, D.

D. VandeGinste, D. De Zutter, D. Deschrijver, T. Dhaene, P. Manfredi, and F. Canavero, “Stochastic modeling-based variability analysis of on-chip interconnects,” IEEE Trans. Compon. Packag. Manuf. Technol. 2, 1182–1192 (2012).
[Crossref]

Dhaene, T.

D. VandeGinste, D. De Zutter, D. Deschrijver, T. Dhaene, P. Manfredi, and F. Canavero, “Stochastic modeling-based variability analysis of on-chip interconnects,” IEEE Trans. Compon. Packag. Manuf. Technol. 2, 1182–1192 (2012).
[Crossref]

Elfadel, I.

Z. Zhang, T. El-Moselhy, I. Elfadel, and L. Daniel, “Stochastic testing method for transistor-level uncertainty quantification based on generalized polynomial chaos,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 32, 1533–1545 (2013).
[Crossref]

Z. Zhang, T. El-Moselhy, P. Maffezzoni, I. Elfadel, and L. Daniel, “Efficient uncertainty quantification for the periodic steady state of forced and autonomous circuits,” IEEE Trans. Circuits Syst. II: Exp. Briefs 60, 687–691 (2013).
[Crossref]

Z. Zhang, I. Elfadel, and L. Daniel, “Uncertainty quantification for integrated circuits: Stochastic spectral methods,” in “IEEE/ACM International Conference on Computer-Aided Design,” (2013), pp. 803–810.

Elfadel, I. M.

Z. Zhang, X. Yang, G. Marucci, P. Maffezzoni, I. M. Elfadel, G. Karnidadakis, and L. Danie, “Stochastic testing simulator for integrated circuits and MEMS: Hierarchical and sparse techniques,” in “IEEE Custom Integrated Circuits Conference,” (2014).

El-Moselhy, T.

Z. Zhang, T. El-Moselhy, P. Maffezzoni, I. Elfadel, and L. Daniel, “Efficient uncertainty quantification for the periodic steady state of forced and autonomous circuits,” IEEE Trans. Circuits Syst. II: Exp. Briefs 60, 687–691 (2013).
[Crossref]

Z. Zhang, T. El-Moselhy, I. Elfadel, and L. Daniel, “Stochastic testing method for transistor-level uncertainty quantification based on generalized polynomial chaos,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 32, 1533–1545 (2013).
[Crossref]

T. El-Moselhy and L. Daniel, “Variation-aware stochastic extraction with large parameter dimensionality: Review and comparison of state of the art intrusive and non-intrusive techniques,” in “International Symposium on Quality Electronic Design (ISQED),” (IEEE, 2011), pp. 1–10.

T. El-Moselhy and L. Daniel, “Stochastic integral equation solver for efficient variation-aware interconnect extraction,” in “Proceedings of Design Automation Conference,” (ACM, 2008), pp. 415–420.

Fard Talebi, S.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Fard Talebi, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in “Optical Fiber Communication Conference,” (OSA, 2014), pp. Th2A-37.

Fernandez, L.

S. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in “2011 8th IEEE International Conference on Group IV Photonics (GFP),” (2011), pp. 71–73.

Flueckiger, J.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Fard Talebi, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in “Optical Fiber Communication Conference,” (OSA, 2014), pp. Th2A-37.

Ghanem, R.

C. Soize and R. Ghanem, “Physical systems with random uncertainties: chaos representations with arbitrary probability measure,” SIAM J. Sci. Comput. 26, 395–410 (2004).
[Crossref]

Ghanem, R. G.

R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, vol. 41 (Springer, 1991).
[Crossref]

Golub, G. H.

G. H. Golub and J. H. Welsch, “Calculation of Gauss quadrature rules,” Math. Comput. 23, 221–230 (1969).
[Crossref]

Grist, S.

Hautala, J.

S. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in “2011 8th IEEE International Conference on Group IV Photonics (GFP),” (2011), pp. 71–73.

Hesthaven, J. S.

D. Xiu and J. S. Hesthaven, “High-order collocation methods for differential equations with random inputs,” SIAM J. Sci. Comput. 27, 1118–1139 (2005).
[Crossref]

Hosseini, E. S.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref] [PubMed]

Jaeger, N. A.

