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Discrete energy analysis of the third-order variable-step BDF time-stepping for diffusion equations

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arXiv 2022年

作者： Liao, Hong-Lin Tang, Tao Zhou, Tao School of Mathematics Nanjing University of Aeronautics and Astronautics Nanjing211106 China MIIT Nanjing211106 China Division of Science and Technology BNU-HKBU United International College Guangdong Province Zhuhai China Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China

This is one of our series works on discrete energy analysis of the variable-step BDF schemes. In this part, we present stability and convergence analysis of the third-order BDF (BDF3) schemes with variable steps for linear diffusion equations, see e.g. [SIAM J. Numer. Anal., 58:2294-2314] and [math. Comp., 90: 1207-1226] for our previous works on the BDF2 scheme. To this aim, we first build up a discrete gradient structure of the variable-step BDF3 formula under the condition that the adjacent step ratios are less than 1.4877, by which we can establish a discrete energy dissipation law. Mesh-robust stability and convergence analysis in the L2 norm are then obtained. Here the mesh robustness means that the solution errors are well controlled by the maximum time-step size but independent of the adjacent time-step ratios. We also present numerical tests to support our theoretical *** Codes 65M06, 65M12 © 2022, CC BY-NC-SA.

关键词： Partial differential equations

Achievable Rate Maximization Pattern Design for Reconfigurable MIMO Antenna Array

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arXiv 2022年

作者： Wang, Haonan Li, Ang Liu, Ya-Feng Qin, Qibo Song, Lingyang Li, Yonghui The School of Information and Communications Engineering Faculty of Electronic and Information Engineering Xi'an Jiaotong University Shaanxi Xi'An710049 China The State Key Laboratory of Integrated Services Networks Xidian University Shaanxi Xi'An China The State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Wireless Network RAN Research Department Shanghai Huawei Technologies Co. Ltd. Shanghai201206 China The School of Electrical Engineering and Computer Science Peking University Beijing100871 China The School of Electrical and Information Engineering The University of Sydney SydneyNSW2006 Australia

Reconfigurable multiple-input multiple-output can provide performance gains over traditional MIMO by reshaping the channels, i.e., introducing more channel realizations. In this paper, we focus on the achievable rate maximization pattern design for reconfigurable MIMO systems. Firstly, we introduce the matrix representation of pattern reconfigurable MIMO (PR-MIMO), based on which a pattern design problem is formulated. To further reveal the effect of the radiation pattern on the wireless channel, we consider pattern design for both the single-pattern case where the optimized radiation pattern is the same for all the antenna elements, and the multi-pattern case where different antenna elements can adopt different radiation patterns. For the single-pattern case, we show that the pattern design is equivalent to a redistribution of gains among all scattering paths, and an eigenvalue optimization based solution is obtained. For the multi-pattern case, we propose a sequential optimization framework with manifold optimization and eigenvalue decomposition to obtain near-optimal solutions. Numerical results validate the superiority of PR-MIMO systems over traditional MIMO in terms of achievable rate, and also show the effectiveness of the proposed solutions. Copyright © 2022, The Authors. All rights reserved.

关键词： MIMO systems

Numerical Investigations of a Class of Biological Models on Unbounded Domain

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Numerical mathematics(Theory,Methods and Applications) 2019年第1期12卷 169-186页

作者： Qiumei Huang Dongfang Li Jiwei Zhang Beijing Institute for Scientific and Engineering Computing Beijing University of TechnologyBeijing 100124China School of Mathematics and Statistics Huazhong University of Science and TechnologyWuhan 430074China Hubei Key Laboratoryof Engineering Modeling and Scientific Computing Huazhong University of Science and TechnologyWuhan 430074China Beijing Computational Science Research Center Beijing1000193China

This paper is concerned with numerical computations of a class of biologi-cal models on unbounded spatial *** overcome the unboundedness of spatial domain,we first construct efficient local absorbing boundary conditions(LABCs)to re-formulate the Cauchy problem into an initial-boundary value(IBV)*** that,we construct a linearized finite difference scheme for the reduced IVB problem,and provide the corresponding error estimates and stability *** delay-dependent dynamical properties on the Nicholson’s blowflies equation and the Mackey-Glass equa-tion are numerically ***,numerical examples are given to demonstrate the efficiency of our LABCs and theoretical results of the numerical scheme.

