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A fast compact finite difference scheme for the fourth-order diffusion-wave equation

INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2024年第2期101卷 170-193页

作者： Wang, Wan Zhang, Haixiang Zhou, Ziyi Yang, Xuehua Hunan Univ Technol Sch Sci Zhuzhou 412007 Peoples R China

In this paper, the H2N2 method and compact finite difference scheme are proposed for the fourth-order time-fractional diffusion-wave equations. In order to improve the efficiency of calculation, a fast scheme is constructed with utilizing the sum-of-exponentials to approximate the kernel t(1-gamma). Based on the discrete energy method, the Cholesky decomposition method and the reduced-order method, we prove the stability and convergence. When K-1 < 3/2, the convergence order is O(tau(3-gamma) + h(4) + epsilon), where K-1 is diffusion coefficient, gamma is the order of fractional derivative, tau is the parameters for the time meshes, h is the parameters for the space meshes and epsilon is tolerance error. Numerical results further verify the theoretical analysis. It is find that the CPU time is extremely little in our scheme.

关键词： Weakly singular kernel H2N2 interpolation fast algorithm stability and convergence

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A fast BDF2 Galerkin finite element method for the one-dimensional time-dependent Schrodinger equation with artificial boundary conditions

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APPLIED NUMERICAL MATHEMATICS 2023年第1期187卷 89-106页

作者： Xie, Jiangming Li, Maojun Chongqing Technol & Business Univ Sch Math & Stat Chongqing 400067 Peoples R China Univ Elect Sci & Technol China Sch Math Sci Chengdu 611731 Sichuan Peoples R China

In this paper, we propose an efficient numerical scheme with linear complexity for the one-dimensional time-dependent Schrodinger equation on unbounded domains. The artificial boundary method is used to address the unboundedness of the domain. By applying the two-step backward difference formula for time discretization and performing the Z- transform, we derive an exact semi-discrete artificial boundary condition of the Dirichlet-to-Neumann type. To expedite the discrete temporal convolution involved in the exact semi-discrete artificial boundary conditions, we design a fast algorithm based on the best relative Chebyshev approximation of the square-root function. The Galerkin finite element method is used for spatial discretization. By introducing a constant damping term to the original Schrodinger equation, we present a complete error estimate for the fully discrete problem. Several numerical examples are provided to demonstrate the accuracy and efficiency of the proposed numerical scheme. (c) 2023 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Schr?dinger equation Unbounded domains fast algorithm Chebyshev approximation Error estimate

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A fast time stepping Legendre spectral method for solving fractional Cable equation with smooth and non-smooth solutions

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MATHEMATICS AND COMPUTERS IN SIMULATION 2023年第1期211卷 154-170页

作者： Xu, Yibin Liu, Yanqin Yin, Xiuling Feng, Libo Wang, Zihua Li, Qiuping Qilu Univ Technol Shandong Acad Sci Sch Math & Stat Jinan 250353 Peoples R China Dezhou Univ Sch Math & Big Data Dezhou 253023 Peoples R China Queensland Univ Technol Sch Math Sci GPO Box 2434 Brisbane Qld 4001 Australia

To improve the calculation accuracy and efficiency, in this article, we develop a fast time stepping Legendre spectral method for solving fractional Cable equation, where in temporal direction the time stepping method is utilized and the spatial variable is discretized by Legendre spectral method. The time stepping method is used to approximate fractional order derivative, and its convergence accuracy in time is O(tau 2). The fast algorithm is applied to the time stepping method and it can reduce the computational complexity from O(M2) to O(M log M), where M denotes the number of time stepping. For non-smooth solutions, we deal with the initial singularity by adding correction terms. We also analyze the numerical stability and convergence in detail. Numerical experiments confirm our theoretical analysis and efficiency of the fast algorithm. (c) 2023 International Association for Mathematics and Computers in Simulation (IMACS). Published by Elsevier B.V. All rights reserved.

关键词： Time stepping method Legendre spectral method Stability and convergence fast algorithm Smooth and non -smooth solutions

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A fast H2N2 Finite Difference Scheme for the Fractional Sine-Gordon Equation

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COMMUNICATIONS ON APPLIED MATHEMATICS AND COMPUTATION 2024年 1-17页

作者： Wang, Baoting Zhang, Guoyu Shanghai Univ Dept Math Shanghai 200444 Peoples R China Inner Mongolia Univ Sch Math Sci Hohhot 010021 Inner Mongolia Peoples R China

In this paper, based on the H2N2 method (a method for approximating the Caputo fractional derivative of order alpha is an element of(1,2)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha \in (1\text {, } 2)$$\end{document} derived by the quadratic Hermite and Newton interpolation polynomials), a direct finite difference scheme with second-order accuracy in space and (3-alpha)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$(3-\alpha )$$\end{document}th order accuracy in time is constructed for the fractional sine-Gordon equation. The stability and convergence of the difference scheme are analyzed theoretically. Further, to improve the computational efficiency, a fast algorithm based on the sum-of-exponentials method is adopted to construct a fast difference scheme. In addition, the initial singularity of the solution is also discussed. Numerical outcomes verify the validity of the direct and fast difference schemes and the correctness of theoretical results.