Karniadakis, G. E.

D. Xiu and G. E. Karniadakis, “The Wiener–Askey polynomial chaos for stochastic differential equations,” SIAM J. Sci. Comput. 24, 619–644 (2002).
[Crossref]

Karnidadakis, G.

Z. Zhang, X. Yang, G. Marucci, P. Maffezzoni, I. M. Elfadel, G. Karnidadakis, and L. Danie, “Stochastic testing simulator for integrated circuits and MEMS: Hierarchical and sparse techniques,” in “IEEE Custom Integrated Circuits Conference,” (2014).

Krishnamoorthy, A.

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

Laird, N. M.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. R. Stat. Soc. Series B 39, 1–38 (1977).

Li, G.

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

Li, Z.

Liu, Q.

Q. Liu and D. A. Pierce, “A note on gausshermite quadrature,” Biometrika 81, 624–629 (1994).

Luo, Y.

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

Maffezzoni, P.

Z. Zhang, T. El-Moselhy, P. Maffezzoni, I. Elfadel, and L. Daniel, “Efficient uncertainty quantification for the periodic steady state of forced and autonomous circuits,” IEEE Trans. Circuits Syst. II: Exp. Briefs 60, 687–691 (2013).
[Crossref]

Z. Zhang, X. Yang, G. Marucci, P. Maffezzoni, I. M. Elfadel, G. Karnidadakis, and L. Danie, “Stochastic testing simulator for integrated circuits and MEMS: Hierarchical and sparse techniques,” in “IEEE Custom Integrated Circuits Conference,” (2014).

Manfredi, P.

I. S. Stievano, P. Manfredi, and F. G. Canavero, “Carbon nanotube interconnects: Process variation via polynomial chaos,” IEEE Trans. Electromagn. Compat. 54, 140–148 (2012).
[Crossref]

D. VandeGinste, D. De Zutter, D. Deschrijver, T. Dhaene, P. Manfredi, and F. Canavero, “Stochastic modeling-based variability analysis of on-chip interconnects,” IEEE Trans. Compon. Packag. Manuf. Technol. 2, 1182–1192 (2012).
[Crossref]

Marucci, G.

Z. Zhang, X. Yang, G. Marucci, P. Maffezzoni, I. M. Elfadel, G. Karnidadakis, and L. Danie, “Stochastic testing simulator for integrated circuits and MEMS: Hierarchical and sparse techniques,” in “IEEE Custom Integrated Circuits Conference,” (2014).

Mekis, A.

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

Melloni, A.

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in “Signal Processing in Photonic Communications,” (OSA, 2013), pp. JT3A–8.

Mickelson, A. R.

Mohamed, M.

Morichetti, F.

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in “Signal Processing in Photonic Communications,” (OSA, 2013), pp. JT3A–8.

Nobile, F.

F. Nobile, R. Tempone, and C. G. Webster, “A sparse grid stochastic collocation method for partial differential equations with random input data,” SIAM J. Numer. Anal. 46, 2309–2345 (2008).
[Crossref]

Pierce, D. A.

Q. Liu and D. A. Pierce, “A note on gausshermite quadrature,” Biometrika 81, 624–629 (1994).

Pinguet, T.

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

Raj, K.

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

Rosseel, E.

S. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in “2011 8th IEEE International Conference on Group IV Photonics (GFP),” (2011), pp. 71–73.

Rubin, D. B.

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. R. Stat. Soc. Series B 39, 1–38 (1977).

Rutenbar, R. A.

A. Singhee and R. A. Rutenbar, “Why quasi-Monte Carlo is better than Monte Carlo or latin hypercube sampling for statistical circuit analysis,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 29, 1763–1776 (2010).
[Crossref]

Selvaraja, S.

S. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in “2011 8th IEEE International Conference on Group IV Photonics (GFP),” (2011), pp. 71–73.

Shang, L.

Shi, W.

Shubin, I.

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

Singhee, A.

A. Singhee and R. A. Rutenbar, “Why quasi-Monte Carlo is better than Monte Carlo or latin hypercube sampling for statistical circuit analysis,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 29, 1763–1776 (2010).
[Crossref]

Soize, C.

