Article
Mathematics, Applied
Xing Zhang, Xiaoyu Jiang, Zhaolin Jiang, Heejung Byun
Summary: This paper implements matrix order-reduction algorithms to solve the CUPL-Toeplitz linear system. Firstly, order-reduction algorithms for the multiplication of real skew-circulant matrix or complex circulant matrix and vector are described. Secondly, based on two fast approaches [1] by splitting the CUPL-Toeplitz matrix into a Toeplitz matrix subtracting a low-rank matrix, new fast Toeplitz solvers are proposed to reduce the amount of calculation. Finally, numerical experiments are conducted to demonstrate the performance of the proposed algorithms.
APPLIED MATHEMATICS AND COMPUTATION
(2022)
Article
Mathematics, Applied
Xin Lu, Zhi-Wei Fang, Hai-Wei Sun
Summary: The study introduces a sine-transform-based splitting preconditioning technique for linear systems in the numerical discretization of fractional diffusion equations, showing efficient acceleration of convergence rate. The fast sine transform helps reduce the computational complexity in practical computations.
JOURNAL OF APPLIED MATHEMATICS AND COMPUTING
(2021)
Article
Mathematics, Applied
Tian-Yi Li, Fang Chen, Hai-Wei Sun, Tao Sun
Summary: We propose two preconditioners based on the fast sine transformation for solving linear systems with ill-conditioned multilevel Toeplitz structure. These matrices are generated by discretizing the two-dimensional nonlocal Helmholtz equations with fractional Laplacian operators via the finite difference method. For complex wave numbers with nonnegative real parts, we give the spectral analysis of the preconditioned matrices. Numerical experiments also indicate that the proposed preconditioners outperform the existing preconditioners.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Computer Science, Information Systems
Lu Zhang, Lei Chen, Xiao Song
Summary: The paper presents a fast numerical method for solving the nonlinear space fractional complex Ginzburg-Landau equations, utilizing a circulant preconditioner and fast Fourier transform to solve the linear system, resulting in computational superiority. Numerical examples are conducted to demonstrate the advantage of the method.
Article
Mathematics, Applied
Lei Zhang, Guo-Feng Zhang, Zhao-Zheng Liang
Summary: This paper proposes a new method for solving d-dimensional fractional partial differential equations, which improves computational efficiency by using low-rank techniques and fast Fourier transform to accelerate matrix-vector multiplication, and constructing a low-rank preconditioner. The numerical results demonstrate the competitiveness of this method.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Mathematics, Applied
Shi-Ping Tang, Yu-Mei Huang
Summary: This paper investigates the initial boundary value problem of the tempered fractional diffusion equations and proposes a new preconditioning method that effectively accelerates the convergence rate of the GMRES method for solving linear systems.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Di Gan, Guo-Feng Zhang, Zhao-Zheng Liang
Summary: This paper considers solutions for discrete systems arising from multi-term time-fractional diffusion equations. By using discrete sine transform techniques, a generalized circulant approximate inverse preconditioner is established for the systems. The effectiveness of the proposed preconditioner is demonstrated through numerical examples.
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics, Applied
Hai-Hua Qin, Hong-Kui Pang, Hai-Wei Sun
Summary: We study the preconditioned iterative methods for solving linear systems arising from the numerical solution of multi-dimensional space fractional diffusion equations. A sine transform based preconditioning technique is developed, and theoretical analyses and numerical experiments demonstrate its correctness and effectiveness.
NUMERICAL LINEAR ALGEBRA WITH APPLICATIONS
(2023)
Article
Mathematics, Applied
V. B. Kiran Kumar, A. Lexy, M. N. N. Namboodiri, A. Noufal
Summary: In this article, an upper bound for the generalized condition number of the covariance matrix is derived, and the concept of preconditioners is extended to singular matrices. Existing preconditioners for non-singular matrices are compared with generalized preconditioners, with numerical experiments showing that the latter exhibit better performance. Additionally, the generating function of the covariance matrix is calculated, and spectral properties are derived using linear algebra techniques.
APPLIED MATHEMATICS AND COMPUTATION
(2023)
Article
Mathematics, Applied
Inna Roitberg, Alexander Sakhnovich
Summary: The article discusses classical results on the inversion of convolution operators and Toeplitz (and block Toeplitz) matrices in the one-dimensional case, and extends the approach to more complicated cases including block TBT matrices and 3-D Toeplitz matrices. Some important special cases are also treated.