关键词： Fractional sine-Gordon equation Finite difference scheme fast algorithm

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A fast implicit difference scheme for solving high-dimensional time-space fractional nonlinear Schrodinger equation

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INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS 2023年第7期100卷 1419-1438页

作者： Mustafa, Almushaira Univ Elect Sci & Technol China Inst Computat Sci Sch Math Sci Chengdu Sichuan Peoples R China Univ Elect Sci & Technol China Inst ComputationalScience Sch Math Sci Chengdu 611731 Sichuan Peoples R China

In this work, an efficient implicit difference scheme is developed for solving the high-dimensional time-space fractional nonlinear Schrodinger equation. The derived scheme is constructed by utilizing a fast evaluation of Caputo fractional derivative based on the L2-1(s) formula;meanwhile, the compact finite difference with matrix transfer technique is adopted for the spatial discretization. Moreover, a linearized iteration method based on the fast discrete sine transform technique is considered to solve the nonlinear system effectively. Because the resultant algorithm does not use matrix inversion, it is computationally efficient in long-time calculations. The stability, truncation error and convergence analysis of the discrete scheme are discussed in detail. Furthermore, a fast iterative algorithm is provided. Finally, several numerical examples are given to verify the efficiency and accuracy of the derived scheme, and a comparison with similar work is presented.

关键词： Time-space fractional Schrodinger equation matrix transfer technique fast algorithm stability

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Partition Map Prediction for fast Block Partitioning in VVC Intra-Frame Coding

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IEEE TRANSACTIONS ON IMAGE PROCESSING 2023年 32卷 2237-2251页

作者： Feng, Aolin Liu, Kang Liu, Dong Li, Li Wu, Feng Univ Sci & Technol China CAS Key Lab Technol Geospatial Informat Proc & App Hefei 230027 Peoples R China Univ Southern Calif Los Angeles CA 90007 USA

The Versatile Video Coding (VVC) standard introduces a block partitioning structure known as quadtree plus nested multi-type tree (QTMTT), which allows more flexible block partitioning compared to its predecessors, like High Efficiency Video Coding (HEVC). Meanwhile, the partition search (PS) process, which is to find out the best partitioning structure for optimizing the rate-distortion cost, becomes far more complicated for VVC than for HEVC. Also, the PS process in VVC reference software (VTM) is not friendly to hardware implementation. We propose a partition map prediction method for fast block partitioning in VVC intra-frame encoding. The proposed method may replace PS totally or be combined with PS partially, thereby achieving adjustable acceleration of the VTM intra-frame encoding. Different from the previous methods for fast block partitioning, we propose to represent a QTMTT-based block partitioning structure by a partition map, which consists of a quadtree (QT) depth map, several multi-type tree (MTT) depth maps, and several MTT direction maps. We then propose to predict the optimal partition map from the pixels through a convolutional neural network (CNN). We propose a CNN structure, known as Down-Up-CNN, for the partition map prediction, where the CNN structure emulates the recursive nature of the PS process. Moreover, we design a post-processing algorithm to adjust the network output partition map, so as to obtain a standard-compliant block partitioning structure. The post-processing algorithm may produce a partial partition tree as well;then based on the partial partition tree, the PS process is performed to obtain the full tree. Experimental results show that the proposed method achieves 1.61 x to 8.64 x encoding acceleration for the VTM-10.0 intra-frame encoder, with the ratio depending on how much PS is performed. Especially, when achieving 3.89 x encoding acceleration, the compression efficiency loss is 2.77% in BD-rate, which is a better tradeoff

关键词： Block partitioning convolutional neural network (CNN) fast algorithm intra-frame coding partition map partition search (PS) QTMTT versatile video coding (VVC)

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Improved fast algorithms for Optimal Min-Max Line Barrier Coverage with Mobile Sensors on the Plane 21

Improved Fast Algorithms for Optimal Min-Max Line Barrier Co...

