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A Riemannian derivative-free Polak-Ribiére-Polyak method for tangent vector field

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arXiv 2019年

作者： Yao, Teng-Teng Zhao, Zhi Bai, Zheng-Jian Jin, Xiao-Qing Department of Mathematics School of Sciences Zhejiang University of Science and Technology Hangzhou310023 China Department of Mathematics School of Sciences Hangzhou Dianzi University Hangzhou310018 China School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing Xiamen University Xiamen361005 China Department of Mathematics University of Macau China

This paper is concerned with the problem of finding a zero of a tangent vector field on a Riemannian manifold. We first reformulate the problem as an equivalent Riemannian optimization problem. Then we propose a Riemannian derivative-free Polak-Ribiére-Polyak method for solving the Riemannian optimization problem, where a non-monotone line search is employed. The global convergence of the proposed method is established under some mild assumptions. To further improve the efficiency, we also provide a hybrid method, which combines the proposed geometric method with the Riemannian Newton method. Finally, some numerical experiments are reported to illustrate the efficiency of the proposed method. Copyright © 2019, The Authors. All rights reserved.

关键词： Newton-Raphson method

EXtended HDG methods for second order elliptic interface problems

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arXiv 2019年

作者： Han, Yihui Chen, Huangxin Wang, Xiao-Ping Xie, Xiaoping School of Mathematics Sichuan University Chengdu610064 China School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling and High Performance Scientific Computing Xiamen University Fujian361005 China Department of Mathematics Hong Kong University of Science and Technology Clear Water Bay Kowloon Hong Kong Hong Kong School of Mathematics Sichuan University Chengdu610064 China

In this paper, we propose two arbitrary order eXtended hybridizable Discontinuous Galerkin (X-HDG) methods for second order elliptic interface problems in two and three dimensions. The first X-HDG method applies to any piecewise C2 smooth interface. It uses piecewise polynomials of degrees k (k ≥ 1) and k − 1 respectively for the potential and flux approximations in the interior of elements inside the subdomains, and piecewise polynomials of degree k for the numerical traces of potential on the inter-element boundaries inside the subdomains. Double value numerical traces on the parts of interface inside elements are adopted to deal with the jump condition. The second X-HDG method is a modified version of the first one and applies to any fold line/plane interface, which uses piecewise polynomials of degree k−1 for the numerical traces of potential. The X-HDG methods are of the local elimination property, then lead to reduced systems which only involve the unknowns of numerical traces of potential on the inter-element boundaries and the interface. Optimal error estimates are derived for the flux approximation in L2 norm and for the potential approximation in piecewise H1 seminorm without requiring "sufficiently large" stabilization parameters in the schemes. In addition, error estimation for the potential approximation in L2 norm is performed using dual arguments. Finally, we provide several numerical examples to verify the theoretical results. Copyright © 2019, The Authors. All rights reserved.

关键词： Galerkin methods

Invariant foliations for stochastic dynamical systems with multiplicative stable Lévy noise

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arXiv 2018年

作者： Chao, Ying Wei, Pingyuan Yuan, Shenglan School of Mathematics and Statistics Center for Mathematical Sciences Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Sciences and Technology Wuhan430074 China

This work deals with the dynamics of a class of stochastic dynamical systems with a multiplicative non-Gaussian Lévy noise. We first establish the existence of stable and unstable foliations for this kind of system via the Lyapunov-Perron method. Then we examine the geometric structure of the invariant foliations, and their relation with invariant manifolds. Finally, we illustrate our results in an example. Copyright © 2018, The Authors. All rights reserved.

关键词： Differential equations

A structure-preserving one-sided jacobi method for computing the SVD of a quaternion matrix

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arXiv 2018年

作者： Ma, Ru-Ru Bai, Zheng-Jian School of Mathematical Sciences Xiamen University Xiamen361005 China School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling & High Performance Scientific Computing Xiamen University Xiamen361005 China

In this paper, we provide a structure-preserving one-sided cyclic Jacobi method for computing the singular value decomposition of a quaternion matrix. In this method, the columns of the quaternion matrix are orthogonalized in pairs by using a sequence of orthogonal JRS-symplectic Jacobi matrices to its real counterpart. The quadratic convergence is also established under some mild conditions. Numerical tests are reported to illustrate the efficiency of the proposed method. Copyright © 2018, The Authors. All rights reserved.