LINEAR ALGEBRA AND ITS APPLICATIONS
(2021)
Article
Mathematics, Applied
Maryam Shams Solary
Summary: This paper studies matrix polynomials of arbitrary degree by using the determinant of block Toeplitz-Hessenberg matrices represented as L(lambda) = lambda(r) I - Sigma(r)(j=1) lambda C-r-j(j), with or without commuting coefficients (CiCj = CjCi, or CiCj not equal CjCi for C-i belonging to C-txt, i, j = 1, ..., r).
NUMERICAL ALGORITHMS
(2023)
Article
Mathematics
Mu-Zheng Zhu, Ya-E Qi, Guo-Feng Zhang
Summary: The paper presents efficient preconditioners based on circulant and skew-circulant approximations to accelerate the convergence of Krylov subspace methods for discretized linear systems of spatial fractional diffusion equations. Numerical experiments show that the new preconditioners can significantly speed up the convergence of CGNR and BiCGSTAB. Results also indicate that there is minimal difference in acceleration effects between the circulant and skew-circulant approximation-based preconditioners for the problems considered.
LINEAR & MULTILINEAR ALGEBRA
(2021)
Article
Mathematics, Applied
Manuel Bogoya, Sven-Erik Ekstrom, Stefano Serra-Capizzano
Summary: This paper discusses the asymptotic expansions of eigenvalues of Toeplitz matrices based on the simple-loop theory and presents a matrix-less algorithm for efficient eigenvalue computation. Numerical experiments demonstrate higher precision and comparable computational cost compared to existing procedures.
NUMERICAL ALGORITHMS
(2022)
Article
Mathematics, Applied
Sean Hon
Summary: In this paper, a preconditioned minimal residual (MINRES) method for non-Hermitian block Toeplitz systems is proposed. The method involves premultiplying the matrix by a simple permutation matrix and constructing a Hermitian positive definite block circulant preconditioner. Theoretical analysis shows that the eigenvalues of the preconditioned matrix are clustered around ±1, ensuring convergence. Useful properties of block circulant matrices with commuting Hermitian blocks are also provided, and the method is generalized to the multilevel block case. The method has potential applications in solving all-at-once systems arising from evolutionary partial differential equations.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Xin Huang, Xue-Lei Lin, Michael K. Ng, Hai-Wei Sun
Summary: This paper analyzes the spectra of preconditioned matrices generated from discretized multi-dimensional Riesz spatial fractional diffusion equations and proposes a preconditioned conjugate gradient method based on the sine transform. Theoretical analysis proves that the spectra of the preconditioned matrices are bounded within a specific range and the proposed method outperforms other methods in terms of convergence.
NUMERICAL MATHEMATICS-THEORY METHODS AND APPLICATIONS
(2022)
Article
Computer Science, Interdisciplinary Applications
Tian Liang, Lin Fu
Summary: In this work, a new shock-capturing framework is proposed based on a new candidate stencil arrangement and the combination of infinitely differentiable non-polynomial RBF-based reconstruction in smooth regions with jump-like non-polynomial interpolation for genuine discontinuities. The resulting scheme achieves high order accuracy and resolves genuine discontinuities with sub-cell resolution.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Lukas Lundgren, Murtazo Nazarov
Summary: In this paper, a high-order accurate finite element method for incompressible variable density flow is introduced. The method addresses the issues of saddle point system and stability problem through Schur complement preconditioning and artificial compressibility approaches, and it is validated to have high-order accuracy for smooth problems and accurately resolve discontinuities.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Gabriele Ciaramella, Laurence Halpern, Luca Mechelli
Summary: This paper presents a novel convergence analysis of the optimized Schwarz waveform relaxation method for solving optimal control problems governed by periodic parabolic PDEs. The analysis is based on a Fourier-type technique applied to a semidiscrete-in-time form of the optimality condition, which enables a precise characterization of the convergence factor at the semidiscrete level. The behavior of the optimal transmission condition parameter is also analyzed in detail as the time discretization approaches zero.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jonas A. Actor, Xiaozhe Hu, Andy Huang, Scott A. Roberts, Nathaniel Trask
Summary: This article introduces a scientific machine learning framework that uses a partition of unity architecture to model physics through control volume analysis. The framework can extract reduced models from full field data while preserving the physics. It is applicable to manifolds in arbitrary dimension and has been demonstrated effective in specific problems.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Nozomi Magome, Naoki Morita, Shigeki Kaneko, Naoto Mitsume
Summary: This paper proposes a novel strategy called B-spline based SFEM to fundamentally solve the problems of the conventional SFEM. It uses different basis functions and cubic B-spline basis functions with C-2-continuity to improve the accuracy of numerical integration and avoid matrix singularity. Numerical results show that the proposed method is superior to conventional methods in terms of accuracy and convergence.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Timothy R. Law, Philip T. Barton
