Abstract

It is well-known that geometrical variations due to manufacturing tolerances can degrade the performance of optical devices. In recent literature, polynomial chaos expansion (PCE) methods were proposed to model this statistical behavior. Nonetheless, traditional PCE solvers require a lot of memory and their computational complexity leads to prohibitively long simulation times, making these methods non-tractable for large optical systems. The uncertainty quantification (UQ) of various types of large, two-dimensional lens systems is presented in this paper, leveraging a novel parallelized Multilevel Fast Multipole Method (MLFMM) based Stochastic Galerkin Method (SGM). It is demonstrated that this technique can handle large optical structures in reasonable time, e.g., a stochastic lens system with more than 10 million unknowns was solved in less than an hour by using 3 compute nodes. The SGM, which is an intrusive PCE method, guarantees the accuracy of the method. By conjunction with MLFMM, usage of a preconditioner and by constructing and implementing a parallelized algorithm, a high efficiency is achieved. This is demonstrated with parallel scalability graphs. The novel approach is illustrated for different types of lens system and numerical results are validated against a collocation method, which relies on reusing a traditional deterministic solver. The last example concerns a Cassegrain system with five random variables, for which a speed-up of more than 12× compared to a collocation method is achieved.

© 2015 Optical Society of America

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References

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  1. 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]
  2. D. B. Xiu and G. E. Karniadakis, “The Wiener-Askey polynomial chaos for stochastic differential equations,” SIAM J. Sci. Comput. 24, 619–644 (2002).
    [Crossref]
  3. 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]
  4. 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 technique,” in 2011 12th International Symposium on Quality Electronic Design (ISQED 2011), 14–16 March 2011, Santa Clara, CA, USA, (2011), pp. 508–517.
  5. C. Chauvière, J. S. Hesthaven, and L. C. Wilcox, “Efficient computation of RCS from scatterers of uncertain shapes,” IEEE Trans. Antennas Propag. 55, 1437–1448 (2007).
    [Crossref]
  6. Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Method of Moments with the Stochastic Galerkin Method,” IEEE Trans. Antennas Propag. 62, 4852–4856 (2014).
    [Crossref]
  7. Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Multilevel Fast Multipole Method (MLFMM) with the Stochastic Galerkin Method (SGM),” IEEE Antennas Wireless Propag. Lett. 13, 1275–1278 (2014).
    [Crossref]
  8. T.-W. Weng, Z. Zhang, Z. Su, Y. Marzouk, A. Melloni, and L. Daniel, “Uncertainty quantification of silicon photonic devices with correlated and non-Gaussian random parameters,” Opt. Express 23, 4242–4254 (2015).
    [Crossref] [PubMed]
  9. R. F. Harrington, Field Computation by Moment Methods (IEEE Press, 1993).
    [Crossref]
  10. A. Keese and H. G. Matthies, “Hierarchical parallelisation for the solution of stochastic finite element equations,” Comput. Struct. 83, 1033–1047 (2005).
    [Crossref]
  11. J. Fostier and F. Olyslager, “An asynchronous parallel MLFMA for scattering at multiple dielectric objects,” IEEE Trans. Antennas Propag. 56, 2346–2355 (2008).
    [Crossref]
  12. A. J. Poggio and E. K. Miller, “Integral equation solutions for three dimensional scattering problems,” in Computer techniques for electromagnetics, R. Mittra, ed. (Pergamon Press, 1973), pp. 159–264.
    [Crossref]
  13. F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microwave Theory Tech. 41, 79–88 (1993).
    [Crossref]
  14. A. Biondi, D. VandeGinste, D. De Zutter, P. Manfredi, and F. Canavero, “Variability analysis of interconnects terminated by general nonlinear loads,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1244–1251 (2013).
    [Crossref]
  15. P. Manfredi, D. Van de Ginste, D. De Zutter, and F. Canavero, “Uncertainty assessment of lossy and dispersive lines in spice-type environments,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1252–1258 (2013).
    [Crossref]
  16. W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).
  17. D. VandeGinste, H. Rogier, F. Olyslager, and D. De Zutter, “A fast multipole method for layered media based on the application of perfectly matched layers–the 2-D case,” IEEE Trans. Antennas Propag. 52, 2631–2640 (2004).
    [Crossref]
  18. D. VandeGinste, E. Michielssen, F. Olyslager, and D. De Zutter, “An efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures,” IEEE Trans. Antennas Propag. 54, 1538–1548 (2006).
    [Crossref]
  19. D. Pissoort, E. Michielssen, D. Vande Ginste, and F. Olyslager, “Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs,” J. Lightwave Technol. 25, 2847–2863 (2007).
    [Crossref]
  20. B. Michiels, J. Fostier, I. Bogaert, and D. De Zutter, “Full-Wave Simulations of Electromagnetic Scattering Problems With Billions of Unknowns,” IEEE Trans. Antennas Propag. 63, 796–799 (2015).
    [Crossref]
  21. R. W. Freund, “A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems,”SIAM J. Sci. Comput. 14, 470–482 (1993).
    [Crossref]
  22. R. D. da Cunha and T. Hopkins, PIM 2.2 The Parallel Iterative Methods Package for Systems of Linear Equations User’s Guide (Fortran 77 version), UKC, University of Kent, Canterbury, UK.
  23. Ö. Ergül and L. Gürel, “Efficient parallelization of the multilevel fast multipole algorithm for the solution of large-scale scattering problems,” IEEE Trans. Antennas Propag. 56, 2335–2345 (2008).
    [Crossref]
  24. J. Fostier and F. Olyslager, “A provably scalable parallel multilevel fast multipole algorithm,” Electron. Lett. 44, 1111–1113 (2008).
    [Crossref]
  25. D. Pissoort, E. Michielssen, D. Vande Ginste, and F. Olyslager, “Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs,” J. Lightwave Technol. 25, 2847–2863 (2007).
    [Crossref]
  26. M. F. Pellissetti and R. G. Ghanem, “Iterative solution of systems of linear equations arising in the context of stochastic finite elements,” Adv. Eng. Softw. 31, 607–616 (2000).
    [Crossref]
  27. I. Ocket, B. Nauwelaers, J. Fostier, L. Meert, F. Olyslager, G. Koers, J. Stiens, R. Vounckx, and I. Jager, “Characterization of speckle/despeckling in active millimeter wave imaging systems using a first order 1.5D model,” in “Millimeter-wave and Terahertz Photonics”, D. Jäger and A. Stöhr, eds., Proc. SPIE6194, 19409–19420 (2006).
  28. J. Fostier and F. Olyslager, “An open-source implementation for full-wave 2D scattering by million-wavelength-size objects,” IEEE Antennas Propag. Mag. 52, 23–34 (2010).
    [Crossref]