关键词： Singular value decomposition

Phase retrieval for sub-Gaussian measurements

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arXiv 2019年

作者： Gao, Bing Liu, Haixia Wang, Yang School of Mathematical Sciences Nankai University Tianjin300071 China School of Mathematics and Statistics Huazhong University of Science and Technology Wuhan430074 China Hubei Key Laboratory of Engineering Modeling and Scientific Computing Huazhong University of Science and Technology Wuhan430074 Department of Mathematics Hong Kong University of Science and Technology Clear Water Bay Kowloon Hong Kong

Generally, phase retrieval problem can be viewed as the reconstruction of a function/signal from only the magnitude of the linear measurements. These measurements can be, for example, the Fourier transform of the density function. Computationally the phase retrieval problem is very challenging. Many algorithms for phase retrieval are based on i.i.d. Gaussian random measurements. However, Gaussian random measurements remain one of the very few classes of measurements. In this paper, we develop an efficient phase retrieval algorithm for sub-gaussian random frames. We provide a general condition for measurements and develop a modified spectral initialization. In the algorithm, we first obtain a good approximation of the solution through the initialization, and from there we use Wirtinger Flow to solve for the solution. We prove that the algorithm converges to the global minimizer linearly. Copyright © 2019, The Authors. All rights reserved.

关键词： Gaussian distribution

A Semi-Langrangian Vortex Penalization Method for3D Incompressible Flows

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数学研究 2019年第3期52卷 277-298页

作者： Chlo Mimeau Iraj Mortazavi Georges-Henri Cottet School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High Performance Scientific Computing Xiamen UniversityXiamen 361005P.R.China Bordeaux INP I2M(UMR CNRS 5295)Université de Bordeaux33607 PessacFrance1 Laboratoire M2N ModélisationMathématique et Num eriqueConservatoire National des Arts et Métiers292 Rue St Martin75003 ParisFrance

A remeshed Vortex method is proposed in this work to simulate three-dimensional incompressible *** convection equation is solved on particles,using a Vortex method,which are then remeshed on a Cartesian underlying *** other differential operators involved in the governing incompressible Navier-Stokes equations are discretized on the grid,through finite differences method or in spectral *** the present work,the redistribution of the particles on the Cartesian mesh is performed using a directional splitting,allowing to save significant computational efforts especially in the case of 3D flows.A coupling of this semi-Lagrangian method with an immersed boundary method,namely the Brinkman penalization technique,is proposed in this paper in order to efficiently take into account the presence of solid and porous obstacles in the fluid flow and then to perform passive flow control us-ing porous *** method,which combines the robustness of particle methods and the flexibility of penalization method,is validated and exploited in the context of different flow physics.

关键词： Semi-Lagrangian method remeshed Vortex method penalization approach bluff body flows DNS.

DeePMD-kit v2: A software package for Deep Potential models

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arXiv 2023年

作者： Zeng, Jinzhe Zhang, Duo Lu, Denghui Mo, Pinghui Li, Zeyu Chen, Yixiao Rynik, Marián Huang, Li'ang Li, Ziyao Shi, Shaochen Wang, Yingze Ye, Haotian Tuo, Ping Yang, Jiabin Ding, Ye Li, Yifan Tisi, Davide Zeng, Qiyu Bao, Han Xia, Yu Huang, Jiameng Muraoka, Koki Wang, Yibo Chang, Junhan Yuan, Fengbo Bore, Sigbjørn Løland Cai, Chun Lin, Yinnian Wang, Bo Xu, Jiayan Zhu, Jia-Xin Luo, Chenxing Zhang, Yuzhi Goodall, Rhys E.A. Liang, Wenshuo Singh, Anurag Kumar Yao, Sikai Zhang, Jingchao Wentzcovitch, Renata Han, Jiequn Liu, Jie Jia, Weile York, Darrin M. Weinan, E. Car, Roberto Zhang, Linfeng Wang, Han Laboratory for Biomolecular Simulation Research Institute for Quantitative Biomedicine Department of Chemistry and Chemical Biology Rutgers University PiscatawayNJ08854 United States AI for Science Institute Beijing100080 China DP Technology Beijing100080 China Academy for Advanced Interdisciplinary Studies Peking University Beijing100871 China HEDPS CAPT College of Engineering Peking University Beijing100871 China College of Electrical and Information Engineering Hunan University Changsha China Yuanpei College Peking University Beijing100871 China Program in Applied and Computational Mathematics Princeton University PrincetonNJ08540 United States Department of Experimental Physics Comenius University Mlynská Dolina F2 Bratislava842 48 Slovakia Center for Quantum Information Institute for Interdisciplinary Information Sciences Tsinghua University Beijing100084 China Center for Data Science Peking University Beijing100871 China ByteDance Research Zhonghang Plaza No. 43 North 3rd Ring West Road Haidian District Beijing China College of Chemistry and Molecular Engineering Peking University Beijing100871 China Baidu Inc. Beijing China Key Laboratory of Structural Biology of Zhejiang Province School of Life Sciences Westlake University Zhejiang Hangzhou China Westlake AI Therapeutics Lab Westlake Laboratory of Life Sciences and Biomedicine Zhejiang Hangzhou China Department of Chemistry Princeton University PrincetonNJ08544 United States SISSA Scuola Internazionale Superiore di Studi Avanzati Trieste34136 Italy Laboratory of Computational Science and Modeling Institute of Materials École Polytechnique Fédérale de Lausanne Lausanne1015 Switzerland Department of Physics National University of Defense Technology Hunan Changsha410073 China State Key Lab of Processors Institute of Computing Technology Chinese Academy of Sciences Beijing China University of Chinese Academy of Sciences Beijing China School of Electronics Engineerin

