北京大学复旦大学厦门大学计算数学联合学术报告会
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：2023-04-07 09:00 —— 2023-04-09 17:30
：厦大海韵园实验楼105报告厅
一、 日程表
4月8日会议地点：厦门大学海韵园实验楼105报告厅
时间 | 报告题目 | 报告人 | |
4 月 8 日 | 08:40-09:00 | 开幕式、合影留念 | |
主持人： 张磊 | |||
09:00-09:30 | Computational Quantum Mechanics in Phase Space— An Attempt to Break the Curse of Dimensionality | 邵嗣烘
| |
09:30-10:00 | Optimal Orbital Selection for Quantum Full Configuration Interaction | 李颖洲 | |
10:00-10:30 | A Geometric Proximal Gradient Method for Sparse Least Squares Regression with Probabilistic Simplex Constraint | 白正简 | |
10:30-10:50 | 休息 | ||
主持人：高卫国 | |||
10:50-11:20 | Numerical approaches to fluid singularities | 黄得 | |
11:20-12:00 | A mixed precision Jacobi SVD algorithm | 邵美悦 | |
12:00-13:30 | 午餐 | ||
主持人：陈黄鑫 | |||
13:30-14:00 | Theoretical Analysis of Inductive Biases in Deep Convolutional Networks | 吴磊 | |
14:00-14:30 | Inversion of trace formulas for inverse spectral problems | 翟剑 | |
14:30-15:00 | Efficient and stable SAV-based methods for the training in deep learning | 毛志平 | |
15:00-18:00 | 自由讨论 | ||
18:30 | 晚餐 | ||
二、学术报告题目与摘要
Computational Quantum Mechanics in Phase Space
— An Attempt to Break the Curse of Dimensionality
邵嗣烘（北京大学）
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 preliminary attempts to break the curse of dimensionality (CoD) which poses a fundamental obstacle to high-dimensional numerical simulations. More specifically, we will report some recent progress in both grid-based deterministic and particle-based stochastic methods for simulating high-dimensional Wigner quantum dynamics. A massively parallel solver, termed the characteristic-spectral-mixed (CHASM) scheme, is proposed to evolve the Wigner-Coulomb system in 6-D phase space. Within particle-based stochastic simulations, CoD, causing the unattainable exponential wall, reappears as the numerical sign problem. To this end, we propose a SPA (Stationary Phase Approximation) + SPADE (Sequential-clustering Particle Annihilation via Discrepancy Estimation) strategy is to overcome the numerical sign problem where it has been translated into a NP-hard problem that may have approximate solutions. 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.
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.
专家简介：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。
Optimal Orbital Selection for Quantum Full Configuration Interaction
李颖洲（复旦大学）
Near-term quantum computers will be limited in the number of qubits on which they can process information as well as the depth of the circuits that they can coherently carry out. To-date, experimental demonstrations of algorithms such as the Variational Quantum Eigensolver (VQE) have been limited to small molecules using minimal basis sets for this reason. In this talk, we propose an orbital optimization scheme and incorporate it into quantum eigensolvers where a parameterized partial unitary transformation is applied to the basis functions set in order to reduce the number of qubits required for a given problem. The optimal transformation is found by minimizing the ground state energy with respect to this partial unitary matrix. Through numerical simulations of small molecules up to 16 spin orbitals, we demonstrate that this method has the ability to greatly extend the capabilities of near-term quantum computers with regard to the electronic structure problem. We find that VQE paired with orbital optimization consistently achieves lower ground state energies than traditional VQE and frequently achieves lower ground state energies than VQE and FCI methods using larger basis sets.
专家简介：李颖洲，复旦大学数学科学学院青年研究员。2012年于复旦大学取得学士学位，2017年于美国斯坦福大学取得计算数学博士学位，之后2017年至2020年在美国杜克大学数学系担任科研助理教授。其科研领域包括：快速算法设计、高性能计算与并行计算、量子计算、机器学习等。已在多个领域的国际顶尖杂志(ACHA, SISC, SIMAX, JCTC等等)发表论文30多篇。
A Geometric Proximal Gradient Method for Sparse Least Squares Regression with Probabilistic Simplex Constraint
白正简（厦门大学）
In this talk, we consider the sparse least squares regression problem with probabilistic simplex constraint. Due to the probabilistic simplex constraint, one could not apply directly the L1-regularization to the considered regression model. To find a sparse solution, we reformulate the sparse least squares regression problem as a nonconvex and nonsmooth L1-regularized minimization problem over the unit sphere. Then we propose a geometric proximal gradient method for solving the regularized problem with a varied regularized parameter, where the explicit expression of the global solution to every involved subproblem is obtained. The global convergence of the proposed method is established under some mild assumptions. Some numerical results are reported to illustrate the effectiveness of the proposed algorithm.