2015 (2)

T.-W. Weng, Z. Zhang, Z. Su, Y. Marzouk, A. Melloni, and L. Daniel, “Uncertainty quantification of silicon photonic devices with correlated and non-Gaussian random parameters,” Opt. Express 23, 4242–4254 (2015).
[Crossref] [PubMed]

B. Michiels, J. Fostier, I. Bogaert, and D. De Zutter, “Full-Wave Simulations of Electromagnetic Scattering Problems With Billions of Unknowns,” IEEE Trans. Antennas Propag. 63, 796–799 (2015).
[Crossref]

2014 (2)

Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Method of Moments with the Stochastic Galerkin Method,” IEEE Trans. Antennas Propag. 62, 4852–4856 (2014).
[Crossref]

Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Multilevel Fast Multipole Method (MLFMM) with the Stochastic Galerkin Method (SGM),” IEEE Antennas Wireless Propag. Lett. 13, 1275–1278 (2014).
[Crossref]

2013 (3)

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]

A. Biondi, D. VandeGinste, D. De Zutter, P. Manfredi, and F. Canavero, “Variability analysis of interconnects terminated by general nonlinear loads,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1244–1251 (2013).
[Crossref]

P. Manfredi, D. Van de Ginste, D. De Zutter, and F. Canavero, “Uncertainty assessment of lossy and dispersive lines in spice-type environments,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1252–1258 (2013).
[Crossref]

2012 (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]

2010 (1)

J. Fostier and F. Olyslager, “An open-source implementation for full-wave 2D scattering by million-wavelength-size objects,” IEEE Antennas Propag. Mag. 52, 23–34 (2010).
[Crossref]

2008 (3)

Ö. Ergül and L. Gürel, “Efficient parallelization of the multilevel fast multipole algorithm for the solution of large-scale scattering problems,” IEEE Trans. Antennas Propag. 56, 2335–2345 (2008).
[Crossref]

J. Fostier and F. Olyslager, “A provably scalable parallel multilevel fast multipole algorithm,” Electron. Lett. 44, 1111–1113 (2008).
[Crossref]

J. Fostier and F. Olyslager, “An asynchronous parallel MLFMA for scattering at multiple dielectric objects,” IEEE Trans. Antennas Propag. 56, 2346–2355 (2008).
[Crossref]

2007 (3)

2006 (1)

D. VandeGinste, E. Michielssen, F. Olyslager, and D. De Zutter, “An efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures,” IEEE Trans. Antennas Propag. 54, 1538–1548 (2006).
[Crossref]

2005 (1)

A. Keese and H. G. Matthies, “Hierarchical parallelisation for the solution of stochastic finite element equations,” Comput. Struct. 83, 1033–1047 (2005).
[Crossref]

2004 (1)

D. VandeGinste, H. Rogier, F. Olyslager, and D. De Zutter, “A fast multipole method for layered media based on the application of perfectly matched layers–the 2-D case,” IEEE Trans. Antennas Propag. 52, 2631–2640 (2004).
[Crossref]

2002 (1)

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

2000 (1)

M. F. Pellissetti and R. G. Ghanem, “Iterative solution of systems of linear equations arising in the context of stochastic finite elements,” Adv. Eng. Softw. 31, 607–616 (2000).
[Crossref]

1993 (2)

F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microwave Theory Tech. 41, 79–88 (1993).
[Crossref]

R. W. Freund, “A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems,”SIAM J. Sci. Comput. 14, 470–482 (1993).
[Crossref]

Biondi, A.