DeePMD-kit is a powerful open-source software package that facilitates molecular dynamics simulations using machine learning potentials (MLP) known as Deep Potential (DP) models. This package, which was released in 2017, has been widely used in the fields of physics, chemistry, biology, and material science for studying atomistic systems. The current version of DeePMD-kit offers numerous advanced features such as DeepPot-SE, attention-based and hybrid descriptors, the ability to fit tensile properties, type embedding, model deviation, Deep Potential - Range Correction (DPRc), Deep Potential Long Range (DPLR), GPU support for customized operators, model compression, non-von Neumann molecular dynamics (NVNMD), and improved usability, including documentation, compiled binary packages, graphical user interfaces (GUI), and application programming interfaces (API). This article presents an overview of the current major version of the DeePMD-kit package, highlighting its features and technical details. Additionally, the article benchmarks the accuracy and efficiency of different models and discusses ongoing developments. © 2023, CC BY-NC-ND.

关键词： Molecular dynamics

Corrigendum to “High order positivity-preserving discontinuous Galerkin schemes for radiative transfer equations on triangular meshes” [J. Comput. Phys. 397 (2019) 108811]

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Journal of Computational Physics 2020年 406卷

作者： Min Zhang Juan Cheng Jianxian Qiu School of Mathematical Sciences Xiamen University Xiamen Fujian 361005 PR China Laboratory of Computational Physics Institute of Applied Physics and Computational Mathematics Beijing 100088 PR China Center for Applied Physics and Technology Peking University Beijing 100871 PR China School of Mathematical Sciences and Fujian Provincial Key Laboratory of Mathematical Modeling and High-Performance Scientific Computing Xiamen University Xiamen Fujian 361005 PR China

An accurate spectral method for maxwell equations in cole-cole dispersive media

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arXiv 2018年

作者： Huang, Can Wang, Li-Lian School of Mathematical Sciences Fujian Provincial Key Laboratory on Mathematical Modeling High Performance Scientific Computing Xiamen University Fujian361005 China Division of Mathematical Sciences School of Physical and Mathematical Sciences Nanyang Technological University Singapore637371 Singapore

In this paper, we propose an accurate numerical means built upon a spectral- Galerkin method in spatial discretization and an enriched multi-step spectral-collocation approach in temporal direction, for Maxwell equations in Cole-Cole dispersive media in twodimensional setting. Our starting point is to derive a new model involving only one unknown field from the original model with three unknown fields: electric, magnetic fields and the induced electric polarisation (described by a global temporal convolution of the electric field). This results in a second-order integral-differential equation with a weakly singular integral kernel expressed by the Mittag-Lefler (ML) function. The most interesting but challenging issue resides in how to efficiently deal with the singularity in time induced by the ML function which is an infinite series of singular power functions with different nature. With this in mind, we introduce a spectral-Galerkin method using Fourier-like basis functions for spatial discretization, leading to a sequence of decoupled temporal integral-differential equations (IDE) with the same weakly singular kernel involving the ML function as the original two-dimensional problem. With a careful study of the regularity of IDE, we incorporate several leading singular terms into the numerical scheme and approximate much regular part of the solution. Then we solve to IDE by a multi-step well-conditioned collocation scheme together with mapping technique to increase the accuracy and enhance the resolution. We show such an enriched collocation method is convergent and accurate. Copyright © 2018, The Authors. All rights reserved.

关键词： Maxwell equations