C. Soize and R. Ghanem, “Physical systems with random uncertainties: chaos representations with arbitrary probability measure,” SIAM J. Sci. Comput. 26, 395–410 (2004).
[Crossref]

Spanos, P. D.

R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, vol. 41 (Springer, 1991).
[Crossref]

Stievano, I. S.

I. S. Stievano, P. Manfredi, and F. G. Canavero, “Carbon nanotube interconnects: Process variation via polynomial chaos,” IEEE Trans. Electromagn. Compat. 54, 140–148 (2012).
[Crossref]

Strunz, K.

K. Strunz and Q. Su, “Stochastic formulation of spice-type electronic circuit simulation with polynomial chaos,” ACM Trans. Model Comput. Simul. 18, 15 (2008).
[Crossref]

Su, Q.

K. Strunz and Q. Su, “Stochastic formulation of spice-type electronic circuit simulation with polynomial chaos,” ACM Trans. Model Comput. Simul. 18, 15 (2008).
[Crossref]

Su, Z.

T.-W. Weng, Z. Zhang, Z. Su, and L. Daniel, “Fast stochastic simulation of silicon waveguide with non-gaussian correlated process variations,” in “Asia Communications and Photonics Conference,” (OSA, 2014), p. AF3B.7.

Sun, J.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref] [PubMed]

Tabat, M.

S. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in “2011 8th IEEE International Conference on Group IV Photonics (GFP),” (2011), pp. 71–73.

Tempone, R.

F. Nobile, R. Tempone, and C. G. Webster, “A sparse grid stochastic collocation method for partial differential equations with random input data,” SIAM J. Numer. Anal. 46, 2309–2345 (2008).
[Crossref]

Thacker, H.

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

Timurdogan, E.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref] [PubMed]

Trotter, D. C.

VandeGinste, D.

D. VandeGinste, D. De Zutter, D. Deschrijver, T. Dhaene, P. Manfredi, and F. Canavero, “Stochastic modeling-based variability analysis of on-chip interconnects,” IEEE Trans. Compon. Packag. Manuf. Technol. 2, 1182–1192 (2012).
[Crossref]

Wang, X.

X. Wang, W. Shi, H. Yun, S. Grist, N. A. Jaeger, and L. Chrostowski, “Narrow-band waveguide bragg gratings on SOI wafers with CMOS-compatible fabrication process,” Opt. Express 20, 15547–15558 (2012).
[Crossref] [PubMed]

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Fard Talebi, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in “Optical Fiber Communication Conference,” (OSA, 2014), pp. Th2A-37.

Wang, Y.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Fard Talebi, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in “Optical Fiber Communication Conference,” (OSA, 2014), pp. Th2A-37.

Watts, M. R.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref] [PubMed]

W. A. Zortman, D. C. Trotter, and M. R. Watts, “Silicon photonics manufacturing,” Opt. Express 18, 23598–23607 (2010).
[Crossref] [PubMed]

Webster, C. G.

F. Nobile, R. Tempone, and C. G. Webster, “A sparse grid stochastic collocation method for partial differential equations with random input data,” SIAM J. Numer. Anal. 46, 2309–2345 (2008).
[Crossref]

Weinzierl, S.

S. Weinzierl, “Introduction to Monte Carlo methods,” arXiv preprint hep-ph/0006269 (2000).

Welsch, J. H.

G. H. Golub and J. H. Welsch, “Calculation of Gauss quadrature rules,” Math. Comput. 23, 221–230 (1969).
[Crossref]

Weng, T.-W.

T.-W. Weng, Z. Zhang, Z. Su, and L. Daniel, “Fast stochastic simulation of silicon waveguide with non-gaussian correlated process variations,” in “Asia Communications and Photonics Conference,” (OSA, 2014), p. AF3B.7.

Wilf, H.

H. Wilf, Mathematics for the Physical Sciences, Problem 9 (Wiley, 1962).

Wu, Y.

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Fard Talebi, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in “Optical Fiber Communication Conference,” (OSA, 2014), pp. Th2A-37.

Xiu, D.