A. Biondi, D. VandeGinste, D. De Zutter, P. Manfredi, and F. Canavero, “Variability analysis of interconnects terminated by general nonlinear loads,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1244–1251 (2013).
[Crossref]

Blomme, K.

F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microwave Theory Tech. 41, 79–88 (1993).
[Crossref]

Bogaert, I.

B. Michiels, J. Fostier, I. Bogaert, and D. De Zutter, “Full-Wave Simulations of Electromagnetic Scattering Problems With Billions of Unknowns,” IEEE Trans. Antennas Propag. 63, 796–799 (2015).
[Crossref]

Canavero, F.

P. Manfredi, D. Van de Ginste, D. De Zutter, and F. Canavero, “Uncertainty assessment of lossy and dispersive lines in spice-type environments,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1252–1258 (2013).
[Crossref]

A. Biondi, D. VandeGinste, D. De Zutter, P. Manfredi, and F. Canavero, “Variability analysis of interconnects terminated by general nonlinear loads,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1244–1251 (2013).
[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]

Chauvière, C.

C. Chauvière, J. S. Hesthaven, and L. C. Wilcox, “Efficient computation of RCS from scatterers of uncertain shapes,” IEEE Trans. Antennas Propag. 55, 1437–1448 (2007).
[Crossref]

Chen, X.

Chew, W. C.

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).

da Cunha, R. D.

R. D. da Cunha and T. Hopkins, PIM 2.2 The Parallel Iterative Methods Package for Systems of Linear Equations User’s Guide (Fortran 77 version), UKC, University of Kent, Canterbury, UK.

Daniel, L.

T.-W. Weng, Z. Zhang, Z. Su, Y. Marzouk, A. Melloni, and L. Daniel, “Uncertainty quantification of silicon photonic devices with correlated and non-Gaussian random parameters,” Opt. Express 23, 4242–4254 (2015).
[Crossref] [PubMed]

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 technique,” in 2011 12th International Symposium on Quality Electronic Design (ISQED 2011), 14–16 March 2011, Santa Clara, CA, USA, (2011), pp. 508–517.

De Zutter, D.

B. Michiels, J. Fostier, I. Bogaert, and D. De Zutter, “Full-Wave Simulations of Electromagnetic Scattering Problems With Billions of Unknowns,” IEEE Trans. Antennas Propag. 63, 796–799 (2015).
[Crossref]

Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Method of Moments with the Stochastic Galerkin Method,” IEEE Trans. Antennas Propag. 62, 4852–4856 (2014).
[Crossref]

Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Multilevel Fast Multipole Method (MLFMM) with the Stochastic Galerkin Method (SGM),” IEEE Antennas Wireless Propag. Lett. 13, 1275–1278 (2014).
[Crossref]

A. Biondi, D. VandeGinste, D. De Zutter, P. Manfredi, and F. Canavero, “Variability analysis of interconnects terminated by general nonlinear loads,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1244–1251 (2013).
[Crossref]

P. Manfredi, D. Van de Ginste, D. De Zutter, and F. Canavero, “Uncertainty assessment of lossy and dispersive lines in spice-type environments,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1252–1258 (2013).
[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]

D. VandeGinste, E. Michielssen, F. Olyslager, and D. De Zutter, “An efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures,” IEEE Trans. Antennas Propag. 54, 1538–1548 (2006).
[Crossref]

D. VandeGinste, H. Rogier, F. Olyslager, and D. De Zutter, “A fast multipole method for layered media based on the application of perfectly matched layers–the 2-D case,” IEEE Trans. Antennas Propag. 52, 2631–2640 (2004).
[Crossref]

F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microwave Theory Tech. 41, 79–88 (1993).
[Crossref]

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]

El-Moselhy, T.

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 technique,” in 2011 12th International Symposium on Quality Electronic Design (ISQED 2011), 14–16 March 2011, Santa Clara, CA, USA, (2011), pp. 508–517.

Ergül, Ö.

Ö. Ergül and L. Gürel, “Efficient parallelization of the multilevel fast multipole algorithm for the solution of large-scale scattering problems,” IEEE Trans. Antennas Propag. 56, 2335–2345 (2008).
[Crossref]

Fostier, J.

