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

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

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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.
Comparison of densities solved with Frensley’s method on gradually refined mesh grids with Nx=720,1440,2880,5760\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_x =720,1440,2880,5760$$\end{document} and Nv=64\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_v = 64$$\end{document}, and ρ(θ∗)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varvec{\rho }(\varvec{\theta }^*)$$\end{document} with respect to the WRSM with Nx=720\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_x=720$$\end{document} and Nv=64\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$N_v=64$$\end{document}. In all of the simulations, boundary condition (39) is used
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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
© 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
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