专家简介：白正简，厦门大学教授、博士生导师。2004年博士毕业于香港中文大学，曾在新加坡国立大学和意大利Insubria 大学作博士后和访问学者。主要研究方向为数值代数、特征值问题及其逆问题、矩阵流形及其在数据科学中的应用等。曾主持国家自然科学基金面上项目和福建省杰出青年基金。在SIAM系列, Numer. Math., Inverse Problems等本学科主流期刊上发表学术论文40余篇。曾获得2009年度福建省科学技术奖二等奖和2010年度”教育部新世纪优秀人才支持计划”入选者。
Numerical approaches to fluid singularities
黄得（北京大学）
Whether the 3D incompressible Euler equations can develop a singularity in finite time from smooth initial data is one of the most challenging problems in mathematical fluid dynamics. In the past decades, numerical approaches have played more and more important roles in the search for potential finite-time singularity, though many new challenges arose as well. We will first review some groundbreaking works that provide insights and evidence of Euler singularity using both classical and modern numerical methods. After that, we will present some recent progresses in this line of research.
专家简介：黄得，2011-2015年就读于北京大学，获信息与计算科学学士学位，物理学双学位。2020年获美国加州理工学院应用与计算数学博士学位。2020-2021年在美国加州理工学院计算与数学科学系从事博士后研究工作。2021年8月加盟北京大学数学科学学院信息与计算科学系，任助理教授。研究领域为随机矩阵理论，流体偏微分方程。
A mixed precision Jacobi SVD algorithm
邵美悦（复旦大学）
We propose a mixed precision Jacobi algorithm for computing the singular value decomposition (SVD) of a dense matrix. After appropriate preconditioning, the proposed algorithm computes the SVD in a lower precision as an initial guess, and then performs one-sided Jacobi rotations in the working precision as iterative refinement. By carefully transforming a lower precision solution to a higher precision one, our algorithm achieves about 2x speedup on the x86-64 architecture compared to the usual one-sided Jacobi SVD algorithm in LAPACK, without sacrificing the accuracy.
专家简介：邵美悦，复旦大学大数据学院青年研究员。2014年毕业于瑞士洛桑联邦理工学院，获得计算数学博士学位。之后在美国劳伦斯伯克利国家实验室从事研究工作，先后担任博士后研究员（2014–2017）和项目科学家（2017–2019）。成果发表在ACM TOMS，SIMAX，SISC，IEEE TPDS，JCTC，CPC等前沿刊物上。主要研究方向：数值线性代数和高性能计算。
Theoretical Analysis of Inductive Biases in Deep Convolutional Networks
吴磊（北京大学）
In this talk, we'll discuss the inductive biases of convolutional neural networks (CNNs), which are believed to be vital drivers behind CNNs' exceptional performance on vision-like tasks. Specifically, we'll analyze the universality of CNNs and show that achieving it requires only a depth of $\cO(\log d)$, where $d$ is the input dimension. Additionally, we'll demonstrate that CNNs can efficiently capture long-range sparse correlations with only $\tilde{\cO}(\log^2d)$ samples. These are achieved through a novel combination of increased network depth and the utilization of multichanneling and downsampling.
We'll also explore the inductive biases of weight sharing and locality through the lens of symmetry by introducing locally-connected networks (LCNs), which can be viewed as CNNs without weight sharing. We'll compare the performance of CNNs, LCNs, and fully-connected networks (FCNs) on a simple regression task and highlight the cruciality of weight sharing and the importance of locality. Our findings demonstrate that weight sharing and locality break different symmetries in the learning process, leading to provable separations between the two biases.
专家简介：吴磊，2012年获南开大学学士学位，2018年获北京大学博士学位。2018-2021年在普林斯顿大学应用与计算数学系从事博士后研究。2021年12月入职北京大学数学科学学院信息与计算科学系，任助理教授。研究领域为深度学习的数学理论，特别侧重于随机梯度算法的隐式正则化和神经网络模型的逼近性质。
Inversion of trace formulas for inverse spectral problems
翟剑（复旦大学）
We propose a new algorithm for inverse spectral problems based on inversion of a sequence of trace formulas. I will demonstrate how this algorithm works for a Sturm-Liouville operator and a damped wave operator.
专家简介：翟剑，现任复旦大学数学科学学院青年副研究员。主要从事波方程，特别是弹性波方程，相关反问题的研究。博士毕业于莱斯大学，曾在华盛顿大学以及香港科技大学从事博士后研究。
Efficient and stable SAV-based methods for the training in deep learning
毛志平（厦门大学）
As we known, the accuracy and efficiency of neural networks approximation in deep learning are significantly affected by the optimization algorithms. Most of existed algorithms are GD-based, in which the learning rate is very important and needed to be carefully set. In this work, we consider developing efficient and stable methods for the training process in deep learning. In particular, we shall consider the gradient flows arising from deep learning and solve the corresponding gradient flows using SAV-based method resulting efficient and energy stable algorithms. We propose several kinds of SAV-based methods and show the energy stabilities. We also consider the adaptive algorithm to improve the accuracy. We demonstrate the effectiveness of the present algorithms with several numerical examples.
专家简介：毛志平，厦门大学数学科学学院教授，2009年本科毕业于重庆大学，2015年博士毕业于厦门大学计算数学专业，国家高层次青年人才，2015年10月至2020年9月在美国布朗大学应用数学系从事博士后研究。毛志平博士主要从事谱方法以及机器学习方面的研究，其目前在SIREV, JCP，SISC，SINUM、CMAME等国际高水平杂志上发表论文20余篇。