关键词： Biological models artificial boundary method absorbing boundary conditions finite difference method stability,convergence

An Augmented Lagrangian Deep Learning Method for Variational Problems with Essential Boundary Conditions

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arXiv 2021年

作者： Huang, Jianguo Wang, Haoqin Zhou, Tao School of Mathematical Sciences MOE-LSC Shanghai Jiao Tong University Shanghai China LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing China

This paper is concerned with a novel deep learning method for variational problems with essential boundary conditions. To this end, we first reformulate the original problem into a minimax problem corresponding to a feasible augmented Lagrangian, which can be solved by the augmented Lagrangian method in an infinite dimensional setting. Based on this, by expressing the primal and dual variables with two individual deep neural network functions, we present an augmented Lagrangian deep learning method for which the parameters are trained by the stochastic optimization method together with a projection technique. Compared to the traditional penalty method, the new method admits two main advantages: i) the choice of the penalty parameter is flexible and robust, and ii) the numerical solution is more accurate in the same magnitude of computational cost. As typical applications, we apply the new approach to solve elliptic problems and (nonlinear) eigenvalue problems with essential boundary conditions, and numerical experiments are presented to show the effectiveness of the new method. © 2021, CC BY.

关键词： Deep neural networks

Convergent and orthogonality preserving schemes for approximating the Kohn-Sham orbitals

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arXiv 2021年

作者： Dai, Xiaoying Zhang, Liwei Zhou, Aihui LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing 100190 Chinaand School of Mathematical Sciences University of Chinese Academy of Sciences Beijing 100049 China

To obtain convergent numerical approximations without using any orthogonalization operations is of great importance in electronic structure calculations. In this paper, we propose and analyze a class of iteration schemes for the discretized Kohn-Sham Density Functional Theory model, with which the iterative approximations are guaranteed to converge to the Kohn-Sham orbitals exponentially without any orthogonalization as long as the initial orbitals are orthogonal and the time step sizes are given properly. In addition, we present a feasible and efficient approach to get suitable time step sizes and report some numerical experiments to validate our *** Codes 37M05, 81Q05, 81Q35 © 2021, CC BY-NC-SA.

关键词： Density functional theory

An inverse averaging finite element method for solving the size-modified Poisson-Nernst-Planck equations in ion channel simulations

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arXiv 2021年

作者： Shen, Ruigang Zhang, Qianru Lu, Benzhuo LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

In this work, we introduce an inverse averaging finite element method (IAFEM) for solving the size-modified Poisson-Nernst-Planck (SMPNP) equations. Comparing with the classical Poisson-Nernst-Planck (PNP) equations, the SMPNP equations add a nonlinear term to each of the Nernst-Planck (NP) fluxes to describe the steric repulsion which can treat multiple nonuniform particle sizes in simulations. Since the new terms include sums and gradients of ion concentrations, the nonlinear coupling of SMPNP equations is much stronger than that of PNP equations. By introducing a generalized Slotboom transform, each of the size-modified NP equation is transformed into a self-adjoint equation with exponentially behaved coefficient, which has similar simple form to the standard NP equation with the Slotboom transformation. This treatment enables employing our recently developed inverse averaging technique to deal with the exponential coefficients of the reformulated formulations, featured with advantages of numerical stability and flux conservation especially in strong nonlinear and convection-dominated cases. Comparing with previous stabilization methods, the IAFEM proposed in this paper can still possess the numerical stability when dealing with convection-dominated problems. And it is more concise and easier to be numerically implemented. Numerical experiments about a model problem with analytic solutions are presented to verify the accuracy and order of IAFEM for SMPNP equations. Studies about the size-effects of a sphere model and an ion channel system are presented to show that our IAFEM is more effective and robust than the traditional finite element method (FEM) when solving SMPNP equations in simulations of biological *** Codes 35J61, 65N30, 92C40 Copyright © 2021, The Authors. All rights reserved.

关键词： Finite element method

Towards Efficient Large-Scale Network Slicing: An LP Dynamic Rounding-and-Refinement Approach

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arXiv 2021年

作者： Chen, Wei-Kun Liu, Ya-Feng Liu, Fan Dai, Yu-Hong Luo, Zhi-Quan School of Mathematics and Statistics Beijing Key Laboratory on MCAACI Beijing Institute of Technology Beijing100081 China State Key Laboratory of Scientific and Engineering Computing Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Chinese Academy of Sciences Beijing100190 China Department of Electrical and Electronic Engineering Southern University of Science and Technology Shenzhen518055 China Shenzhen Research Institute of Big Data The Chinese University of Hong Kong Shenzhen518172 China

In this paper, we propose an efficient algorithm for the network slicing problem which attempts to map multiple customized virtual network requests (also called services) to a common shared network infrastructure and allocate network resources to meet diverse service requirements. The problem has been formulated as a mixed integer linear programming (MILP) formulation in the literature. We first propose a novel linear programming (LP) relaxation of the MILP formulation. We show that compared with a natural LP relaxation of the MILP formulation, the novel LP relaxation is much more compact in terms of smaller numbers of variables and constraints, and much stronger in terms of providing a better LP bound, which makes it particularly suitable to be embedded in an LP relaxation based algorithm. Then we design an efficient two-stage LP dynamic rounding-and-refinement algorithm based on this novel LP relaxation. In the first stage, the proposed algorithm uses an LP dynamic rounding procedure to place the virtual network functions of all services into cloud nodes while taking traffic routing of all services into consideration;in the second stage, the proposed algorithm uses an iterative LP refinement procedure to obtain a solution for traffic routing of all services with their end-to-end delay constraints being satisfied. Compared with the existing algorithms which either have an exponential complexity or return a low-quality solution, our proposed algorithm achieves a better trade-off between the solution quality and the computational complexity. In particular, the worst-case complexity of our proposed algorithm is polynomial, which makes it suitable for solving large-scale problems. Numerical results demonstrate the effectiveness and efficiency of our proposed algorithm. Copyright © 2021, The Authors. All rights reserved.

