Article
Automation & Control Systems
Soheila Ghoroghi Shafiei, Masoud Hajarian
Summary: This paper introduces the application of the Kaczmarz method in solving large-scale linear algebraic systems and Sylvester matrix equations. By deriving the matrix form and extending the algorithm, a new iterative algorithm is proposed and validated through numerical examples.
JOURNAL OF THE FRANKLIN INSTITUTE-ENGINEERING AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Shi-Ping Tang, Yu-Mei Huang
Summary: In this paper, fast second-order numerical methods are proposed for solving one- and two-dimensional space-fractional advection-diffusion equations defined on a finite domain. We analyze the stability and convergence of the proposed methods, and propose new approximate preconditioners for solving the discretized linear systems. The numerical results demonstrate the effectiveness of the proposed methods.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2024)
Article
Mathematics, Applied
Jing Li, Hao Cheng
Summary: In this paper, the inverse problem for the time-fractional two-dimensional Cahn-Hilliard equation is investigated. The ill-posedness and conditional stability of the inverse problem are proved. A new iterative variational regularization method is proposed to solve it, and the convergence rates of the regularized solutions under different parameter choice rules are obtained. Numerical examples demonstrate the effectiveness and stability of the proposed method.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Li Chen, Shujuan Lu
Summary: In this paper, the numerical approximation of the time fractional Cahn-Hilliard equation with initial singularity is studied. A nonlinear fully discrete scheme is proposed and its stability and convergence are rigorously proved. The results provide insights into the numerical simulation of related equations.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2022)
Article
Computer Science, Interdisciplinary Applications
David Kay, Vanessa Styles
Summary: This study introduces an efficient solver for saddle point problems in the finite element approximation of nonlocal multi-phase Allen-Cahn variational inequalities. The solver exhibits mesh independence and mild dependence on phase field variables, and converges to the two-phase problem solution within three GMRES iterations regardless of mesh size or interfacial width. Numerical results demonstrate the competitiveness of this approach.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Interdisciplinary Applications
Eylem Ozturk, Joseph L. Shomberg
Summary: In this study, a viscous Cahn-Hilliard phase-separation model with memory and a nonlocal fractional Laplacian operator in the chemical potential is examined. The existence of global weak solutions is proven using a Galerkin approximation scheme. Continuous dependence estimate provides uniqueness of weak solutions and existence of a compact connected global attractor in the weak energy phase space.
FRACTAL AND FRACTIONAL
(2022)
Article
Mathematics, Applied
Yuka Hashimoto, Takashi Nodera
Summary: This paper proposes a novel technique for accelerating the Krylov subspace methods for transfer operators by replacing positive definite kernels in RKHS, which is equivalent to preconditioning the transfer operator with a specific linear operator.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Shi-Ping Tang, Yu-Mei Huang
Summary: The paper examines an optimal control problem constrained by a fractional diffusion equation, developing closed-form solutions for the optimal control variable and utilizing the decoupled gradient projection method to solve the system of equations. By discretizing the state and costate equations, applying a DRCS preconditioner, and analyzing the spectral distributions of the preconditioned matrix, the study demonstrates significant improvements in convergence for Krylov subspace iteration methods.
COMPUTATIONAL & APPLIED MATHEMATICS
(2021)
Article
Mathematics
Shengjie Xu, Fei Xue
Summary: This paper introduces a flexible extended Krylov subspace method (F-EKSM) for numerical approximation of the action of a matrix function f(A) to a vector b. By replacing the zero pole in EKSM with a properly chosen fixed nonzero pole, the optimal fixed pole for F-EKSM is derived for symmetric positive definite matrices to achieve a lower asymptotic convergence factor upper bound than that of EKSM. Numerical experiments show that for large and sparse matrices efficiently handled by LU factorizations, F-EKSM and a variant of RKSM based on a small number of fixed poles outperform EKSM in both storage and runtime, and usually have advantages over adaptive RKSM in runtime.
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)
Correction
Mathematics
Sebastian Scholtes, Maria G. Westdickenberg
Summary: This article corrects a dissipation estimate from the original article, but the main result remains unchanged.
JOURNAL OF DIFFERENTIAL EQUATIONS
(2023)
Article
Mathematics, Applied
Min Zhang, Guo-Feng Zhang
Summary: This study investigated the solution strategy for the space-fractional modified Cahn-Hilliard equation as a tool for gray value image inpainting model. A fast solver was developed for solving the linear systems arising from the 2D space-fractional modified Cahn-Hilliard equation. The theoretical analysis showed a fast convergence rate of the proposed preconditioner, which was confirmed in numerical examples.
