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A fast finite volume method for spatial fractional diffusion equations on nonuniform meshes

COMPUTERS & MATHEMATICS WITH APPLICATIONS 2022年第0期108卷 175-184页

作者： Fang, Zhi-Wei Zhang, Jia-Li Sun, Hai-Wei Foshan Univ Sch Math & Big Data Foshan 528000 Peoples R China Univ Macau Dept Math Macau Peoples R China

In this paper, a fast finite volume method is proposed for the initial and boundary value problems of spatial fractional diffusion equations on nonuniform meshes. The discretizations of the Riemann-Liouville fractional derivatives lead to unstructured dense coefficient matrices, differing from the Toeplitz-like structure under the uniform mesh. The fast algorithm is proposed by using the sum-of-exponentials (SOE) technique to the spatial kernel x(a-1), alpha is an element of(0, 1). Then, the matrix-vector multiplications of the resulting coefficient matrices could be implemented in O(m log(2) m) operations, where m denotes the size of matrices. Iterative solvers are preferably applied to obtain the numerical solution. The proposed fast scheme is proved to be unconditionally stable for sufficiently accurate SOE approximation. Meanwhile, a banded preconditioner is exploited to accelerate the Krylov subspace method. Numerical experiments are provided to demonstrate the efficiency of the proposed fast algorithm.

关键词： Spatial fractional diffusion equations Finite volume method Sum-of-exponentials technique fast algorithm Banded preconditioner

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fast difference scheme for the reaction-diffusion-advection equation with exact artificial boundary conditions

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APPLIED NUMERICAL MATHEMATICS 2022年 173卷 395-417页

作者： Li, Can Wang, Haihong Yue, Hongyun Guo, Shimin Xian Univ Technol Dept Appl Math Xian 710054 Shaanxi Peoples R China Henan Univ Technol Coll Sci Zhengzhou 450001 Henan Peoples R China Xi An Jiao Tong Univ Sch Math & Stat Xian 710049 Shaanxi Peoples R China

In this paper, we derive the exact artificial boundary conditions for one-dimensional reaction-diffusion-advection equation on an unbounded domain. By employing the Laplace transform, we reduce the original unbound domain problem into a bounded domain problem. The exact artificial boundary conditions are given by Caputo-tempered fractional derivatives in the reduced initial-boundary value problem. We show that the reduced initial-boundary value problem is stable with the exact artificial boundary conditions. We design a finite difference scheme for the reduced finite domain problem. To save the computational cost, we developed a fast algorithm to solve Caputo-tempered derivatives arise in the boundary conditions. We prove that the present difference schemes are uniquely solvable and unconditionally stable in the energy norm. Finally, we demonstrate the effectiveness of the proposed methods by some numerical examples. (C) 2021 IMACS. Published by Elsevier B.V. All rights reserved.

关键词： Reaction-diffusion-advection equation Artificial boundary conditions Tempered fractional derivative fast algorithm

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Reconstruction of shear force in Atomic Force Microscopy from measured displacement of the cone-shaped cantilever tip

MATHEMATICS IN ENGINEERING

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MATHEMATICS IN ENGINEERING 2024年第1期6卷 137-154页

作者： Hasanov, Alemdar Kawano, Alexandre Baysal, Onur Kocaeli Univ Dept Math Kocaeli Turkiye Univ Sao Paulo Escola Politecn BR-05508900 Sao Paulo Brazil Univ Malta Dept Math Msida Malta

We present a new comprehensive mathematical model of the cone-shaped cantilever tipsample interaction in Atomic Force Microscopy (AFM). The importance of such AFMs with coneshaped cantilevers can be appreciated when its ability to provide high-resolution information at the nanoscale is recalled. It is an indispensable tool in a wide range of scientific and industrial fields. The interaction of the cone-shaped cantilever tip with the surface of the specimen (sample) is modeled by the damped Euler-Bernoulli beam equation rho A(x)utt +mu(x)ut + (r(x)uxx + kappa(x)uxxt)xx = 0, (x, t) E (0, euro) x (0, T), subject to the following initial, u(x, 0) = 0, ut(x, 0) = 0 and boundary, u(0, t) = 0, ux(0, t) = 0, (r(x)uxx(x, t) + kappa(x)uxxt)x=e = M(t), (-(r(x)uxx + kappa(x)uxxt)x)x=e = g(t) conditions, where M(t) := 2h cos theta g(t)/pi is the moment generated by the transverse shear force g(t). Based on this model, we propose an inversion algorithm for the reconstruction of an unknown shear force in the AFM cantilever. The measured displacement nu(t) := u(euro, t) is used as additional data for the reconstruction of gradient formula for the Frechet derivative of the cost functional is derived via the weak solution contained in this formula. This enables us to construct a gradient based numerical algorithm for the reconstructions of the shear force from noise free as well as from random noisy measured output nu(t). Computational experiments show that the proposed algorithm is very fast and robust. This creates the basis for developing a numerical "gadget" for computational experiments with generic AFMs.