B. Michiels, J. Fostier, I. Bogaert, and D. De Zutter, “Full-Wave Simulations of Electromagnetic Scattering Problems With Billions of Unknowns,” IEEE Trans. Antennas Propag. 63, 796–799 (2015).
[Crossref]

J. Fostier and F. Olyslager, “An open-source implementation for full-wave 2D scattering by million-wavelength-size objects,” IEEE Antennas Propag. Mag. 52, 23–34 (2010).
[Crossref]

J. Fostier and F. Olyslager, “A provably scalable parallel multilevel fast multipole algorithm,” Electron. Lett. 44, 1111–1113 (2008).
[Crossref]

J. Fostier and F. Olyslager, “An asynchronous parallel MLFMA for scattering at multiple dielectric objects,” IEEE Trans. Antennas Propag. 56, 2346–2355 (2008).
[Crossref]

I. Ocket, B. Nauwelaers, J. Fostier, L. Meert, F. Olyslager, G. Koers, J. Stiens, R. Vounckx, and I. Jager, “Characterization of speckle/despeckling in active millimeter wave imaging systems using a first order 1.5D model,” in “Millimeter-wave and Terahertz Photonics”, D. Jäger and A. Stöhr, eds., Proc. SPIE6194, 19409–19420 (2006).

Freund, R. W.

R. W. Freund, “A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems,”SIAM J. Sci. Comput. 14, 470–482 (1993).
[Crossref]

Ghanem, R. G.

M. F. Pellissetti and R. G. Ghanem, “Iterative solution of systems of linear equations arising in the context of stochastic finite elements,” Adv. Eng. Softw. 31, 607–616 (2000).
[Crossref]

Gürel, L.

Ö. Ergül and L. Gürel, “Efficient parallelization of the multilevel fast multipole algorithm for the solution of large-scale scattering problems,” IEEE Trans. Antennas Propag. 56, 2335–2345 (2008).
[Crossref]

Harrington, R. F.

R. F. Harrington, Field Computation by Moment Methods (IEEE Press, 1993).
[Crossref]

Hesthaven, J. S.

C. Chauvière, J. S. Hesthaven, and L. C. Wilcox, “Efficient computation of RCS from scatterers of uncertain shapes,” IEEE Trans. Antennas Propag. 55, 1437–1448 (2007).
[Crossref]

Hopkins, T.

R. D. da Cunha and T. Hopkins, PIM 2.2 The Parallel Iterative Methods Package for Systems of Linear Equations User’s Guide (Fortran 77 version), UKC, University of Kent, Canterbury, UK.

Jager, I.

I. Ocket, B. Nauwelaers, J. Fostier, L. Meert, F. Olyslager, G. Koers, J. Stiens, R. Vounckx, and I. Jager, “Characterization of speckle/despeckling in active millimeter wave imaging systems using a first order 1.5D model,” in “Millimeter-wave and Terahertz Photonics”, D. Jäger and A. Stöhr, eds., Proc. SPIE6194, 19409–19420 (2006).

Jin, J. M.

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).

Karniadakis, G. E.

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

Keese, A.

A. Keese and H. G. Matthies, “Hierarchical parallelisation for the solution of stochastic finite element equations,” Comput. Struct. 83, 1033–1047 (2005).
[Crossref]

Koers, G.

I. Ocket, B. Nauwelaers, J. Fostier, L. Meert, F. Olyslager, G. Koers, J. Stiens, R. Vounckx, and I. Jager, “Characterization of speckle/despeckling in active millimeter wave imaging systems using a first order 1.5D model,” in “Millimeter-wave and Terahertz Photonics”, D. Jäger and A. Stöhr, eds., Proc. SPIE6194, 19409–19420 (2006).

Li, Z.

Manfredi, P.

A. Biondi, D. VandeGinste, D. De Zutter, P. Manfredi, and F. Canavero, “Variability analysis of interconnects terminated by general nonlinear loads,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1244–1251 (2013).
[Crossref]

P. Manfredi, D. Van de Ginste, D. De Zutter, and F. Canavero, “Uncertainty assessment of lossy and dispersive lines in spice-type environments,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1252–1258 (2013).
[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]

Marzouk, Y.

Matthies, H. G.

A. Keese and H. G. Matthies, “Hierarchical parallelisation for the solution of stochastic finite element equations,” Comput. Struct. 83, 1033–1047 (2005).
[Crossref]

Meert, L.

I. Ocket, B. Nauwelaers, J. Fostier, L. Meert, F. Olyslager, G. Koers, J. Stiens, R. Vounckx, and I. Jager, “Characterization of speckle/despeckling in active millimeter wave imaging systems using a first order 1.5D model,” in “Millimeter-wave and Terahertz Photonics”, D. Jäger and A. Stöhr, eds., Proc. SPIE6194, 19409–19420 (2006).

Melloni, A.

Michiels, B.

B. Michiels, J. Fostier, I. Bogaert, and D. De Zutter, “Full-Wave Simulations of Electromagnetic Scattering Problems With Billions of Unknowns,” IEEE Trans. Antennas Propag. 63, 796–799 (2015).
[Crossref]

Michielssen, E.