D. Xiu and J. S. Hesthaven, “High-order collocation methods for differential equations with random inputs,” SIAM J. Sci. Comput. 27, 1118–1139 (2005).
[Crossref]

D. Xiu and G. E. Karniadakis, “The Wiener–Askey polynomial chaos for stochastic differential equations,” SIAM J. Sci. Comput. 24, 619–644 (2002).
[Crossref]

Yaacobi, A.

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref] [PubMed]

Yang, X.

Z. Zhang, X. Yang, G. Marucci, P. Maffezzoni, I. M. Elfadel, G. Karnidadakis, and L. Danie, “Stochastic testing simulator for integrated circuits and MEMS: Hierarchical and sparse techniques,” in “IEEE Custom Integrated Circuits Conference,” (2014).

Yao, J.

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

Yun, H.

Zhang, Z.

Z. Zhang, T. El-Moselhy, I. Elfadel, and L. Daniel, “Stochastic testing method for transistor-level uncertainty quantification based on generalized polynomial chaos,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 32, 1533–1545 (2013).
[Crossref]

Z. Zhang, T. El-Moselhy, P. Maffezzoni, I. Elfadel, and L. Daniel, “Efficient uncertainty quantification for the periodic steady state of forced and autonomous circuits,” IEEE Trans. Circuits Syst. II: Exp. Briefs 60, 687–691 (2013).
[Crossref]

Z. Zhang, I. Elfadel, and L. Daniel, “Uncertainty quantification for integrated circuits: Stochastic spectral methods,” in “IEEE/ACM International Conference on Computer-Aided Design,” (2013), pp. 803–810.

Z. Zhang, X. Yang, G. Marucci, P. Maffezzoni, I. M. Elfadel, G. Karnidadakis, and L. Danie, “Stochastic testing simulator for integrated circuits and MEMS: Hierarchical and sparse techniques,” in “IEEE Custom Integrated Circuits Conference,” (2014).

T.-W. Weng, Z. Zhang, Z. Su, and L. Daniel, “Fast stochastic simulation of silicon waveguide with non-gaussian correlated process variations,” in “Asia Communications and Photonics Conference,” (OSA, 2014), p. AF3B.7.

Zheng, X.

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

Zortman, W. A.

ACM Trans. Model Comput. Simul. (1)

K. Strunz and Q. Su, “Stochastic formulation of spice-type electronic circuit simulation with polynomial chaos,” ACM Trans. Model Comput. Simul. 18, 15 (2008).
[Crossref]

Appl. Opt. (1)

Biometrika (1)

Q. Liu and D. A. Pierce, “A note on gausshermite quadrature,” Biometrika 81, 624–629 (1994).

IEEE Photon. J. (1)

A. Krishnamoorthy, X. Zheng, G. Li, J. Yao, T. Pinguet, A. Mekis, H. Thacker, I. Shubin, Y. Luo, K. Raj, and J. Cunningham, “Exploiting CMOS manufacturing to reduce tuning requirements for resonant optical devices,” IEEE Photon. J. 3, 567–579 (2011).
[Crossref]

IEEE Trans. Circuits Syst. II: Exp. Briefs (1)

Z. Zhang, T. El-Moselhy, P. Maffezzoni, I. Elfadel, and L. Daniel, “Efficient uncertainty quantification for the periodic steady state of forced and autonomous circuits,” IEEE Trans. Circuits Syst. II: Exp. Briefs 60, 687–691 (2013).
[Crossref]

IEEE Trans. Compon. Packag. Manuf. Technol. (1)

D. VandeGinste, D. De Zutter, D. Deschrijver, T. Dhaene, P. Manfredi, and F. Canavero, “Stochastic modeling-based variability analysis of on-chip interconnects,” IEEE Trans. Compon. Packag. Manuf. Technol. 2, 1182–1192 (2012).
[Crossref]

IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. (2)

Z. Zhang, T. El-Moselhy, I. Elfadel, and L. Daniel, “Stochastic testing method for transistor-level uncertainty quantification based on generalized polynomial chaos,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 32, 1533–1545 (2013).
[Crossref]

A. Singhee and R. A. Rutenbar, “Why quasi-Monte Carlo is better than Monte Carlo or latin hypercube sampling for statistical circuit analysis,” IEEE Trans. Comput.-Aided Design Integr. Circuits Syst. 29, 1763–1776 (2010).
[Crossref]

