volume | PIER Journals

Abstract

A novel polynomial-chaos (PC) techni_nullque is implemented based on anisotropic index sets. The proposed scheme takes advantage of the effect of each random variable on the output parameter of interest and adaptively constructs the PC expansion. Particularly, the algorithm starts by generating bases via low and high reliability heuristics and builds a PC representation, until an error criterion is satisfied or until the maximum desired polynomial order is reached. Our method is tested on a variety of uncertainty problems, where the statistical moments of the outputs of interest are estimated. Numerical results prove the efficiency of the proposed approach, since accurate outcomes are obtained in lower computational times than other techniques.

1. Austin, A. C. M. and C. D. Sarris, "Efficient analysis of geometrical uncertainty in the FDTD method using polynomial chaos with application to microwave circuits," IEEE Trans. Microw. Theory Techn., Vol. 61, No. 12, 4293-4301, Dec. 2013.
doi:10.1109/TMTT.2013.2281777

2. Hastings, F. D., J. B. Schneider, and S. L. Broschat, "A Monte-Carlo FDTD technique for rough surface scattering," IEEE Trans. Antennas Propag., Vol. 43, No. 11, 1183-1191, Nov. 1995.
doi:10.1109/8.475089

3. Xiu, D. and G. E. Karniadakis, "The Wiener-Askey polynomial chaos for stochastic differential equations," SIAM J. Sci. Comput., Vol. 24, No. 2, 619-644, 2002.
doi:10.1137/S1064827501387826

4. Rong, A. and A. C. Cangellaris, "Transient analysis of distributed electromagnetic systems exhibiting stochastic variability in material parameters," 2011 XXXth URSI General Assembly and Scientific Symposium, 1-4, Istanbul, Turkey, Aug. 2011.

5. Shen, J. and J. Chen, "An efficient polynomial chaos method for uncertainty quantification in electromagnetic simulations," 2010 IEEE Antennas and Propagation Society International Symposium, 1-4, Jul. 2010.

6. Salis, C., N. Kantartzis, and T. Zygiridis, "Efficient uncertainty assessment in EM problems via dimensionality reduction of polynomial-chaos expansions," Technologies, Vol. 7, No. 2, 2019.
doi:10.3390/technologies7020037

7. Spina, D., F. Ferranti, T. Dhaene, L. Knockaert, G. Antonini, and D. Vande Ginste, "Variability analysis of multiport systems via polynomial-chaos expansion," IEEE Trans. Microw. Theory Tech., Vol. 60, No. 8, 2329-2338, Aug. 2012.
doi:10.1109/TMTT.2012.2202685

8. Parussini, L. and V. Pediroda, "Investigation of multi geometric uncertainties by different polynomial chaos methodologies using a fictitious domain solver," CMES Comp. Model. Eng., Vol. 23, No. 1, 29-52, 2008.

9. Salis, C. I., T. T. Zygiridis, N. V. Kantartzis, and C. S. Antonopoulos, "An anisotropic polynomial-chaos technique for assessing uncertainties in microwave circuits," 2019 22nd International Conference on the Computation of Electromagnetic Fields (COMPUMAG), 1-2, Paris, France, Jul. 2019.

10. Blatman, G., "Adaptive sparse polynomial chaos expansions for uncertainty propagaton and sensitivity analysis,", Ph.D. dissertation, Universite Blaise Pascal, Clermont-Ferrand, France, 2009.

11. Smolyak, S., "Quadrature and interpolation formulas for tensor products of certain classes of functions," Dokl. Akad. Nauk SSSR, Vol. 148, No. 5, 1042-1045, 1963.

12. Peng, J., J. Hampton, and A. Doostan, "A weighted ℓ₁-minimization approach for sparse polynomial chaos expansions," J. Comput. Phys., Vol. 267, 92-111, 2014.
doi:10.1016/j.jcp.2014.02.024

13. Salis, C. and T. Zygiridis, "Dimensionality reduction of the polynomial chaos technique based on the method of moments," IEEE Antennas Wirel. Propag. Lett., Vol. 17, No. 12, 2349-2353, Dec. 2018.
doi:10.1109/LAWP.2018.2874521