D. Pissoort, E. Michielssen, D. Vande Ginste, and F. Olyslager, “Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs,” J. Lightwave Technol. 25, 2847–2863 (2007).
[Crossref]

D. Pissoort, E. Michielssen, D. Vande Ginste, and F. Olyslager, “Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs,” J. Lightwave Technol. 25, 2847–2863 (2007).
[Crossref]

D. VandeGinste, E. Michielssen, F. Olyslager, and D. De Zutter, “An efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures,” IEEE Trans. Antennas Propag. 54, 1538–1548 (2006).
[Crossref]

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).

Mickelson, A. R.

Miller, E. K.

A. J. Poggio and E. K. Miller, “Integral equation solutions for three dimensional scattering problems,” in Computer techniques for electromagnetics, R. Mittra, ed. (Pergamon Press, 1973), pp. 159–264.
[Crossref]

Mohamed, M.

Nauwelaers, B.

I. Ocket, B. Nauwelaers, J. Fostier, L. Meert, F. Olyslager, G. Koers, J. Stiens, R. Vounckx, and I. Jager, “Characterization of speckle/despeckling in active millimeter wave imaging systems using a first order 1.5D model,” in “Millimeter-wave and Terahertz Photonics”, D. Jäger and A. Stöhr, eds., Proc. SPIE6194, 19409–19420 (2006).

Ocket, I.

I. Ocket, B. Nauwelaers, J. Fostier, L. Meert, F. Olyslager, G. Koers, J. Stiens, R. Vounckx, and I. Jager, “Characterization of speckle/despeckling in active millimeter wave imaging systems using a first order 1.5D model,” in “Millimeter-wave and Terahertz Photonics”, D. Jäger and A. Stöhr, eds., Proc. SPIE6194, 19409–19420 (2006).

Olyslager, F.

J. Fostier and F. Olyslager, “An open-source implementation for full-wave 2D scattering by million-wavelength-size objects,” IEEE Antennas Propag. Mag. 52, 23–34 (2010).
[Crossref]

J. Fostier and F. Olyslager, “A provably scalable parallel multilevel fast multipole algorithm,” Electron. Lett. 44, 1111–1113 (2008).
[Crossref]

J. Fostier and F. Olyslager, “An asynchronous parallel MLFMA for scattering at multiple dielectric objects,” IEEE Trans. Antennas Propag. 56, 2346–2355 (2008).
[Crossref]

D. Pissoort, E. Michielssen, D. Vande Ginste, and F. Olyslager, “Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs,” J. Lightwave Technol. 25, 2847–2863 (2007).
[Crossref]

D. Pissoort, E. Michielssen, D. Vande Ginste, and F. Olyslager, “Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs,” J. Lightwave Technol. 25, 2847–2863 (2007).
[Crossref]

D. VandeGinste, E. Michielssen, F. Olyslager, and D. De Zutter, “An efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures,” IEEE Trans. Antennas Propag. 54, 1538–1548 (2006).
[Crossref]

D. VandeGinste, H. Rogier, F. Olyslager, and D. De Zutter, “A fast multipole method for layered media based on the application of perfectly matched layers–the 2-D case,” IEEE Trans. Antennas Propag. 52, 2631–2640 (2004).
[Crossref]

F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microwave Theory Tech. 41, 79–88 (1993).
[Crossref]

I. Ocket, B. Nauwelaers, J. Fostier, L. Meert, F. Olyslager, G. Koers, J. Stiens, R. Vounckx, and I. Jager, “Characterization of speckle/despeckling in active millimeter wave imaging systems using a first order 1.5D model,” in “Millimeter-wave and Terahertz Photonics”, D. Jäger and A. Stöhr, eds., Proc. SPIE6194, 19409–19420 (2006).

Pellissetti, M. F.

M. F. Pellissetti and R. G. Ghanem, “Iterative solution of systems of linear equations arising in the context of stochastic finite elements,” Adv. Eng. Softw. 31, 607–616 (2000).
[Crossref]

Pissoort, D.

Poggio, A. J.

A. J. Poggio and E. K. Miller, “Integral equation solutions for three dimensional scattering problems,” in Computer techniques for electromagnetics, R. Mittra, ed. (Pergamon Press, 1973), pp. 159–264.
[Crossref]

Rogier, H.

D. VandeGinste, H. Rogier, F. Olyslager, and D. De Zutter, “A fast multipole method for layered media based on the application of perfectly matched layers–the 2-D case,” IEEE Trans. Antennas Propag. 52, 2631–2640 (2004).
[Crossref]

Shang, L.

Song, J.

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).

Stiens, J.

I. Ocket, B. Nauwelaers, J. Fostier, L. Meert, F. Olyslager, G. Koers, J. Stiens, R. Vounckx, and I. Jager, “Characterization of speckle/despeckling in active millimeter wave imaging systems using a first order 1.5D model,” in “Millimeter-wave and Terahertz Photonics”, D. Jäger and A. Stöhr, eds., Proc. SPIE6194, 19409–19420 (2006).

Su, Z.

Van de Ginste, D.