IEEE Trans. Electromagn. Compat. (1)

I. S. Stievano, P. Manfredi, and F. G. Canavero, “Carbon nanotube interconnects: Process variation via polynomial chaos,” IEEE Trans. Electromagn. Compat. 54, 140–148 (2012).
[Crossref]

J. Microelectromech. Syst. (1)

N. Agarwal and N. Aluru, “Stochastic analysis of electrostatic MEMS subjected to parameter variations,” J. Microelectromech. Syst. 18, 1454–1468 (2009).
[Crossref]

J. R. Stat. Soc. Series B (1)

A. P. Dempster, N. M. Laird, and D. B. Rubin, “Maximum likelihood from incomplete data via the em algorithm,” J. R. Stat. Soc. Series B 39, 1–38 (1977).

Math. Comput. (1)

G. H. Golub and J. H. Welsch, “Calculation of Gauss quadrature rules,” Math. Comput. 23, 221–230 (1969).
[Crossref]

Nature (1)

J. Sun, E. Timurdogan, A. Yaacobi, E. S. Hosseini, and M. R. Watts, “Large-scale nanophotonic phased array,” Nature 493, 195–199 (2013).
[Crossref] [PubMed]

Opt. Express (2)

Reports on Progress in Physics (1)

A. Biberman and K. Bergman, “Optical interconnection networks for high-performance computing systems,” Reports on Progress in Physics 75, 046402 (2012).
[Crossref] [PubMed]

SIAM J. Numer. Anal. (1)

F. Nobile, R. Tempone, and C. G. Webster, “A sparse grid stochastic collocation method for partial differential equations with random input data,” SIAM J. Numer. Anal. 46, 2309–2345 (2008).
[Crossref]

SIAM J. Sci. Comput. (3)

C. Soize and R. Ghanem, “Physical systems with random uncertainties: chaos representations with arbitrary probability measure,” SIAM J. Sci. Comput. 26, 395–410 (2004).
[Crossref]

D. Xiu and J. S. Hesthaven, “High-order collocation methods for differential equations with random inputs,” SIAM J. Sci. Comput. 27, 1118–1139 (2005).
[Crossref]

D. Xiu and G. E. Karniadakis, “The Wiener–Askey polynomial chaos for stochastic differential equations,” SIAM J. Sci. Comput. 24, 619–644 (2002).
[Crossref]

Other (11)

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in “Signal Processing in Photonic Communications,” (OSA, 2013), pp. JT3A–8.

R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, vol. 41 (Springer, 1991).
[Crossref]

Z. Zhang, X. Yang, G. Marucci, P. Maffezzoni, I. M. Elfadel, G. Karnidadakis, and L. Danie, “Stochastic testing simulator for integrated circuits and MEMS: Hierarchical and sparse techniques,” in “IEEE Custom Integrated Circuits Conference,” (2014).

S. Weinzierl, “Introduction to Monte Carlo methods,” arXiv preprint hep-ph/0006269 (2000).

L. Chrostowski, X. Wang, J. Flueckiger, Y. Wu, Y. Wang, and S. Fard Talebi, “Impact of fabrication non-uniformity on chip-scale silicon photonic integrated circuits,” in “Optical Fiber Communication Conference,” (OSA, 2014), pp. Th2A-37.

S. Selvaraja, E. Rosseel, L. Fernandez, M. Tabat, W. Bogaerts, J. Hautala, and P. Absil, “SOI thickness uniformity improvement using corrective etching for silicon nano-photonic device,” in “2011 8th IEEE International Conference on Group IV Photonics (GFP),” (2011), pp. 71–73.

Z. Zhang, I. Elfadel, and L. Daniel, “Uncertainty quantification for integrated circuits: Stochastic spectral methods,” in “IEEE/ACM International Conference on Computer-Aided Design,” (2013), pp. 803–810.

T.-W. Weng, Z. Zhang, Z. Su, and L. Daniel, “Fast stochastic simulation of silicon waveguide with non-gaussian correlated process variations,” in “Asia Communications and Photonics Conference,” (OSA, 2014), p. AF3B.7.