Summary: This paper presents a practical cell-centred volume-of-fluid method for simulating compressible solid-fluid problems within a pure Eulerian setting. The method incorporates a mixed-cell update to maintain sharp interfaces, and can be easily extended to include other coupled physics. Various challenging test problems are used to validate the method, and its robustness and application in a multi-physics context are demonstrated.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Xing Ji, Fengxiang Zhao, Wei Shyy, Kun Xu
Summary: This paper presents the development of a third-order compact gas-kinetic scheme for compressible Euler and Navier-Stokes solutions, constructed particularly for an unstructured tetrahedral mesh. The scheme demonstrates robustness in high-speed flow computation and exhibits excellent adaptability to meshes with complex geometrical configurations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Alsadig Ali, Abdullah Al-Mamun, Felipe Pereira, Arunasalam Rahunanthan
Summary: This paper presents a novel Bayesian statistical framework for the characterization of natural subsurface formations, and introduces the concept of multiscale sampling to localize the search in the stochastic space. The results show that the proposed framework performs well in solving inverse problems related to porous media flows.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jacob Rains, Yi Wang, Alec House, Andrew L. Kaminsky, Nathan A. Tison, Vamshi M. Korivi
Summary: This paper presents a novel method called constrained optimized DMD with Control (cOptDMDc), which extends the optimized DMD method to systems with exogenous inputs and can enforce the stability of the resulting reduced order model (ROM). The proposed method optimally places eigenvalues within the stable region, thus mitigating spurious eigenvalue issues. Comparative studies show that cOptDMDc achieves high accuracy and robustness.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Andrea La Spina, Jacob Fish
Summary: This work introduces a hybridizable discontinuous Galerkin formulation for simulating ideal plasmas. The proposed method couples the fluid and electromagnetic subproblems monolithically based on source and employs a fully implicit time integration scheme. The approach also utilizes a projection-based divergence correction method to enforce the Gauss laws in challenging scenarios. Numerical examples demonstrate the high-order accuracy, efficiency, and robustness of the proposed formulation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Junhong Yue, Peijun Li
Summary: This paper proposes two numerical methods (IP-FEM and BP-FEM) to study the flexural wave scattering problem of an arbitrary-shaped cavity on an infinite thin plate. These methods successfully decompose the fourth-order plate wave equation into the Helmholtz and modified Helmholtz equations with coupled conditions on the cavity boundary, providing an effective solution to this challenging problem.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
William Anderson, Mohammad Farazmand
Summary: We develop fast and scalable methods, called RONS, for computing reduced-order nonlinear solutions. These methods have been proven to be highly effective in tackling challenging problems, but become computationally prohibitive as the number of parameters grows. To address this issue, three separate methods are proposed and their efficacy is demonstrated through examples. The application of RONS to neural networks is also discussed.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Marco Caliari, Fabio Cassini
Summary: In this paper, a second order exponential scheme for stiff evolutionary advection-diffusion-reaction equations is proposed. The scheme is based on a directional splitting approach and uses computation of small sized exponential-like functions and tensor-matrix products for efficient implementation. Numerical examples demonstrate the advantage of the proposed approach over state-of-the-art techniques.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Sebastiano Boscarino, Seung Yeon Cho, Giovanni Russo
Summary: This work proposes a high order conservative semi-Lagrangian method for the inhomogeneous Boltzmann equation of rarefied gas dynamics. The method combines a semi-Lagrangian scheme for the convection term, a fast spectral method for computation of the collision operator, and a high order conservative reconstruction and a weighted optimization technique to preserve conservative quantities. Numerical tests demonstrate the accuracy and efficiency of the proposed method.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)
Article
Computer Science, Interdisciplinary Applications
Jialei Li, Xiaodong Liu, Qingxiang Shi
Summary: This study shows that the number, centers, scattering strengths, inner and outer diameters of spherical shell-structured sources can be uniquely determined from the far field patterns. A numerical scheme is proposed for reconstructing the spherical shell-structured sources, which includes a migration series method for locating the centers and an iterative method for computing the inner and outer diameters without computing derivatives.
JOURNAL OF COMPUTATIONAL PHYSICS
(2024)