Speaker: Dr. Sihong Shao (邵嗣烘博士)

Time: 11:15-12:15, 8 March 2023 (Wednesday) (Beijing time)

Venue: A103,Lijiao Building

Tencent Meeting ID: 459-714-620

Abstract

The Wigner function has provided an equivalent and convenient way to render quantum mechanics in phase space. It allows one to express macroscopically measurable quantities, such as currents and heat fluxes, in statistical forms as usually does in classical statistical mechanics, thereby facilitating its applications in nanoelectronics, quantum optics and etc. Distinct from the Schrödinger equation, the most appealing feature of the Wigner equation, which governs the dynamics of the Wigner function, is that it shares many analogies to the classical mechanism and simply reduces to the classical counterpart when the reduced Planck constant vanishes. Despite the theoretical advantages, numerical resolution for the Wigner equation is notoriously difficult and remains one of the most challenging problems in computational physics, mainly because of its high dimensionality and nonlocal pseudo-differential operator. On one hand, the commonly used finite difference methods fail to capture the highly oscillatory structure accurately. On the other hand, all existing stochastic algorithms, including the affinity-based Wigner Monte Carlo and signed particle Wigner Monte Carlo methods, have been confined to 2-D phase space. Few results have been reported for higher dimensional simulations. My group has made substantial progress in both grid-based deterministic and particle-based stochastic methods.

We attempted to solve the Wigner equation in 4-D and 6-D phase space with gird-based deterministic methods by exploiting its intriguing mathematical structure. For 4-D simulations, we succeeded to detail the quantum dynamics of a Helium-like system and the quantum interference fringes in the double-slit experiment. For the 6-D Wigner-Coulomb system, we proposed a massively parallel solver, termed the characteristic-spectral-mixed scheme (CHASM), which utilizes the locally distributed cubic B-spline basis to interpolate the local spatial advection and the truncated kernel method to approximate the pseudodifferential operator with weakly singular symbol under the Coulomb interaction. Several typical numerical experiments demonstrate the accuracy and efficiency of CHASM, as well as its scalability up to 16000 cores.

We built the bridge between the Wigner equation and a stochastic particle method in a rigorous manner and proposed a SPA (Stationary Phase Approximation) + SPADE (Sequential-clustering Particle Annihilation via Discrepancy Estimation) strategy to overcome the sign problem. The curse of dimensionality which causes the unattainable exponential wall is translated into the NP-hard problems that may have approximate solutions. SPADE follows a divide-and-conquer strategy: Adaptive clustering of particles via controlling their number-theoretic discrepancies and independent random matching in each group, and it may learn the minimal amount of particles that can accurately capture the non-classicality of the Wigner function. A thorough performance benchmark of SPADE is provided with the reference solutions in 6-D phase space produced by CHASM under a 733*803 uniform grid, which fully explores the limit of grid-based deterministic Wigner solvers. Simulations of the proton-electron couplings in 6-D and 12-D phase space demonstrate the accuracy and the efficiency of our particle-based stochastic methods.

As a permanent goal and a tireless direction of computational mathematics, developing an accurate and stable high-dimensional solver has been attracting more and more attentions in recent years due to the urgent need in e.g., quantum science and high energy density physics. This talk represents our recent attempts to break the curse of dimensionality which poses a fundamental obstacle to high-dimensional numerical simulations.

Reference

[1] Y. Xiong, Y. Zhang, S. Shao, A characteristic-spectral-mixed scheme for six-dimensional Wigner-Coulomb dynamics, arXiv:2205.02380v1, 2022.

[2] Y. Xiong, S. Shao, Overcoming the numerical sign problem in the Wigner dynamics via adaptive particle annihilation, arXiv:2008.05161v2, 2022.

[3] S. Shao, Y. Xiong, SPADE: Sequential-clustering particle annihilation via discrepancy estimation, arXiv:2005.05129v1, 2020.

[4] S. Shao, Lili Su, Nonlocalization of singular potentials in quantum dynamics, arXiv:2301.07298v1, 2023.

[5] Y. Xiong, Y. Zhang, S. Shao, Performance evaluations on the parallel CHAracteristic-Spectral-Mixed (CHASM) scheme, arXiv:2205.01922v1, 2022.

[6] Z. Chen, H. Jiang, S. Shao, A higher-order accurate operator splitting spectral method for the Wigner-Poisson system, Journal of Computational Electronics 21 (2022) 756.

[7] S. Shao, Y. Xiong, Branching random walk solutions to the Wigner equation, SIAM Journal on Numerical Analysis 58 (2020) 2589.

[8] S. Shao, Y. Xiong, A branching random walk method for many-body Wigner quantum dynamics, Numerical Mathematics: Theory, Methods and Applications 12 (2019) 21.

[9] Z. Chen, S. Shao, W. Cai, A high order efficient numerical method for 4-D Wigner equation of quantum double-slit interferences, Journal of Computational Physics 396 (2019) 54.

[10] Y. Xiong, S. Shao, The Wigner branching random walk: Efficient implementation and performance evaluation, Communications in Computational Physics 25 (2019) 871.

[11] Z. Chen, Y. Xiong, S. Shao, Numerical methods for the Wigner equation with unbounded potential, Journal of Scientific Computing 79 (2019) 345.

[12] Y. Xiong, Z. Chen, and S. Shao, An advective-spectral-mixed method for time-dependent many-body Wigner simulations, SIAM Journal on Scientific Computing 38 (2016) B491.

[13] S. Shao and J. M. Sellier, Comparison of deterministic and stochastic methods for time-dependent Wigner simulations, Journal of Computational Physics 300 (2015) 167.

[14] S. Shao, T. Lu, and W. Cai, Adaptive conservative cell average spectral element methods for transient Wigner equation in quantum transport, Communications in Computational Physics 9 (2011) 711.

About Dr. Shao

邵嗣烘，北京大学数学科学学院副教授，毕业于北京大学数学科学学院并获得理学学士和博士学位，先后到访过北卡罗莱那大学夏洛特分校，香港科技大学，普林斯顿大学、塞维利亚大学和香港中文大学等。主要开展面向智能、量子和计算的交叉融合研究，落脚点在基础的数学理论和高效的算法设计，强调离散数学结构的设计、分析和应用。具体研究领域包括：高维数值方法、组合优化、计算量子力学、图（网络）上的数学及其算法、微分方程数值解和脑科学等，获国家自然科学基金青年，面上和优青连续资助。2019年入选北京智源人工智能研究院“智源青年科学家”。2020年获北京大学优秀博士学位论文指导老师。2021年获北京大学黄廷芳/信和青年杰出学者奖。曾获中国计算数学学会优秀青年论文一等奖，北京大学学术类创新奖，北京大学优秀博士学位论文三等奖，宝洁教师奖和北京大学优秀班主任等。