Numerical comparison of three stochastic methods for nonlinear PN junction problems

doi:10.1007/s11464-013-0327-5

Front. Math. China

2014, Vol. 9

Issue (3) : 659-698 https://doi.org/10.1007/s11464-013-0327-5

RESEARCH ARTICLE

Numerical comparison of three stochastic methods for nonlinear PN junction problems

Wenqi YAO¹,Tiao LU^1,^2,^*(

)

1. School of Mathematical Sciences, Peking University, Beijing 100871, China
2. HEDPS, CAPT and LMAM, Peking University, Beijing 100871, China

Download: PDF(1338 KB)
Export: BibTeX | EndNote | Reference Manager | ProCite | RefWorks

Abstract

We apply the Monte Carlo, stochastic Galerkin, and stochastic collocation methods to solving the drift-diffusion equations coupled with the Poisson equation arising in semiconductor devices with random rough surfaces. Instead of dividing the rough surface into slices, we use stochastic mapping to transform the original deterministic equations in a random domain into stochastic equations in the corresponding deterministic domain. A finite element discretization with the help of AFEPack is applied to the physical space, and the equations obtained are solved by the approximate Newton iterative method. Comparison of the three stochastic methods through numerical experiment on different PN junctions are given. The numerical results show that, for such a complicated nonlinear problem, the stochastic Galerkin method has no obvious advantages on efficiency except accuracy over the other two methods, and the stochastic collocation method combines the accuracy of the stochastic Galerkin method and the easy implementation of the Monte Carlo method.

Keywords Stochastic partial differential equation stochastic Galerkin method stochastic collocation method Monte Carlo method

Corresponding Author(s): Tiao LU

Issue Date: 24 June 2014

Cite this article:

Wenqi YAO,Tiao LU. Numerical comparison of three stochastic methods for nonlinear PN junction problems[J]. Front. Math. China, 2014, 9(3): 659-698.

URL:

https://academic.hep.com.cn/fmc/EN/10.1007/s11464-013-0327-5
https://academic.hep.com.cn/fmc/EN/Y2014/V9/I3/659

