- A preview of this full-text is provided by Springer Nature.
- Learn more

Preview content only

Content available from Journal of Scientific Computing

This content is subject to copyright. Terms and conditions apply.

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

© Springer Science+Business Media, LLC, part of Springer Nature 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

123

Content courtesy of Springer Nature, terms of use apply. Rights reserved.