关键词： Atomic Force Microscopy cone-shaped cantilever reconstruction of shear force damped Euler-Bernoulli cantilever beam inverse problem gradient formula fast algorithm

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Effective image registration model using optimized KAZE algorithm

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MULTIMEDIA TOOLS AND APPLICATIONS 2024年第11期83卷 33959-33984页

作者： Zhang, Sheng Shen, Jie Zheng, Shengnan Tang, Jingjing Hohai Univ Coll Comp & Informat Nanjing 211100 Peoples R China Nanjing Inst Technol Sch Comp Engn Nanjing 211167 Peoples R China

The incompatible problem with velocity and accuracy has been restricting the application of the KAZE algorithm. In order to resolve this shortage, we propose the effective image registration model using the optimized KAZE algorithm. This effective image registration model consist of four stages. First of all, to reduce the input data of image registration, the original registration images are preprocessed by the fusion preprocessing method based on the average and the perceptual hashing algorithms. Second, to extract image features quickly, we utilize the fast algorithm to extract image features instead of the local extremum based on the Hessian matrix and the Taylor principle. Third, in order to accelerate the velocity of image features matching, the compressed sensing principle is used to reduce the dimension of the image feature descriptors. Finally, the two-step strategy is adopted to ensure the accuracy of image registration, the step one is that the hybrid matching method based on the FLANN and the KNN algorithms is used to rough matching, and the step two is that adopt the RANSAC algorithm to further accurate matching. This paper utilizes two groups of the experiments to verify the effective model, the experiment results show that the effective model has velocity advantage compared with other current image registration methods, and also achieves the compatible with velocity and accuracy in the case of the highest matching score. This model provides an effective solution for the application of image registration, and also has great significance for the development of image registration.

关键词： Image registration KAZE algorithm Hashing algorithm fast algorithm RANSAC algorithm

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A fast Time Two-Mesh algorithm for Allen-Cahn Equation

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BULLETIN OF THE MALAYSIAN MATHEMATICAL SCIENCES SOCIETY 2020年第3期43卷 2417-2441页

作者： Wang, Danxia Du, Qingqing Zhang, Jianwen Jia, Hongen Taiyuan Univ Technol Coll Math Taiyuan 030024 Peoples R China Xiangtan Univ Sch Math & Computat Sci Xiangtan 411105 Peoples R China

In this article, we consider numerical methods for solving Allen-Cahn equation involving strong nonlinearities. A fast algorithm based on time two-mesh (TT-M) finite element (FE) scheme is proposed to overcome the time-consuming computation of nonlinear terms. The TT-M FE algorithm includes the following three main steps: Firstly, a nonlinear FE scheme is solved on a coarse time mesh tau c. Here, the FE method is used for spatial discretization and the implicit second-order backward difference scheme is used for temporal discretization. Secondly, the Lagrange's interpolation is used to obtain the interpolation result on the fine grid. Finally, a linearized FE system is solved on a fine time mesh tau(tau

关键词： fast algorithm TT-M FE method Allen-Cahn equation Stability Priori error estimate CPU time 35Q30 74S05

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fast feature selection algorithm for neighborhood rough set model based on Bucket and Trie structures

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GRANULAR COMPUTING 2020年第3期5卷 329-347页

作者： Benouini, Rachid Batioua, Imad Ezghari, Soufiane Zenkouar, Khalid Zahi, Azeddine Sidi Mohamed Ben Abdellah Univ Fac Sci & Technol Lab Intelligent Syst & Applicat LSIA Fes Morocco

Feature selection is viewed as the problem of finding the minimal number of features from an original set with the minimum information loss. Due to its high importance in the fields of pattern recognition and data mining, it is necessary to investigate fast and effective search algorithms. In this paper, we introduce a novel fast feature selection algorithm for neighborhood rough set model based on Bucket and Trie structures. This proposed algorithm can guarantee to find the optimal minimal reduct by adopting a global search strategy. In addition, the dependence degree is subsequently used to evaluate the relevance of the attribute subset. Consequently, the proposed algorithm is tested on several standard data sets from UCI repository and compared with the most recent related approaches. The obtained theoretical and experimental results reveal that the present algorithm is very effective and convenient for the problem of feature selection, indicating that it could be useful for many pattern recognition and data mining systems.