COMPUTERS & MATHEMATICS WITH APPLICATIONS
(2021)
Article
Mathematics, Applied
Min-Li Zeng, Jun-Feng Yang, Guo-Feng Zhang
Summary: In this paper, we investigate the structure of discretized linear systems derived from spatial fractional diffusion equations. We propose a new approximate inverse preconditioner based on the diagonal-plus-Toeplitz structure of the coefficient matrices. The tau matrix-based approximate inverse (TAI) preconditioning technique, implemented using discrete sine transforms (DST), is shown to achieve fast convergence in Krylov subspace methods. Numerical experiments demonstrate that the performance of the tau-matrix based preconditioning technique is superior to other tested preconditioners.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2022)
Article
Mathematics, Applied
Yue Yu, Jiansong Zhang, Rong Qin
Summary: In this paper, the authors study a class of time-fractional phase-field models, including the Allen-Cahn and Cahn-Hilliard equations. Two explicit time-stepping schemes are proposed based on the exponential scalar auxiliary variable (ESAV) approach, where the fractional derivative is discretized using L1 and L1+ formulas respectively. These novel schemes allow for the decoupled computations of the phase variable phi and the auxiliary variable R. Furthermore, the schemes exhibit energy dissipation law on general nonuniform meshes, which is inherent in the continuous level. Numerical experiments are conducted to demonstrate the accuracy and efficiency of the proposed methods.
JOURNAL OF SCIENTIFIC COMPUTING
(2023)
Article
Mathematics, Applied
Zhong-Zhi Bai, Kang-Ya Lu
Summary: This study proposes a method for optimal control problems constrained with certain time and space-fractional diffusive equations, achieving specially structured linear systems with positive definiteness. Both theoretical analysis and numerical experiments show that incorporating rotated block-diagonal preconditioners with preconditioned Krylov subspace iteration methods can exhibit optimal convergence properties.
APPLIED NUMERICAL MATHEMATICS
(2021)
Article
Computer Science, Theory & Methods
Alexander Ostermann, Frederic Rousset, Katharina Schratz
Summary: This study introduces a new filtered low-regularity Fourier integrator for the cubic nonlinear Schrödinger equation, which achieves better convergence rates at low regularity. By combining the better local error properties of the new scheme with a stability analysis based on general discrete Strichartz-type estimates, it can handle a much rougher class of solutions.
FOUNDATIONS OF COMPUTATIONAL MATHEMATICS
(2021)
Article
Mathematics, Applied
Lukas Einkemmer, Alexander Ostermann, Mirko Residori
Summary: The goal of this work is to solve a linear dispersive equation with variable coefficient advection on an unbounded domain. By discretizing space using a spectral method, the number of grid points required for accuracy is drastically reduced. By performing an operator splitting scheme and treating the dispersive part implicitly, an efficient numerical scheme is achieved despite the challenge of non-homogeneous transparent boundary conditions.
JOURNAL OF COMPUTATIONAL AND APPLIED MATHEMATICS
(2021)
Article
Computer Science, Interdisciplinary Applications
Marco Caliari, Lukas Einkemmer, Alexander Moriggl, Alexander Ostermann
Summary: Rational Exponential Integrators (REXI) are numerical methods suitable for time integration of linear partial differential equations with imaginary eigenvalues. A novel REXI scheme proposed in this paper drastically improves accuracy and efficiency, along with the ability to easily determine the required number of terms for accurate results. Comparative numerical simulations for a shallow water equation show the efficiency of the approach, indicating that REXI schemes can be efficiently implemented on graphic processing units.
JOURNAL OF COMPUTATIONAL PHYSICS
(2021)
Article
Mathematics, Applied
Yong-Liang Zhao, Xian-Ming Gu, Alexander Ostermann
Summary: This paper proposes a modification for dealing with Volterra subdiffusion problems with weakly singular kernel, saving computational resources and efficiently handling the singularity of the solution.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
Article
Mathematics, Applied
Alexander Ostermann, Frederic Rousset, Katharina Schratz
Summary: We study a filtered Lie splitting scheme for the cubic nonlinear Schrodinger equation and establish error estimates at low regularity by using discrete Bourgain spaces. This allows us to handle data in H-s with 0 < s < 1, overcoming the standard stability restriction to smooth Sobolev spaces with index s > 1/2. More precisely, we prove convergence rates of order tau(s/2) in L-2 at this level of regularity.
