W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 1–8 (2014).

[Crossref]

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).

[Crossref]

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).

[Crossref]

N. Agarwal and N. R. Aluru, “Weighted Smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures,” Int. J. Numer. Meth. Eng. 85, 1365–1389 (2011).

[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).

[Crossref]

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

[Crossref]

D. Xiu, “Fast numerical methods for stochastic computations: a review,” Commun. Comput. Phys. 5, 242–272 (2009).

B. Ganapathysubramanian and N. Zabaras, “Sparse grid collocation schemes for stochastic natural convection problems,” J. Comput. Phys. 225, 652–685 (2007).

[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

V. Barthelmann, E. Novak, and K. Ritter, “High dimensional polynomial interpolation on sparse grid,” Adv. Comput. Math. 12, 273–288 (2000).

[Crossref]

E. Novak and K. Ritter, “Simple cubature formulas with high polynomial exactness,” Constr. Approx. 15, 499–522 (1999).

[Crossref]

E. Novak and K. Ritter, “High dimensional integration of smooth functions over cubes,” Numer. Math. 75, 79–97 (1996).

[Crossref]

W. A. Weiser and S. E. Zarantonello, “A note on piecewise linear and multilinear table interpolation in many dimensions,” Math. Comput. 50, 189–196 (1988).

[Crossref]

A. Der Kiureghian and P. L. Liu, “Structural reliability under incomplete probability information,” J. Eng. Mech. 112, 85–104 (1986).

[Crossref]

N. Agarwal and N. R. Aluru, “Weighted Smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures,” Int. J. Numer. Meth. Eng. 85, 1365–1389 (2011).

[Crossref]

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

[Crossref]

N. Agarwal and N. R. Aluru, “Weighted Smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures,” Int. J. Numer. Meth. Eng. 85, 1365–1389 (2011).

[Crossref]

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

[Crossref]

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).

[Crossref]

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).

[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).

[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

V. Barthelmann, E. Novak, and K. Ritter, “High dimensional polynomial interpolation on sparse grid,” Adv. Comput. Math. 12, 273–288 (2000).

[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 1–8 (2014).

[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).

[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

K.-K. K. Kim and R. D. Braatz, “Generalized polynomial chaos expansion approaches to approximate stochastic receding horizon control with applications to probabilistic collision checking and avoidance,” in IEEE International Conference on Control Applications, Dubrovnik, Croatia (Oct.3–5, 2012).

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in Advanced Photonics (Optical Society of America, 2013), paper JT3A.8.

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

L. Chrostowski and M. Hochberg, Silicon Photonics Design: From Devices to Systems (Cambridge University, 2015).

A. Der Kiureghian and P. L. Liu, “Structural reliability under incomplete probability information,” J. Eng. Mech. 112, 85–104 (1986).

[Crossref]

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).

[Crossref]

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).

[Crossref]

W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 1–8 (2014).

[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).

[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

M. S. Eldred, “Recent advance in non-intrusive polynomial-chaos and stochastic collocation methods for uncertainty analysis and design,” in Proceedings of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, California (May, 2009).

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

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).

[Crossref]

W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 1–8 (2014).

[Crossref]

G. S. Fishman, Monte Carlo: Concepts, Algorithms, and Applications (Springer-Verlag, 1996).

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

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).

[Crossref]

B. Ganapathysubramanian and N. Zabaras, “Sparse grid collocation schemes for stochastic natural convection problems,” J. Comput. Phys. 225, 652–685 (2007).

[Crossref]

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).

[Crossref]

L. Chrostowski and M. Hochberg, Silicon Photonics Design: From Devices to Systems (Cambridge University, 2015).

P. R. Johnston, “Defibrillation thresholds: a generalised polynomial chaos study,” in Conference in Computing in Cardiology, Cambridge, MA (Sept.7–10, 2014).

K.-K. K. Kim and R. D. Braatz, “Generalized polynomial chaos expansion approaches to approximate stochastic receding horizon control with applications to probabilistic collision checking and avoidance,” in IEEE International Conference on Control Applications, Dubrovnik, Croatia (Oct.3–5, 2012).

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).

[Crossref]

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).

[Crossref]

A. Der Kiureghian and P. L. Liu, “Structural reliability under incomplete probability information,” J. Eng. Mech. 112, 85–104 (1986).

[Crossref]

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).

[Crossref]

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).

[Crossref]

M. Loeve, Probability Theory, 4th ed. (Springer-Verlag, 1977).

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

T. 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]

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in Advanced Photonics (Optical Society of America, 2013), paper JT3A.8.

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in Advanced Photonics (Optical Society of America, 2013), paper JT3A.8.

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).

[Crossref]

V. Barthelmann, E. Novak, and K. Ritter, “High dimensional polynomial interpolation on sparse grid,” Adv. Comput. Math. 12, 273–288 (2000).

[Crossref]

E. Novak and K. Ritter, “Simple cubature formulas with high polynomial exactness,” Constr. Approx. 15, 499–522 (1999).

[Crossref]

E. Novak and K. Ritter, “High dimensional integration of smooth functions over cubes,” Numer. Math. 75, 79–97 (1996).

[Crossref]

V. Barthelmann, E. Novak, and K. Ritter, “High dimensional polynomial interpolation on sparse grid,” Adv. Comput. Math. 12, 273–288 (2000).