P. Manfredi, D. Van de Ginste, D. De Zutter, and F. Canavero, “Uncertainty assessment of lossy and dispersive lines in spice-type environments,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1252–1258 (2013).
[Crossref]

Vande Ginste, D.

Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Multilevel Fast Multipole Method (MLFMM) with the Stochastic Galerkin Method (SGM),” IEEE Antennas Wireless Propag. Lett. 13, 1275–1278 (2014).
[Crossref]

Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Method of Moments with the Stochastic Galerkin Method,” IEEE Trans. Antennas Propag. 62, 4852–4856 (2014).
[Crossref]

D. Pissoort, E. Michielssen, D. Vande Ginste, and F. Olyslager, “Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs,” J. Lightwave Technol. 25, 2847–2863 (2007).
[Crossref]

D. Pissoort, E. Michielssen, D. Vande Ginste, and F. Olyslager, “Fast-multipole analysis of electromagnetic scattering by photonic crystal slabs,” J. Lightwave Technol. 25, 2847–2863 (2007).
[Crossref]

VandeGinste, D.

A. Biondi, D. VandeGinste, D. De Zutter, P. Manfredi, and F. Canavero, “Variability analysis of interconnects terminated by general nonlinear loads,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1244–1251 (2013).
[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]

D. VandeGinste, E. Michielssen, F. Olyslager, and D. De Zutter, “An efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures,” IEEE Trans. Antennas Propag. 54, 1538–1548 (2006).
[Crossref]

D. VandeGinste, H. Rogier, F. Olyslager, and D. De Zutter, “A fast multipole method for layered media based on the application of perfectly matched layers–the 2-D case,” IEEE Trans. Antennas Propag. 52, 2631–2640 (2004).
[Crossref]

Vounckx, R.

I. Ocket, B. Nauwelaers, J. Fostier, L. Meert, F. Olyslager, G. Koers, J. Stiens, R. Vounckx, and I. Jager, “Characterization of speckle/despeckling in active millimeter wave imaging systems using a first order 1.5D model,” in “Millimeter-wave and Terahertz Photonics”, D. Jäger and A. Stöhr, eds., Proc. SPIE6194, 19409–19420 (2006).

Weng, T.-W.

Wilcox, L. C.

C. Chauvière, J. S. Hesthaven, and L. C. Wilcox, “Efficient computation of RCS from scatterers of uncertain shapes,” IEEE Trans. Antennas Propag. 55, 1437–1448 (2007).
[Crossref]

Xiu, D. B.

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

Zhang, Z.

Zubac, Z.

Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Method of Moments with the Stochastic Galerkin Method,” IEEE Trans. Antennas Propag. 62, 4852–4856 (2014).
[Crossref]

Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Multilevel Fast Multipole Method (MLFMM) with the Stochastic Galerkin Method (SGM),” IEEE Antennas Wireless Propag. Lett. 13, 1275–1278 (2014).
[Crossref]

Adv. Eng. Softw. (1)

M. F. Pellissetti and R. G. Ghanem, “Iterative solution of systems of linear equations arising in the context of stochastic finite elements,” Adv. Eng. Softw. 31, 607–616 (2000).
[Crossref]

Appl. Opt. (1)

Comput. Struct. (1)

A. Keese and H. G. Matthies, “Hierarchical parallelisation for the solution of stochastic finite element equations,” Comput. Struct. 83, 1033–1047 (2005).
[Crossref]

Electron. Lett. (1)

J. Fostier and F. Olyslager, “A provably scalable parallel multilevel fast multipole algorithm,” Electron. Lett. 44, 1111–1113 (2008).
[Crossref]

IEEE Antennas Propag. Mag. (1)

J. Fostier and F. Olyslager, “An open-source implementation for full-wave 2D scattering by million-wavelength-size objects,” IEEE Antennas Propag. Mag. 52, 23–34 (2010).
[Crossref]

IEEE Antennas Wireless Propag. Lett. (1)

Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Multilevel Fast Multipole Method (MLFMM) with the Stochastic Galerkin Method (SGM),” IEEE Antennas Wireless Propag. Lett. 13, 1275–1278 (2014).
[Crossref]

IEEE Trans. Antennas Propag. (7)

B. Michiels, J. Fostier, I. Bogaert, and D. De Zutter, “Full-Wave Simulations of Electromagnetic Scattering Problems With Billions of Unknowns,” IEEE Trans. Antennas Propag. 63, 796–799 (2015).
[Crossref]

C. Chauvière, J. S. Hesthaven, and L. C. Wilcox, “Efficient computation of RCS from scatterers of uncertain shapes,” IEEE Trans. Antennas Propag. 55, 1437–1448 (2007).
[Crossref]

Z. Zubac, D. De Zutter, and D. Vande Ginste, “Scattering from two-dimensional objects of varying shape combining the Method of Moments with the Stochastic Galerkin Method,” IEEE Trans. Antennas Propag. 62, 4852–4856 (2014).
[Crossref]

