Article
Mathematics, Applied
Yun-Chi Huang, Lot-Kei Chou, Siu-Long Lei
Summary: A divide-and-conquer solver coupled with Tensor-Train (TT) format is proposed for solving the d-dimensional time-space fractional diffusion equations with alternating direction implicit finite difference scheme. The complexity and storage of the proposed solver grow slowly with dimension d if the TT-ranks r is low, which is achieved by an efficient approximated Toeplitz inversion. TT rounding is performed to reduce the increase of TT-ranks, and a criterion for preserving convergence rate of the numerical scheme is given. Numerical experiments demonstrate the accuracy and efficiency of the proposed solver for low TT-ranks initial conditions and source terms, with dimension d up to 20.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Ying Wang, Fawang Liu, Liquan Mei, Vo V. Anh
Summary: In this paper, an efficient spectral Galerkin method for the three-dimensional multi-term time-space fractional diffusion equation is developed. The fully discrete numerical scheme is shown to be unconditionally stable, with second-order accuracy in time and optimal error estimation in space. The proposed method is validated through numerical experiments and applied to solve the fractional Bloch-Torrey model.
NUMERICAL ALGORITHMS
(2021)
Article
Mathematics, Applied
Xuehua Yang, Wenlin Qiu, Haixiang Zhang, Liang Tang
Summary: In this work, an efficient alternating direction implicit (ADI) finite difference scheme is proposed to solve the three-dimensional time-fractional telegraph equation. Stability and convergence of the scheme are proved via the energy method in L-2 and H-1 norms, and several numerical examples are provided to validate the theoretical results.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Yue Wang, Hu Chen
Summary: The study focuses on using an alternating direction implicit (ADI) difference method to solve a two-dimensional time-fractional diffusion equation, discussing the temporal convergence and error estimation of the method, proving that the method has a certain accuracy globally.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Mostafa Abbaszadeh, Mehdi Dehghan
Summary: The paper introduces a new high-order finite difference scheme with low computational complexity to solve the space-time fractional tempered diffusion equation. The stability analysis and convergence order proof demonstrate the effectiveness and feasibility of the technique.
INTERNATIONAL JOURNAL OF COMPUTER MATHEMATICS
(2021)
Article
Mathematics, Applied
T. Guo, O. Nikan, Z. Avazzadeh, W. Qiu
Summary: This paper proposes an alternating direction implicit (ADI) numerical method for solving multi-dimensional distributed-order fractional integrodifferential problems. The method discretizes the unknown solution, Riemann-Liouville fractional integral term, and distributed-order time-fractional derivative to reduce the computational burden. Additionally, it applies general centered finite difference (FD) technique for spatial discretization. Convergence analysis is performed using the energy method, and numerical examples are provided to verify the accuracy of the proposed method.
COMPUTATIONAL & APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Pieter Decleer, Arne Van Londersele, Hendrik Rogier, Dries Vande Ginste
Summary: This paper proposes a novel hybrid FDTD method for solving the time-dependent Schrodinger equation, which combines the unconditional stability of the ADI scheme with fast explicit calculations. Several numerical experiments demonstrate the high accuracy and decreased CPU time of the scheme compared to traditional methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Xiao Qin, Xiaozhong Yang, Peng Lyu
Summary: The paper presents a class of explicit implicit alternating difference schemes for the generalized time fractional Fisher equation, which have both unconditional stability and convergence with order O(tau(2-alpha) + h(2)). The proposed schemes reduce the calculation cost by almost 60% compared to the classical implicit difference scheme, and have been shown to be efficient for solving the equation with initial weak singularity.
Article
Mathematics, Applied
Ali Ruhsen Cete, Oguz Kaan Onay
Summary: The study combines a novel fast-implicit iteration scheme called the alternating cell direction implicit (ACDI) method with the approximate factorization scheme to increase the accuracy of numerical solutions for partial differential equations. By applying the ACDI method to unstructured grids, the study shows improvement in the method's capabilities. The research results validate the enhancements brought by the ACDI method and demonstrate its potential for broader applications beyond structured grids.
ADVANCES IN APPLIED MATHEMATICS AND MECHANICS
(2022)
Article
Mathematics, Applied
Di Gan, Guo-Feng Zhang
Summary: In this paper, the alternating direction implicit (ADI) finite difference method and preconditioned Krylov subspace method are combined to solve high-dimensional spatial fractional diffusion equations with variable diffusion coefficients. The unconditional stability and convergence rate of the ADI finite difference method are proven under certain conditions on the diffusion coefficients. A circulant approximate inverse preconditioner is established to accelerate the Krylov subspace method for the linear system in each spatial direction. Matrix-free algorithms and fast Fourier transforms (FFT) are used to speed up the solution of linear systems. Numerical experiments demonstrate the effectiveness of the ADI method and the preconditioner.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2023)
Article
Optics
Jiankai Xiong, Jiaqing Shen, Yuan Gao, Yingshi Chen, Jun-Yu Ou, Qing Huo Liu, Jinfeng Zhu
Summary: This article presents a divide-and-conquer deep learning model for the design of plasmonic stack metamaterials (PSMs). The model demonstrates significant reduction in prediction error and training parameters in the forward network, supporting powerful inverse design from spectra to PSM structures. Additionally, a flexible tool based on free customer definition is developed for real-time design of metamaterials with various circuit-analog functions.
LASER & PHOTONICS REVIEWS
(2023)
Article
Mathematics, Applied
Yanqin Liu, Xiuling Yin, Fawang Liu, Xiaoyi Xin, Yanfeng Shen, Libo Feng
Summary: In this paper, an ADILS method is developed for solving the 2D multi-term time fractional Oldroyd-B fluid type diffusion equation using a Legendre spectral approximation and a new time difference discretization. The method shows improved convergence accuracy for non-smooth solutions with the help of a correction scheme.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Statistics & Probability
Zhaoxing Gao, Ruey S. Tsay
Summary: This article proposes a hierarchical approximate-factor approach to analyze high-dimensional, large-scale heterogeneous time series data. The method uses Principal Component Analysis (PCA) for multiple-fold dimension reduction and is suitable for modeling large-scale data that cannot be handled by a single machine. By transferring factors hierarchically, the method selects global common factors.
JOURNAL OF THE AMERICAN STATISTICAL ASSOCIATION
(2022)
Article
Statistics & Probability
Lanjue Chen, Jin Su, Alan T. K. Wan, Yong Zhou
Summary: The accelerated failure time (AFT) model is a simple and interpretable tool in survival analysis. However, the computational demands of fitting the model and performing inference can be significant when dealing with large volumes of data. This article proposes a distributed method based on the divide-and-conquer strategy to address this issue, offering both statistical efficiency and computational economy. The method is validated through simulation studies and applied to a kidney transplantation dataset.
JOURNAL OF COMPUTATIONAL AND GRAPHICAL STATISTICS
(2023)
Article
Computer Science, Interdisciplinary Applications
Mingrong Cui
Summary: This paper presents an alternating direction implicit (ADI) compact finite difference scheme for the two-dimensional multi-term time-fractional mixed diffusion and diffusion-wave equation. By applying the Riemann-Liouville fractional integral operator, a time-fractional integro-differential equation is obtained. The scheme is fully discrete in both time and space using weighted and shifted Grunwald formulas, Crank-Nicolson approximation, and fourth-order compact approximation. The stability and convergence properties are proved using energy method. Numerical examples demonstrate the efficiency and accuracy of the proposed method.
MATHEMATICS AND COMPUTERS IN SIMULATION
(2023)