[Crossref]

E. Novak and K. Ritter, “Simple cubature formulas with high polynomial exactness,” Constr. Approx. 15, 499–522 (1999).

[Crossref]

E. Novak and K. Ritter, “High dimensional integration of smooth functions over cubes,” Numer. Math. 75, 79–97 (1996).

[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).

[Crossref]

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).

[Crossref]

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).

[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).

[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

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

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

W. A. Weiser and S. E. Zarantonello, “A note on piecewise linear and multilinear table interpolation in many dimensions,” Math. Comput. 50, 189–196 (1988).

[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

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

D. Xiu, “Fast numerical methods for stochastic computations: a review,” Commun. Comput. Phys. 5, 242–272 (2009).

B. Ganapathysubramanian and N. Zabaras, “Sparse grid collocation schemes for stochastic natural convection problems,” J. Comput. Phys. 225, 652–685 (2007).

[Crossref]

W. A. Weiser and S. E. Zarantonello, “A note on piecewise linear and multilinear table interpolation in many dimensions,” Math. Comput. 50, 189–196 (1988).

[Crossref]

V. Barthelmann, E. Novak, and K. Ritter, “High dimensional polynomial interpolation on sparse grid,” Adv. Comput. Math. 12, 273–288 (2000).

[Crossref]

D. Xiu, “Fast numerical methods for stochastic computations: a review,” Commun. Comput. Phys. 5, 242–272 (2009).

E. Novak and K. Ritter, “Simple cubature formulas with high polynomial exactness,” Constr. Approx. 15, 499–522 (1999).

[Crossref]

W. Bogaerts, M. Fiers, and P. Dumon, “Design challenges in silicon photonics,” IEEE J. Sel. Top. Quantum Electron. 20, 1–8 (2014).

[Crossref]

S. K. Selvaraja, W. Bogaerts, P. Dumon, D. Van Thourhout, and R. Baets, “Subnanometer linewidth uniformity in silicon nanophotonic waveguide devices using CMOS fabrication technology,” IEEE J. Sel. Top. Quantum Electron. 16, 316–324 (2010).

[Crossref]

D. Spina, F. Ferranti, G. Antonini, T. Dhaene, and L. Knockaert, “Efficient variability analysis of electromagnetic systems via polynomial chaos and model order reduction,” IEEE Trans. Compon. Packag. Manuf. Technol. 4, 1038–1051 (2014).

[Crossref]

N. Agarwal and N. R. Aluru, “Weighted Smolyak algorithm for solution of stochastic differential equations on non-uniform probability measures,” Int. J. Numer. Meth. Eng. 85, 1365–1389 (2011).

[Crossref]

B. Ganapathysubramanian and N. Zabaras, “Sparse grid collocation schemes for stochastic natural convection problems,” J. Comput. Phys. 225, 652–685 (2007).

[Crossref]

A. Der Kiureghian and P. L. Liu, “Structural reliability under incomplete probability information,” J. Eng. Mech. 112, 85–104 (1986).

[Crossref]

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

[Crossref]

W. A. Weiser and S. E. Zarantonello, “A note on piecewise linear and multilinear table interpolation in many dimensions,” Math. Comput. 50, 189–196 (1988).

[Crossref]

E. Novak and K. Ritter, “High dimensional integration of smooth functions over cubes,” Numer. Math. 75, 79–97 (1996).

[Crossref]

W. Bogaerts, D. Taillaert, B. Luyssaert, P. Dumon, J. Van Campenhout, P. Bienstman, D. Van Thourhout, R. Baets, V. Wiaux, and S. Beckx, “Basic structures for photonic integrated circuits in silicon-on-insulator,” Opt. Express 12, 1583–1591 (2004).

[Crossref]

T. 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]

M. Streshinsky, R. Ding, Y. Liu, A. Novack, C. Galland, A.-J. Lim, G. Q. Lo, T. Baehr-Jones, and M. Hochberg, “The road to affordable, large-scale silicon photonics,” Opt. Photon. News 24(9), 32–39 (2013).

[Crossref]

P. R. Johnston, “Defibrillation thresholds: a generalised polynomial chaos study,” in Conference in Computing in Cardiology, Cambridge, MA (Sept.7–10, 2014).

K.-K. K. Kim and R. D. Braatz, “Generalized polynomial chaos expansion approaches to approximate stochastic receding horizon control with applications to probabilistic collision checking and avoidance,” in IEEE International Conference on Control Applications, Dubrovnik, Croatia (Oct.3–5, 2012).

D. Cassano, F. Morichetti, and A. Melloni, “Statistical analysis of photonic integrated circuits via polynomial-chaos expansion,” in Advanced Photonics (Optical Society of America, 2013), paper JT3A.8.

M. S. Eldred, “Recent advance in non-intrusive polynomial-chaos and stochastic collocation methods for uncertainty analysis and design,” in Proceedings of 50th AIAA/ASME/ASCE/AHS/ASC Structures, Structural Dynamics, and Materials Conference, Palm Springs, California (May, 2009).

G. S. Fishman, Monte Carlo: Concepts, Algorithms, and Applications (Springer-Verlag, 1996).

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

M. Loeve, Probability Theory, 4th ed. (Springer-Verlag, 1977).

L. Chrostowski and M. Hochberg, Silicon Photonics Design: From Devices to Systems (Cambridge University, 2015).