14. Beddek, K., S. Clenet, O. Moreau, V. Costan, Y. Le Menach, and A. Benabou, "Adaptive method for non-intrusive spectral projection --- Application on a stochastic eddy current NDT problem," IEEE Trans. Magn., Vol. 48, No. 2, 759-762, 2012.
doi:10.1109/TMAG.2011.2175204

15. Thapa, M., S. B. Mulani, and R. W. Walters, "Adaptive weighted least-squares polynomial chaos expansion with basis adaptivity and sequential adaptive sampling," Comput. Methods Appl. Mech. Eng., Vol. 360, 112759, 2020.
doi:10.1016/j.cma.2019.112759

16. Ahadi, M. and S. Roy, "Sparse linear regression (SPLINER) approach for efficient multidimensional uncertainty quantification of high-speed circuits," IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., Vol. 35, No. 10, 1640-1652, Oct. 2016.
doi:10.1109/TCAD.2016.2527711

17. Salis, C., N. Kantartzis, and T. Zygiridis, "An adaptive sparse polynomialchaos technique based on anisotropic indices," COMPEL, Vol. 39, No. 3, 691-707, May 2020.
doi:10.1108/COMPEL-10-2019-0392

18. Yan, L. and T. Zhou, "Adaptive multi-fidelity polynomial chaos approach to bayesian inference in inverse problems," J. Comput. Phys., Vol. 381, 110-128, 2019.
doi:10.1016/j.jcp.2018.12.025

19. Yangtian, L., H. Li, and G. Wei, "Dimension-adaptive algorithm-based PCE for models with many model parameters," Eng. Comput., Vol. 37, No. 2, 522-545, 2019.
doi:10.1108/EC-12-2018-0595

20. Thapa, M., S. B. Mulani, and R. W. Walters, "Adaptive weighted leastsquares polynomial chaos expansion with basis adaptivity and sequential adaptive sampling," Comput. Methods Appl. Mech. Eng., Vol. 360, 112759, 2020.
doi:10.1016/j.cma.2019.112759

21. Zhang, Z., T. A. El-Moselhy, I. M. Elfadel, and L. Daniel, "Stochastic testing method for transistor-level uncertainty quantification based on generalized polynomial chaos," IEEE Trans. Comput.-Aided Des. Integr. Circuits Syst., Vol. 32, No. 10, 1533-1545, 2013.
doi:10.1109/TCAD.2013.2263039

22. Zygiridis, T., A. Papadopoulos, N. Kantartzis, and E. Glytsis, "Sparse polynomial-chaos models for stochastic problems with filtering structures," AEM, Vol. 8, No. 5, 51-58, 2019.
doi:10.7716/aem.v8i5.1328

23. Blatman, G. and B. Sudret, "Adaptive sparse polynomial chaos expansion based on least angle regression," J. Comput. Phys., Vol. 230, No. 6, 2345-2367, 2011.
doi:10.1016/j.jcp.2010.12.021

24. Ishigami, T. and T. Homma, "An importance quantification technique in uncertainty analysis for computer models," Proceedings --- First International Symposium on Uncertainty Modeling and Analysis, 398-403, 1990.

25. Taflove, A. and S. C. Hagness, Computational Electrodynamics: The Finite-Difference Time-Domain Method, 3rd Ed., Artech House, Norwood, 2005.

26. Mur, G., "Absorbing boundary conditions for the finite-difference approximation of the time-domain electromagnetic-field equations," IEEE Trans. Electromagn. Compat., Vol. 23, No. 4, 377-382, Nov. 1981.
doi:10.1109/TEMC.1981.303970

27. Shorbagy, M. E., R. M. Shubair, M. I. AlHajri, and N. K. Mallat, "On the design of millimetre-wave antennas for 5G," 2016 16th Mediterranean Microwave Symposium (MMS), 1-4, Nov. 2016.

28. "3ds.com, 2020, Electromagnetic systems --- Cst Studio Suite,", https://www.3ds.com/products-services/simulia/products/cst-studiosuite/electromagnetic-systems, accessed: 2020-03-10.

29. Berenger, J.-P., "A perfectly matched layer for the absorption of electromagnetic waves," J. Comput. Phys., Vol. 114, No. 2, 185-200, 1994.
doi:10.1006/jcph.1994.1159

Dr. Christos I. Salis

Prof. Nikolaos V. Kantartzis

Prof. Theodoros T. Zygiridis