J. Fostier and F. Olyslager, “An asynchronous parallel MLFMA for scattering at multiple dielectric objects,” IEEE Trans. Antennas Propag. 56, 2346–2355 (2008).
[Crossref]

D. VandeGinste, H. Rogier, F. Olyslager, and D. De Zutter, “A fast multipole method for layered media based on the application of perfectly matched layers–the 2-D case,” IEEE Trans. Antennas Propag. 52, 2631–2640 (2004).
[Crossref]

D. VandeGinste, E. Michielssen, F. Olyslager, and D. De Zutter, “An efficient perfectly matched layer based multilevel fast multipole algorithm for large planar microwave structures,” IEEE Trans. Antennas Propag. 54, 1538–1548 (2006).
[Crossref]

Ö. Ergül and L. Gürel, “Efficient parallelization of the multilevel fast multipole algorithm for the solution of large-scale scattering problems,” IEEE Trans. Antennas Propag. 56, 2335–2345 (2008).
[Crossref]

IEEE Trans. Compon., Packag., Manuf. Technol. (3)

A. Biondi, D. VandeGinste, D. De Zutter, P. Manfredi, and F. Canavero, “Variability analysis of interconnects terminated by general nonlinear loads,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1244–1251 (2013).
[Crossref]

P. Manfredi, D. Van de Ginste, D. De Zutter, and F. Canavero, “Uncertainty assessment of lossy and dispersive lines in spice-type environments,” IEEE Trans. Compon., Packag., Manuf. Technol. 3, 1252–1258 (2013).
[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]

IEEE Trans. Microwave Theory Tech. (1)

F. Olyslager, D. De Zutter, and K. Blomme, “Rigorous analysis of the propagation characteristics of general lossless and lossy multiconductor transmission lines in multilayered media,” IEEE Trans. Microwave Theory Tech. 41, 79–88 (1993).
[Crossref]

J. Lightwave Technol. (2)

Opt. Express (1)

SIAM J. Sci. Comput. (2)

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

R. W. Freund, “A transpose-free quasi-minimal residual algorithm for non-Hermitian linear systems,”SIAM J. Sci. Comput. 14, 470–482 (1993).
[Crossref]

Other (6)

R. D. da Cunha and T. Hopkins, PIM 2.2 The Parallel Iterative Methods Package for Systems of Linear Equations User’s Guide (Fortran 77 version), UKC, University of Kent, Canterbury, UK.

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 technique,” in 2011 12th International Symposium on Quality Electronic Design (ISQED 2011), 14–16 March 2011, Santa Clara, CA, USA, (2011), pp. 508–517.

W. C. Chew, J. M. Jin, E. Michielssen, and J. Song, Fast and Efficient Algorithms in Computational Electromagnetics (Artech House, 2001).

R. F. Harrington, Field Computation by Moment Methods (IEEE Press, 1993).
[Crossref]

A. J. Poggio and E. K. Miller, “Integral equation solutions for three dimensional scattering problems,” in Computer techniques for electromagnetics, R. Mittra, ed. (Pergamon Press, 1973), pp. 159–264.
[Crossref]

I. Ocket, B. Nauwelaers, J. Fostier, L. Meert, F. Olyslager, G. Koers, J. Stiens, R. Vounckx, and I. Jager, “Characterization of speckle/despeckling in active millimeter wave imaging systems using a first order 1.5D model,” in “Millimeter-wave and Terahertz Photonics”, D. Jäger and A. Stöhr, eds., Proc. SPIE6194, 19409–19420 (2006).

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Figures (12)

Fig. 1
Fig. 1 Canonical problem geometry. Objects are described by their electrical properties µi, εi and their geometries are defined with contours Ci. Stochastic variations of the geometry are introduced and indicated by means of a set of random variables ξi, i = 1, …, M.
Fig. 2
Fig. 2 Typical MLFMM constellation of a source box B′ and an observation box B.
Fig. 3
Fig. 3 Organization of MLFMM boxes and partitioning scheme for an arbitrary structure. On the lowest level each box is handled by one process; on higher levels, K radiation samples are shared among all processes. O P W k B i is k-th PCE coefficient of the box on the level i.
Fig. 4
Fig. 4 The lens system setup.
Fig. 5
Fig. 5 Field density |Ez|(V/m) for the deterministic simulation for the configuration of Fig. 4.
Fig. 6
Fig. 6 Mean and standard deviation of the field density |Ez|(V/m) around the focal points.
Fig. 7
Fig. 7 Speedup and parallel efficiency for a varying number of parallel processes.
Fig. 8
Fig. 8 Standard deviation of the field density |Ez|(V/m) around the focal points.
Fig. 9
Fig. 9 PDF of the local intensity I (mW/m2) for two points behind the rightmost lens.
Fig. 10
Fig. 10 The average total field density |Ez| (V/m) of the Cassegrain antenna system illuminated with a Gaussian beam from the bottom onto the lens. The results are obtained with a polynomial order P = 3. coated incident
Fig. 11
Fig. 11 The standard deviation of the total field density |Ez|(V/m) of the Cassegrain antenna system illuminated with a Gaussian beam incident from the bottom onto the coated lens. The results are obtained with polynomial order P = 3.
Fig. 12
Fig. 12 The average total field density |Ez| (V/m) radiated away from the Cassegrain antenna system along an 3800λ long line segment, 1400λ above the center of the main parabolic reflector.