1	AskeyR, WilsonJ. Some Basic Hypergeometric Orthogonal Polynomials that Generalize Jacobi Polynomials. Mem Amer Math Soc, No 319. Providence: Amer Math Soc, 1985
2	BabuškaI, TemponeR, ZourarisG E. Galerkin finite element approximations of stochastic elliptic partial differential equations. SIAM J Numer Anal, 2004, 42(2): 800-825 doi: 10.1137/S0036142902418680
3	Bäck,J, NobileF, TamelliniL, TemponeR. Stochastic spectral Galerkin and collocation methods for PDEs with random coefficients: a numerical comparison. In: Proceedings of the International Conference on Spectral and High Order Methods (ICOSAHOM 09). Berlin: Springer-Verlag, 2010
4	BankR E, RoseD J. Global approximate Newton methods. Numer Math, 1981, 37: 279-295 doi: 10.1007/BF01398257
5	BankR E, RoseD J, FichtnerW. Numerical methods for semiconductor device simulation. SIAM J Sci Stat Comput, 1983, 4: 416-435 doi: 10.1137/0904032
6	BejanA. Shape and Structure, from Engineering to Nature. New York: Cambridge Univ Press, 2000
7	BürglerJ F, BankR E, FichtnerW, SmithR K. A new discretization for the semiconductor current continuity equations. IEEE Trans Comput-Aided Design Integrat Circuits Sys, 1989, 8: 479-489 doi: 10.1109/43.24876
8	BürglerJ F, ConghranW M, FichtnerJr W. An adaptive grid refinement strategy for the drift-diffusion equations. IEEE Trans Comput-Aided Design Integrat Circuits Sys, 1991, 10: 1251-1258 doi: 10.1109/43.88921
9	DebM K, BabuškaI M, OdenJ T. Solution of stochastic partial differential equations using Galerkin finite element techniques. Comput Meth Appl Mech Engrg, 2001, 190: 6359-6372 doi: 10.1016/S0045-7825(01)00237-7
10	ElmanH C, MillerC W, PhippsE T, TuminaroR S. Assessment of collocation and Galerkin approaches to linear diffusion equations with random data. Int J Uncertainty Quant, 2011, 1(1): 19-33 doi: 10.1615/Int.J.UncertaintyQuantification.v1.i1.20
11	FishmanG. Monte Carlo: Concepts, Algorithms, and Applications. New York: Springer-Verlag, 1996 doi: 10.1007/978-1-4757-2553-7
12	GanapathysubramanianB, ZabarasN. Sparse grid collocation methods for stochastic natural convection problems. J Comput Phys, 2007, 225(1): 652-685 doi: 10.1016/j.jcp.2006.12.014
13	GhanemR G, SpanosP. Stochastic Finite Elements: a Spectral Approach. New York: Springer, 1991 doi: 10.1007/978-1-4612-3094-6
14	LiR. On multi-mesh h-adaptive methods. J Sci Comput, 2005, 24(3): 321-341 doi: 10.1007/s10915-004-4793-5
15	LoèveM. Probability Theory. 4th ed. Berlin: Springer-Verlag, 1977
16	NovakE, RitterK. High dimensional integration of smooth functions over cubes. Numer Math, 1996, 75: 79-97 doi: 10.1007/s002110050231
17	NovakE, RitterK. Simple cubature formulas with high polynomial exactness. Constructive Approx, 1999, 15: 499-522 doi: 10.1007/s003659900119
18	SmolyakS A. Quadrature and interpolation formulas for tensor products of certain classes of functions. Dokl Akad Nauk SSSR, 1963, 148: 1042-1045(in Russian); Soviet Math Dokl, 1963, 4: 240-243
19	TaurY, NingH. Fundamentals of Modern VLSI Devices. 2nd ed. Cambridge: Cambridge Univ Press, 2009
20	Van TreesH L. Detection, Estimation, and Modulation Theory, Part I. New York: Wiley, 1968
21	WasilkowskiG, WozniakowskiH. Explicit cost bounds of algorithms for multivariate tensor product problems. J Complexity, 1995, 11: 1-56 doi: 10.1006/jcom.1995.1001
22	WienerN. The homogeneous chaos. Amer J Math, 1938, 60: 897-936 doi: 10.2307/2371268
23	XiuD, HesthavenJ. High-order collocation methods for differential equations with random inputs. SIAM J Sci Comput, 2005, 27: 1118-1139 doi: 10.1137/040615201
24	XiuD, KarniadakisG E. The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J Sci Comput, 2002, 24(2): 619-644 doi: 10.1137/S1064827501387826
25	XiuD, KarniadakisG E. Modeling uncertainty in steady state diffusion problems via generalized polynomial chaos. Comput Methods Appl Math Engrg, 2002, 191: 4927-4948 doi: 10.1016/S0045-7825(02)00421-8
26	XiuD, KarniadakisG E. Modeling uncertainty in flow simulations via generalized polynomial chaos. J Comput Phys, 2003, 187: 137-167 doi: 10.1016/S0021-9991(03)00092-5
27	XiuD, TartakovskyD M. Numerical methods for differential equations in random domain. SIAM J Sci Comput, 2006, 28: 1167-1185 doi: 10.1137/040613160
28	YuS, ZhaoY, ZengL, DuG, KangJ, HanR, LiuX. Impact of line-edge roughness on double-gate Schottky-barrier filed-effect transistors. IEEE Trans Electron Devices, 2009, 56(6): 1211-1219 doi: 10.1109/TED.2009.2017644

[1]	Shaoqin ZHANG. Shift Harnack inequality and integration by parts formula for semilinear stochastic partial differential equations[J]. Front. Math. China, 2016, 11(2): 461-496.
[2]	Zhi LI,Jiaowan LUO. Transportation inequalities for stochastic delay evolution equations driven by fractional Brownian motion[J]. Front. Math. China, 2015, 10(2): 303-321.
[3]	Zhen-Qing CHEN,Wai-Tong(Louis) FAN. Scaling limits of interacting diffusions in domains[J]. Front. Math. China, 2014, 9(4): 717-736.
[4]	Jinqiao DUAN. Stochastic modeling of unresolved scales in complex systems[J]. Front Math Chin, 2009, 4(3): 425-436.
[5]	BO Lijun, WANG Yongjin, YAN Liqing. Higher-order stochastic partial differential equations with branching noises[J]. Front. Math. China, 2008, 3(1): 15-35.

Viewed

Full text

Abstract

Cited

Shared

Discussed