T. El-Moselhy and L. Daniel, “Stochastic integral equation solver for efficient variation-aware interconnect extraction,” in “Proceedings of Design Automation Conference,” (ACM, 2008), pp. 415–420.

T. El-Moselhy and L. Daniel, “Variation-aware stochastic extraction with large parameter dimensionality: Review and comparison of state of the art intrusive and non-intrusive techniques,” in “International Symposium on Quality Electronic Design (ISQED),” (IEEE, 2011), pp. 1–10.

H. Wilf, Mathematics for the Physical Sciences, Problem 9 (Wiley, 1962).

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Figures (7)

Algorithm 1
Algorithm 1 Univariate basis function construction and Quadrature points computation

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Algorithm 2
Algorithm 2 Compute gPCe coefficients with Stochastic collocation

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Fig. 1
Fig. 1 The cross section of a SOI-based directional coupler with nominal width W0, nominal gap g0, height H0, and refractive indices nSi = 3.48,nSiO2 = 1.445.

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Fig. 2
Fig. 2 Top plot: Histograms of ΔW in the example. Bottom plot: Histogram of Δg in the example.

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Fig. 3
Fig. 3 The simulated PDF of δ (λ, ΔWe,ΔWi) with λ = 1.55 μm. The solid line is the Stochastic Collocation (SC) result, whereas the dash line represents Monte Carlo (MC) result. The nominal value of δ is 0.11.

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Fig. 4
Fig. 4 The simulated pdf of power coupling coefficient K(z). The solid line is the Stochastic Collocation (SC) result, whereas the dash line represents Monte Carlo (MC) result.

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Fig. 5
Fig. 5 The truncation error in terms of the level m.

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Tables (1)

Tables Icon

Table 1 Performance summary of Stochastic Collocation and Monte Carlo simulation

View Table

Equations (33)

Equations on this page are rendered with MathJax. Learn more.