Tables (2)

Tables Icon

Table 1 Mean and standard deviation for the point indicated on Fig. 4, i.e. the point close to the second focal point with maximum variance.

Tables Icon

Table 2 Mean and standard deviation for the point close to the second focal point with maximum variance.

Equations (34)

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E z i lim r C + C [ E z G 0 n j k 0 2 ω ε 0 G 0 H t ] d c = lim r C C [ E z G n j k 2 ω ε G H t ] d c ,
H t i lim r C + C [ j ω ε 0 k 0 2 E z 2 G 0 n n G 0 n H t ] d c = lim r C C [ j ω ε k 2 E z 2 G 0 n n G n H t ] d c ,
G 0 ( ρ , ρ ) = j 4 H 0 ( 2 ) ( k 0 | ρ ρ | ) ,
H t = i = 1 N s e g H t , i b i ( ρ ) ,
E z = i = 1 N s e g E z , i t i ( ρ ) .
V i = j = 1 N Z i j I j , for all i = 1 , , N ,
V i = l i E z i b i ( ρ ) d ρ
V i = l i H t i t i ( ρ ) d ρ ,
Z i j = l i l j E z , j b i ( ρ ) t j ( ρ ) G 0 n d ρ d ρ + l i l j E z , j b i ( ρ ) t j ( ρ ) G n d ρ d ρ ,
Z i j = l i l j H t , j b i ( ρ ) b j ( ρ ) j k 0 2 ω ε 0 G 0 d ρ d ρ l i l j H t , j b i ( ρ ) b j ( ρ ) j k 2 ω ε G d ρ d ρ ,
Z i j = l i l j E z , j t i ( ρ ) t j ( ρ ) j ω ε 0 k 0 2 2 G 0 n n d ρ d ρ l i l j E z , j t i ( ρ ) t j ( ρ ) j ω ε k 2 2 G n n d ρ d ρ ,
Z i j = l i l j H t , j t i ( ρ ) b j ( ρ ) G 0 n d ρ d ρ l i l j H t , j t i ( ρ ) b j ( ρ ) G n d ρ d ρ ,
Z ¯ ( ξ ) I ( ξ ) = V ( ξ ) ,
Z ¯ ( ξ ) k = 0 K Z ¯ k ϕ k ( ξ ) ,
V ( ξ ) k = 0 K V k ϕ k ( ξ ) ,
I ( ξ ) k = 0 K I k ϕ k ( ξ ) ,
< ϕ j ( ξ ) , ϕ k ( ξ ) > = δ j k
< f ( ξ ) , g ( ξ ) > = Ω f ( ξ ) g ( ξ ) W ( ξ ) d ξ .
K + 1 = ( M + P ) ! M ! P ! .
X k = < X ( ξ ) , ϕ k ( ξ ) > ,
k = 0 K V k ϕ k ( ξ ) = k = 0 K l = 0 K Z ¯ k I l ϕ k ( ξ ) ϕ l ( ξ ) ,
< k = 0 K V k ϕ k ( ξ ) , ϕ m ( ξ ) > = < k = 0 K l = 0 K Z ¯ k I l ϕ k ( ξ ) ϕ l ( ξ ) , ϕ m ( ξ ) > , for all m = 0 , , K ,
V m = m : γ k ; l 0 Z ¯ k I l γ k l m
O P W q B ( ξ ) = s i e j k ( φ q ) ( ρ ( ξ s ) ρ s c ) J s i ( ξ ) ,
Q = 2 k R + 1.8 d 0 2 / 3 ( 2 k R ) 1 / 3 .
O P W B ( ξ ) = k = 0 K O P W k B ϕ k ( ξ )
O P W m B = m : γ k l m 0 A ¯ k I l γ k l m
A ¯ ( ξ s ) q , s i = e j k ( φ q ) ( ρ ( ξ s ) ρ s c ) , q = Q , , Q ,
I P W k , q B = T q q ( k , | ρ s o c c | , φ s o c c ) O P W k , q B , q = Q , , Q , k = 0 , , K ,
T q q ( k , ρ , φ ) = 1 2 Q + 1 q = Q Q H q ( 2 ) ( k ρ ) e j q ( φ φ q π 2 )
G ( ρ ( ξ o ) ; ρ ( ξ s ) ) = j 4 q = Q Q e j k ( φ q ) ( ρ ( ξ o ) ρ o ) I P W q B ( ξ )
S p = T 1 T p .
η p = S p p = T 1 p T p .
I = c n ε 0 2 | E z | 2 ,

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