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24th International Conference on Modeling Analysis and Simulation of Wireless and Mobile Systems

作者： Yao, Pei Guo, Longkun Li, Peng Lin, Jiawei Fuzhou Univ Fuzhou Peoples R China Google Incorp Kirkland WA USA

ISBN: (纸本)9781450390774

Emerging applications raise the min-max line barrier coverage (LBC) problem that aims to minimize the maximum movement of the sensors for the sake of balancing energy consumption. In this paper, we devise an exact algorithm to optimally solve LBC within a runtime of O(n(2)), comparing favorably to the previous state-of-art runtime O(n(2) log n), where n is the number of sensors. To achieve the improvement, we accelerate the computation of optimum solutions by using a novel approximation algorithm. Numerical experiments demonstrated that our algorithms outperform all the other baselines, including the previous state-of-art algorithm.

关键词： fast algorithm barrier coverage optimal solution mobile sensor approximation algorithm

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fast Matrix Multiplication algorithm for a Bank of Digital Filters

Fast Matrix Multiplication Algorithm for a Bank of Digital F...

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Conference on Systems of Signal Synchronization, Generating and Processing in Telecommunications (SYNCHROINFO)

作者： Kreyndelin, V. B. Grigorieva, E. D. Moscow Tech Univ Telecommun & Informat Moscow Russia

ISBN: (纸本)9781665415491

A method of implementing matrix multiplication for use in digital filter banks (sets) is presented. This method allows to obtain noticeable savings of computational costs compared to standard methods. Reduction of computational complexity of digital filter banks (sets) is achieved without any performance loss. The method proposed in the report is based on the use of a combination of the known method of 3M multiplication of complex matrices and the Strassen method for fast multiplication of matrices. A feature of the application of the Strassen method in this case is that it is recursively applied, starting with some dimension of blocks. Multiplication of lower dimension blocks is carried out by the traditional method. In order to reduce computational complexity, the minimum dimension of blocks was selected, starting from which it is advisable to use the Strassen method. An analysis of the computational complexity of the proposed method has shown that its use in implementing a bank (set) of digital filters allows to obtain a gain in complexity compared to a traditional algorithm by about 1.8-1.9 times with large dimensions of matrices. An approximate analysis of the sensitivity of the method proposed in the article to rounding errors that occur during digital processing has been carried out. As a result of the analysis, it was found that the proposed method has approximately the same sensitivity to rounding errors as the traditional method.

关键词： bank (set) of digital filters fast algorithm sensitivity to rounding errors 3M method Strassen method computational complexity

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A fast Gateway Placement algorithm for Flying Networks 93

A Fast Gateway Placement Algorithm for Flying Networks

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IEEE 93rd Vehicular Technology Conference (VTC-Spring)

作者： Santos, Goncalo Martins, Joao Coelho, Andre Fontes, Helder Ricardo, Manuel Campos, Rui Univ Porto INESC TEC Porto Portugal Univ Porto Fac Engn Porto Portugal

ISBN: (纸本)9781728189642

The ability to operate anywhere, anytime, as well as their capability to hover and carry cargo on board make Unmanned Aerial Vehicles (UAVs) suitable platforms to act as Flying Gateways (FGWs) to the Internet. The problem is the optimal placement of the FGWs within the flying network, such that the Quality of Service (QoS) offered is maximized. The literature has been focused on optimizing the placement of the Flying Access Points (FAPs), which establish high-capacity small cells to serve the users on the ground, overlooking the backhaul network design, including the FGW placement. The FGW placement problem is exacerbated in highly dynamic flying networks, where the dynamic traffic demand and the movements of the users may induce frequent changes in the placement of the FAPs. The main contribution of this paper is a fast gateway placement (F-GWP) algorithm for flying networks that determines the optimal position of a FGW. With F-GWP, backhaul communications paths with high enough capacity are established between the FAPs and the FGW, in order to accommodate the traffic demand of the users on the ground. Simulation and experimental results show F-GWP is two orders of magnitude faster than its state of the art counterpart, while ensuring the same flying network performance.

关键词： Unmanned Aerial Vehicles Flying Networks Aerial Networks Gateway Placement fast algorithm Quality of Service

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A high-order and fast scheme with variable time steps for the time-fractional Black-Scholes equation

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MATHEMATICAL METHODS IN THE APPLIED SCIENCES 2023年第2期46卷 1990-2011页

作者： Song, Kerui Lyu, Pin Southwestern Univ Finance & Econ Sch Math Chengdu Peoples R China

In this paper, a high-order and fast numerical method is investigated for the time-fractional Black-Scholes equation. In order to deal with the typical weak initial singularity of the solution, we construct a finite difference scheme with variable time steps, where the fractional derivative is approximated by the nonuniform Alikhanov formula and the sum-of-exponentials (SOE) technique. In the spatial direction, an average approximation with fourth-order accuracy is employed. The stability and the convergence with second order in time and fourth order in space of the proposed scheme are religiously derived by the energy method. Numerical examples are given to demonstrate the theoretical statement.

关键词： fast algorithm high-order method time-fractional Black-Scholes equation variable time steps

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