p ξ → ( ξ → ) = 0 , if ξ → ∉ Ξ .
p i ( ξ i ) = ∫ Ξ p ξ → ( ξ 1 , ⋯ , ξ N ) d ξ 1 ⋯ d ξ i − 1 d ξ i + 1 ⋯ d ξ N .
〈 ϕ k ( i ) , ϕ k ′ ( i ) 〉 p i = ∫ ℝ ϕ k ( i ) ( ξ i ) ϕ k ′ ( i ) ( ξ i ) p i ( ξ i ) d ξ i = | | ϕ k ( i ) 2 | | δ k k ′
Ψ α → ( ξ → ) = [ p 1 ( ξ 1 ) ⋯ p N ( ξ N ) p ξ → ( ξ → ) ] 1 / 2 ϕ α 1 ( 1 ) ( ξ 1 ) ⋯ ϕ α N ( N ) ( ξ N )
[ ξ 1 ξ 2 ] ~ a · N ( μ → A , Σ A ) + b · N ( μ → B , Σ B )
a + b = 1 , 0 ≤ a , b ≤ 1
μ A = [ μ A 1 μ A 2 ] , Σ A = [ σ A 1 2 ρ A σ A 1 σ A 2 ρ A σ A 1 σ A 2 σ A 2 2 ] μ B = [ μ B 1 μ B 2 ] , Σ B = [ σ B 1 2 ρ B σ B 1 σ B 2 ρ B σ B 1 σ B 2 σ B 2 2 ]
ξ i ~ a · N ( μ A i , σ A i 2 ) + b · N ( μ B i , σ B i 2 ) ,
p i ( ξ i ) = a · exp ( ( ξ i − μ A i ) 2 − 2 σ A i 2 ) 2 π σ A i 2 + b · exp ( ( ξ i − μ B i ) 2 − 2 σ B i 2 ) 2 π σ B i 2 .
u ( x , ξ → ) ≈ ∑ | | α → | | ∞ ≤ t → C a → ( x ) Ψ α → ( ξ 1 , ⋯ , ξ N )
ϕ k + 1 ( ξ ) = ( ξ − a k ) ϕ k ( ξ ) + b k ϕ k − 1 ( ξ ) , k ≥ 1
a k = 〈 ξ ϕ k , ϕ k 〉 p 〈 ϕ k , ϕ k 〉 p
b k = 〈 ϕ k , ϕ k 〉 p 〈 ϕ k − 1 , ϕ k − 1 〉 p .
〈 ξ ϕ k , ϕ k 〉 p = ∫ ℝ ξ ϕ k 2 ( ξ ) p ( ξ ) d ξ
a ∫ ℝ ( μ A i + σ A i y i ) ϕ k 2 ( μ A i + σ A i y i ) exp ( y i 2 − 2 ) 2 π d y i + b ∫ ℝ ( μ B i + σ B i z i ) ϕ k 2 ( μ B i + σ B i z i ) exp ( z i 2 − 2 ) 2 π d z i .
a ∑ j = 1 q w j ( μ A i + σ A i x j ) ϕ k 2 ( μ A i + σ A i x j ) + b ∑ j = 1 q w j ( μ B i + σ B i x j ) ϕ k 2 ( μ B i + σ B i x j )
〈 ϕ k , ϕ k 〉 p = ∫ ℝ ϕ k 2 ( ξ ) p ( ξ ) d ξ ≈ a ∑ j = 1 q w j ϕ k 2 ( μ A i + σ A i x j ) + b ∑ j = 1 q w j ϕ k 2 ( μ B i + σ B i x j )
C α → ( x ) = 〈 u ( x , ξ → ) , Ψ α → ( ξ → ) 〉 p ξ → 〈 Ψ α → ( ξ → ) , Ψ α → ( ξ → ) 〉 p ξ → = 〈 u ( x , ξ → ) , Ψ α → ( ξ → ) 〉 p ξ → | | Ψ α → ( ξ → ) | | 2
∫ Ξ u ( x , ξ → ) [ p 1 ( ξ 1 ) ⋯ p N ( ξ N ) p ξ → ( ξ → ) ] 1 / 2 ∏ i = 1 N ϕ α i ( i ) ( ξ i ) p ξ → ( ξ → ) d ξ →
g ( x , ξ → ) = u ( x , ξ → ) [ p ξ → ( ξ → ) p 1 ( ξ 1 ) ⋯ p N ( ξ N ) ] 1 / 2 ∏ i = 1 N ϕ α i ( i ) ( ξ i ) .
∫ Ξ g ( x , ξ 1 , ⋯ , ξ N ) p 1 ( ξ 1 ) ⋯ p N ( ξ N ) d ξ 1 ⋯ d ξ N
∑ i 1 = 1 q 1 ∑ i 2 = 1 q 2 ⋯ ∑ i N = 1 q N g ( x , ξ 1 ( i 1 ) , ⋯ , ξ N ( i N ) ) w 1 ( i 1 ) ⋯ w N ( i N )
J m k = { a m − 1 , if k = m , m = 1 , … , n b m − 1 , if k = m − 1 , m = 2 , … , n b m , if k = m + 1 , m = 1 , … , n − 1
w j ( i j ) = μ j ( v j ( i j , 1 ) ) 2
W = W 0 + Δ W e + Δ W i , g = g 0 − 2 Δ W i
K ( z ) = sin 2 ( δ z )
[ Δ W e Δ W i ] ~ 0.6 · N ( μ → A , Σ A ) + 0.4 · N ( μ → B , Σ B )
μ A = [ 9 6 ] nm, Σ A = [ 6 0 0 3 ] nm 2
μ B = [ 8 7 ] nm, Σ B = [ 5 1 1 4 ] nm 2 .
q i = 2 m i − 1
〈 Ψ α → , Ψ γ → 〉 p ξ → = ∫ Ξ Ψ α → ( ξ → ) Ψ γ → ( ξ → ) p ξ → ( ξ → ) d ξ → .
〈 Ψ α → ( ξ → ) , Ψ γ → ( ξ → ) 〉 p ξ → = ∏ i = 1 N ∫ ℝ ϕ α i ( i ) ( ξ i ) ϕ γ i ( i ) ( ξ i ) d ξ i = ∏ i = 1 N | | ϕ α i ( i ) ( ξ i ) | | 2 δ α i γ i = | | Ψ α → ( ξ → ) | | 2 δ α → γ → .
δ α → γ → = { 1 , if α → = γ → , 0 otherwise .

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