关键词： Feature selection Rough set theory Neighborhood rough set fast algorithm Trie structure Bucket structure

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Super Fourier analysis: A highly efficient framework for multivariate signal processing

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SIGNAL PROCESSING 2025年 231卷

作者： Li, Tianqi Chen, Nan Peng, Zhike He, Qingbo Shanghai Jiao Tong Univ State Key Lab Mech Syst & Vibrat Shanghai 200240 Peoples R China Ningxia Univ Sch Mech Engn Yinchuan 750021 Peoples R China Natl Univ Singapore Dept Ind Syst Engn & Management Singapore 117576 Singapore

The escalating integration of multi-sensor systems across diverse high-dimensional monitoring areas generates a large amount of large-scale multivariate signals. However, the theoretical foundation of efficient multivariate signal processing methods is currently lacking, making it challenging to fully exploit the value of multivariate signals under the requirement of rapid response. Here, we introduce super Fourier analysis (SFA), which innovates traditional Fourier analysis with the principle of multivariate statistics for highly efficient processing of multivariate signals. By integrating multi-channel information and reducing the data dimensionality, SFA can inherently handle the correlation across channels and has low time complexity. In the framework of SFA, we deduce and define the super Fourier series, super Fourier transform, and discrete super Fourier transform. Mode alignment property and noise resilience property of the SFA are analyzed. As an example, variational mode decomposition, a classic univariate signal processing method, is extended to multivariate context based on SFA. Our demonstrations include simulated signals, multi-channel electroencephalography, global sea surface temperature, and motion microscopy, highlighting SFA's potential in rapid and large-scale multivariate signal processing. SFA's efficiency and effectiveness promise its applications in various areas with a large number of sensors or channels, making the processing of multivariate signals as simple as univariate signals.

关键词： Multivariate signals Fourier analysis Multivariate statistics fast algorithm

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fast Second-Order Evaluation for Variable-Order Caputo Fractional Derivative with Applications to Fractional Sub-Diffusion Equations

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Numerical Mathematics(Theory,Methods and Applications) 2022年第1期15卷 200-226页

作者： Jia-Li Zhang Zhi-Wei Fang Hai-Wei Sun Department of Mathematics University of MacaoMacaoSARChina

In this paper,we propose a fast second-order approximation to the variable-order(VO)Caputo fractional derivative,which is developed based on L2-1σformula and the exponential-sum-approximation *** fast evaluation method can achieve the second-order accuracy and further reduce the computational cost and the acting memory for the VO Caputo fractional *** fast algorithm is applied to construct a relevant fast temporal second-order and spatial fourth-order scheme(F L2-1σscheme)for the multi-dimensional VO time-fractional sub-diffusion ***,F L2-1σscheme is proved to fulfill the similar properties of the coefficients as those of the well-studied L2-1σ***,F L2-1σscheme is strictly proved to be unconditionally stable and convergent.A sharp decrease in the computational cost and the acting memory is shown in the numerical examples to demonstrate the efficiency of the proposed method.

关键词： Variable-order Caputo fractional derivative exponential-sum-approximation method fast algorithm time-fractional sub-diffusion equation stability and convergence

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A POD based reduced-order local RBF collocation approach for time-dependent nonlocal diffusion problems

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APPLIED MATHEMATICS LETTERS 2025年 160卷

作者： Lu, Jiashu Zhang, Lei Guo, Xuncheng Qi, Qiong Xian Modern Control Technol Res Inst Natl Key Lab Land & Air Based Informat Percept & C Xian 710065 Shaanxi Peoples R China

A fast algorithm based on reduced-order model (ROM) is proposed for unsteady nonlocal diffusion models. It combines proper orthogonal decomposition (POD) approach and collocation method with local radial basis functions (RBFs), which makes it possible for using ROM to solve nonlocal models. Several numerical experiments showed that this approach significantly reduce the computational cost of nonlocal models while keep the similar convergent behavior compared with the RBF collocation methods.

关键词： Nonlocal models Radial basis functions Model reduction fast algorithm

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Acceleration of the frequency-shift demodulation in phase-sensitive OTDR

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MEASUREMENT SCIENCE AND TECHNOLOGY 2024年第2期35卷 025101-025101页

作者： Pu, Zhengyu He, Haijun Zhou, Yin Jiang, Lin Pan, Wei Yan, Lianshan Southwest Jiaotong Univ Sch Informat Sci & Technol Ctr Informat Photon & Commun Chengdu 611756 Sichuan Peoples R China

The frequency-shift demodulation is a primary demodulation method in phase-sensitive optical time domain reflectometry (phi-OTDR) with intrinsic resistance to interference fading. So far, the least mean squares (LMS) estimation method has the optimal demodulation accuracy and robustness. However, it takes much processing time due to the step-by-step sliding operation. In this work, we propose a fast LMS estimation method based on cross-correlation calculation to accelerate the demodulation while maintaining accuracy. Experiments are performed along a 9 km sensing fiber with a 4 m spatial resolution. The performance of the fast LMS, LMS, and cross-correlation methods are compared by using the same parameters. Compared with the LMS method, the fast LMS achieves a 12-time improvement in processing speed while remaining the same demodulation accuracy. Although the proposed fast LMS method takes slightly more time than the cross-correlation method (1.6 times), it improves the demodulation accuracy similar to 6 dB for the vibration signal and similar to 2.1 dB for the overall demodulation accuracy.

关键词： distributed optical fiber sensor phase-sensitive optical time domain reflectometry frequency-shift demodulation fast algorithm high-resolution measurement

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