MATHEMATICS OF COMPUTATION
(2022)
Article
Mathematics, Applied
Alexander Ostermann, Fangyan Yao
Summary: In this paper, we propose and analyze a fully discrete low-regularity integrator for the one-dimensional cubic nonlinear Schrodinger equation on the torus. The scheme is explicit and implemented using the fast Fourier transform with a complexity of O(N log N) operations per time step. Numerical examples demonstrate the convergence behavior of the proposed scheme.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
L. Einkemmer, A. Ostermann, M. Residori
Summary: The present work introduces a second-order time splitting scheme for a linear dispersive equation with a variable advection coefficient, using a dual Petrov-Galerkin method for spatial discretization. A modified Strang splitting scheme is proposed to overcome order reduction caused by an inflow boundary condition, maintaining second-order accuracy. Stability analysis of this numerical scheme confirms the theoretical derivations.
NUMERISCHE MATHEMATIK
(2022)
Article
Computer Science, Interdisciplinary Applications
Marco Caliari, Fabio Cassini, Lukas Einkemmer, Alexander Ostermann, Franco Zivcovich
Summary: In this paper, a mu-mode integrator is proposed for solving stiff evolution equations. The integrator is based on a d-dimensional splitting approach and utilizes exact one-dimensional matrix exponentials. It demonstrates efficient computation of spectral transforms and outperforms established numerical methods in solving three-dimensional linear and nonlinear Schrodinger equations.
JOURNAL OF COMPUTATIONAL PHYSICS
(2022)
Review
Mathematics, Applied
Duy Phan, Alexander Ostermann
Summary: This paper considers two types of second-order in time partial differential equations, namely semilinear wave equations and semilinear beam equations. To solve these equations with exponential integrators, an approach to efficiently compute the action of the matrix exponential and related matrix functions is presented. Various numerical simulations are provided to illustrate the effectiveness of this approach.
JOURNAL OF SCIENTIFIC COMPUTING
(2022)
Article
Mathematics, Applied
Alexander Ostermann, Frederic Rousset, Katharina Schratz
Summary: In this paper, a new integration scheme is proposed for the nonlinear Schrodinger equation, which guarantees convergence rates at low regularity and exhibits superiority over standard schemes in numerical experiments.
JOURNAL OF THE EUROPEAN MATHEMATICAL SOCIETY
(2023)
Article
Mathematics, Applied
Alexander Ostermann, Fardin Saedpanah, Nasrin Vaisi
Summary: The aim of this paper is to construct and analyze explicit exponential Runge-Kutta methods for the temporal discretization of linear and semilinear integro-differential equations. The order conditions are derived and used for constructing numerical methods. Convergence analysis is performed in a Hilbert space setting, and both linear and semilinear cases are considered.
SIAM JOURNAL ON NUMERICAL ANALYSIS
(2023)
Article
Computer Science, Interdisciplinary Applications
Lukas Einkemmer, Alexander Ostermann, Carmela Scalone
Summary: Dynamical low-rank approximation has shown great efficiency in solving kinetic equations, but it fails to preserve the structure of the underlying physical problem. This study proposes a robust low-rank integrator that conserves mass and momentum and significantly improves energy conservation.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Alexander Ostermann, Yifei Wu, Fangyan Yao
Summary: This paper analyzes a new exponential-type integrator for the nonlinear cubic Schrodinger equation. The integrator is explicit, efficient, and capable of dealing with complex resonance structures in arbitrary dimensions.
ADVANCES IN CONTINUOUS AND DISCRETE MODELS
(2022)
Article
Mathematics, Applied
Huan-Yan Jian, Ting-Zhu Huang, Alexander Ostermann, Xian-Ming Gu, Yong-Liang Zhao
Summary: This article aims to establish fast and efficient numerical methods for nonlinear space-fractional convection-diffusion-reaction equations. The use of WENO scheme and fractional centered difference formula results in a nonlinear system of ODEs. The proposed fast solution algorithm based on IIF-WENO scheme and adaptive restarting Krylov subspace method shows promising results in terms of computational complexity and memory storage.
JOURNAL OF SCIENTIFIC COMPUTING
(2021)