关键词： Iterative methods

Vaccine efficacy and SARS-CoV-2 control in California and *** the session 2020e2026:A modeling study

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Infectious Disease Modelling 2022年第1期7卷 62-81页

作者： Md Shahriar Mahmud Md Kamrujjaman Md Mashih Ibn Yasin Adan Md Alamgir Hossain Md Mizanur Rahman Md Shahidul Islam Muhammad Mohebujjaman Md Mamun Molla Department of Computer Science and Engineering State University of BangladeshDhaka1205Bangladesh Department of Mathematics University of DhakaDhaka1000Bangladesh Department of Mathematics and Statistics University of CalgaryCalgaryABCanada Computational Biology Research Lab(CBRL) Department of PharmacyJagannath UniversityDhaka1100Bangladesh Hitotsubashi Institute for Advanced Study Hitotsubashi UniversityNaka KunitachiTokyo186-8601Japan Department of Mathematics and Physics Texas A&M International UniversityLaredoTX78041USA Department of Mathematics&Physics North South UniversityDhaka1229Bangladesh Center for Applied Scientific Computing(CASC) North South UniversityDhaka1229Bangladesh

Background:Besides maintaining health precautions,vaccination has been the only prevention from SARS-CoV-2,though no clinically proved 100%effective vaccine has been developed till *** this stage,to withhold the debris of this pandemic-experts need to know the impact of the vaccine efficacy rates,the threshold level of vaccine effectiveness and how long this pandemic may extent with vaccines that have different efficacy *** this article,a mathematical model study has been done on the importance of vaccination and vaccine efficiency rate during an ongoing ***:We simulated a five compartment mathematical model to analyze the pandemic scenario in both California,and whole *** considered four vaccines,Pfizer(95%),Moderna(94%),AstraZeneca(79%),and Johnson&Johnson(72%),which are being used rigorously to control the SARS-CoV-2 pandemic,in addition with two special cases:a vaccine with 100%efficacy rate and no vaccine under ***-CoV-2 related data of California,and *** used in this ***:Both the infection and death rates are very high in *** model suggests that the pandemic situation in California will be under control in the last quartile of the year 2023 if vaccination program is continued with the Pfizer *** this time,six waves may happen from the beginning of the immunization where the case fatality and recovery rates will be 1.697%and 98.30%,***,according to the considered model,this period might be extended to the mid of 2024 when vaccines with lower efficacy rates are *** the other hand,the daily cases and deaths in the *** be under control at the end of 2026 with multiple *** the number of susceptible people will fall down to none in the beginning of 2027,there is less chance to stop the vaccination program if vaccinated with a vaccine other than a 100%effective vaccine or Pfizer,and at that case vaccination program must run till the mid of *** to this s

关键词： Vaccine Model California Control measurement SARS-CoV-2

Effective boundary conditions for dynamic contact angle hysteresis on chemically inhomogeneous surfaces

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arXiv 2021年

作者： Zhang, Zhen Xu, Xianmin Shenzhen518055 China NCMIS & LSEC Institute of Computational Mathematics and Scientific/Engineering Computing Academy of Mathematics and Systems Science Beijing100190 China School of Mathematical Sciences University of Chinese Academy of Sciences Beijing100049 China

Recent experiments (Guan et_al. 2016a,b) showed many interesting phenomena on dynamic contact angle hysteresis while there is still a lack of complete theoretical interpretation. In this work, we study the time averaging of the apparent advancing and receding contact angles on surfaces with periodic chemical patterns. We first derive a Cox-type boundary condition for the apparent dynamic contact angle on homogeneous surfaces using Onsager variational principle. Based on this condition, we propose a reduced model for some typical moving contact line problems on chemically inhomogeneous surfaces in two dimensions. Multiscale expansion and averaging techniques are employed to approximate the model for asymptotically small chemical patterns. We obtain a quantitative formula for the averaged dynamic contact angles. It gives explicitly how the advancing and receding contact angles depend on the velocity and the chemical inhomogeneity of the substrate. The formula is a coarse-graining version of the Cox-type boundary condition on inhomogeneous surfaces. Numerical simulations are presented to validate the analytical results. The numerical results also show that the formula characterizes very well the complicated behaviour of dynamic contact angle hysteresis observed in the experiments. © 2021, CC BY-NC-ND.

关键词： Boundary conditions