Optimization Modeling and Simulating of the Stationary Wigner Inflow Boundary Value Problem
- PDF / 1,560,782 Bytes
- 19 Pages / 439.37 x 666.142 pts Page_size
- 13 Downloads / 180 Views
Optimization Modeling and Simulating of the Stationary Wigner Inflow Boundary Value Problem Zhangpeng Sun1 · Wenqi Yao2
· Tiao Lu3
Received: 20 December 2019 / Revised: 29 September 2020 / Accepted: 8 October 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract The stationary Wigner inflow boundary value problem (SWIBVP) is modeled as an optimization problem by using the idea of shooting method in this paper. To remove the singularity at v = 0, we consider a regularized SWIBVP, where a regularization constraint is considered along with the original SWIBVP, and a modified optimization problem is established for it. A shooting algorithm is proposed to solve the two optimization problems, involving the limited-memory BFGS (L-BFGS) algorithm as the optimization solver. Numerical results show that solving the optimization problems with respect to the SWIBVP with the shooting algorithm is as effective as solving the SWIBVP with Frensley’s numerical method (Frensley in Phys Rev B 36:1570–1580, 1987). Furthermore, the modified optimization problem gets rid of the singularity at v = 0, and preserves symmetry of the Wigner function, which implies the optimization modeling with respect to the regularized SWIBVP is successful. Keywords The stationary Wigner equation · Inflow boundary condition · Shooting method · Optimization problem · Regularization Mathematics Subject Classification 45J05 · 82B05 · 34K28
1 Introduction The Wigner equation was firstly proposed as the quantum correction of the classical Boltzmann equation by Wigner in 1932 [1]. It has been widely used in the fields of quantum information process, quantum physics, quantum electronics, etc.(for a review, e.g., [2]).
B
Wenqi Yao [email protected] Zhangpeng Sun [email protected] Tiao Lu [email protected]
1
School of Mathematics and Computational Science, XiangTan University, Xiangtan, Hunan, China
2
School of Mathematics, South China University of Technology, Guangzhou, Guangdong, China
3
CAPT, HEDPS, LMAM, School of Mathematical Sciences, Peking University, Beijing, China 0123456789().: V,-vol
123
21
Page 2 of 19
Journal of Scientific Computing
(2020) 85:21
Compared with other quantum pictures, such as the Schrödinger picture and the Heisenberg picture, the Wigner equation as an inheritance from the Boltzmann equation can be investigated with plenty of numerical methods that have been performed very well in solving the Boltzmann equation [3–12]. Especially, the stationary Wigner equation with inflow boundary condition has become a popular model for simulating nanoscale semiconductor devices, ever since Frensley reproduced the negative differential resistance phenomenon of resonant tunneling diodes in 1987 [13]. The wellposedness of the stationary Wigner inflow boundary value problem (SWIBVP), i.e., (1)–(3), is still an open problem, where many efforts have been made to solve it [14– 17]. In [16], the wellposedness of the SWIBVP with periodic potential has been proved. In [17], the authors have discussed the wellpo
Data Loading...
