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# Optimization Modeling and Simulating of the Stationary Wigner Inflow Boundary Value Problem

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Journal of Scientiﬁc Computing (2020) 85:21
https://doi.org/10.1007/s10915-020-01338-2
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 /
Published online: 15 October 2020
Abstract
The stationary Wigner inﬂow boundary value problem (SWIBVP) is modeled as an optimiza-
tion 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 Boltz-
mann 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]).
BWenqi Yao
yaowq@scut.edu.cn
Zhangpeng Sun
zpsun@xtu.edu.cn
Tiao Lu
tlu@math.pku.edu.cn
1School of Mathematics and Computational Science, XiangTan University, Xiangtan, Hunan, China
2School of Mathematics, South China University of Technology, Guangzhou, Guangdong, China
3CAPT, HEDPS, LMAM, School of Mathematical Sciences, Peking University, Beijing, China
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