Xian-Ming Gu

Xian-Ming Gu
Southwestern University of Finance and Economics · School of Mathematics

Doctor of Philosophy
(tenured) Associate Professor

About

102
Publications
32,330
Reads
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899
Citations
Introduction
1. My group (including two PhD students and one MSc student) will focus on the fast and parallel numerical methods (maybe non-uniform meshes) with low storage requirement for time and space fractional/nonlocal partial differential equations (e.g., advection-diffusion equations, Schrödinger equation, Allen-Cahn equation, Black-Scholes Option Pricing Model, etc.). Discussions will be highly welcome. 2. Recently, we contribute us to do a test matrices library for scientific computing, so if you have some interesting test matrices from realistic simulation applications and you would like to share with us, please email me.
Additional affiliations
February 2021 - present
Southwestern University of Finance and Economics
Position
  • Professor (Associate)
July 2019 - November 2019
University of Macau
Position
  • PostDoc Position
Description
  • Work with Prof. Hai-Wei Sun for efficient numerical solutions of nonlocal PDEs, including the integral fractional Laplacian.
July 2018 - July 2018
University of Macau
Position
  • Visiting Scholar
Description
  • Time period: 2-July-2018 ~ 27-July-2018. Working topic: 1) Fast and parallel numerical scheme for fractional differential equations; 2) Novel Krylov subspace methods for computing PageRank.
Education
August 2014 - August 2016
University of Groningen
Field of study
  • Numerical Mathematics
September 2011 - June 2017
University of Electronic Science and Technology of China
Field of study
  • Computational Mathematics
September 2007 - June 2011
Tangshan Normal Unviversity
Field of study
  • Mathematics and Applied Mathematics

Publications

Publications (102)
Article
Full-text available
In this paper we want to propose practical numerical methods to solve a class of initial-boundary problem of time-space fractional convection-diffusion equations (TSFCDEs). To start with, an implicit difference method based on two-sided weighted shifted Gr¨unwald formulae is proposed with a discussion of the stability and convergence. We construct...
Article
In this paper, a fast linearized conservative finite element method is studied for solving the strongly coupled nonlinear fractional Schr{\"o}dinger equations. We prove that the scheme preserves both the mass and energy, which are defined by virtue of some recursion relationships. Using the Sobolev inequalities and then employing the mathematical i...
Article
Full-text available
An all-at-once linear system arising from the nonlinear tempered fractional diffusion equation with variable coefficients is studied. Firstly, the nonlinear and linearized implicit schemes are proposed to approximate such the nonlinear equation with continuous/discontinuous coefficients. The stabilities and convergences of the two schemes are prove...
Article
Full-text available
Volterra subdiffusion problems with weakly singular kernel describe the dynamics of subdiffusion processes well. The graded L1 scheme is often chosen to discretize such problems since it can handle the singularity of the solution near \(t = 0\). In this paper, we propose a modification. We first split the time interval [0, T] into \([0, T_0]\) and...
Article
PageRank is one of the most important ranking techniques in modern search engines. Many great interesting researches focus on developing efficient numerical methods to compute PageRank problems. In this paper, we consider a simpler generalized minimal residual (SGMRES) algorithm for computing PageRank. The main features of the SGMRES algorithm lie...
Article
Full-text available
PageRank is a greatly essential ranking algorithm in web information retrieval or search engine. In the current paper, we present a cost-effective Hessenberg-type method built upon the Hessenberg process for the computation of PageRank vector, which is better suited than the Arnoldi-type algorithm when the damping factor becomes high and especially...
Article
Full-text available
We propose a fast solution method for banded linear systems that transforms the original system into an equivalent one with an almost block triangular coefficient matrix, and then constructs a preconditioner based on this formulation. We analyze the algorithmic complexity of the new method and the eigenvalue distribution of the resulting preconditi...
Article
Full-text available
In this work, two fully novel finite difference schemes for two-dimensional time-fractional mixed diffusion and diffusion-wave equation (TFMDDWEs) are presented. Firstly, a Hermite and Newton quadratic interpolation polynomial have been used for time discretization and central quotient has used in spatial direction. The H2N2 finite difference is co...
Article
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In this paper, three energy preserving numerical methods are proposed, including the Crank–Nicolson Galerkin–Legendre spectral (CN–GLS) method, the SAV Galerkin–Legendre spectral (SAV–GLS) method, and the ESAV Galerkin–Legendre spectral (ESAV–GLS) method, for the space fractional nonlinear Schrödinger equation with wave operator. In theoretical ana...
Preprint
Full-text available
Fractional Ginzburg-Landau equations as the generalization of the classical one have been used to describe various physical phenomena. In this paper, we propose a numerical integration method for solving space fractional Ginzburg-Landau equations based on a dynamical low-rank approximation. We first approximate the space fractional derivatives by u...
Article
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In this paper, we propose a semi-implicit difference scheme for solving one-dimensional nonlinear space-fractional diffusion equations. The method is first-order accurate in time and second-order accurate in space. It uses a fractional central difference formula and the backward Euler method to approximate its space and time derivatives, respective...
Article
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We put forward and analyze the high-order (up to fourth) strong stability-preserving implicit-explicit Runge-Kutta schemes for the time integration of the space-fractional Allen-Cahn equation, which inherits the maximum principle preserving and energy stability. The space-fractional Allen-Cahn equation with homogeneous Dirichlet boundary condition...
Article
Event detection on social platforms can help people perceive essential events and make actionable decisions. Existing document-pivot streaming social event detection methods generally embed documents and perform text clustering. They face the challenges of constantly changing context and unknown event categories and struggle by designing compound t...
Article
Full-text available
In this article, our goal is to establish fast and efficient numerical methods for nonlinear space-fractional convection-diffusion-reaction (CDR) equations in the 1, 2, and 3 dimensions. For the spatial discretization of the CDR equations, the weighted essentially non-oscillatory (WENO) scheme is used to approximate the convection term, and the fra...
Preprint
Full-text available
Time-space fractional Bloch-Torrey equations are developed by some researchers to investigate the relationship between diffusion and fractional-order dynamics. In this paper, we first propose a second-order scheme for this equation by employing the recently proposed L2-type formula [A. A. Alikhanov, C. Huang, Appl. Math. Comput. (2021) 126545]. The...
Article
Image deblurring is an important pre-processing step in image analysis. The research for efficient image deblurring methods is still a great challenge. Most of the currently used methods are based on integer-order derivatives, but they typically lead to texture elimination and staircase effects. To overcome these drawbacks, some researchers have pr...
Article
Full-text available
The PageRank model computes the stationary distribution of a Markov random walk on the linking structure of a network, and it uses the values within to represent the importance or centrality of each node. This model is first proposed by Google for ranking web pages, then it is widely applied as a centrality measure for networks arising in various f...
Article
In this paper, fast numerical methods are established to solve a class of time distributed-order and Riesz space fractional diffusion-wave equations. We derive new difference schemes by a weighted and shifted Grünwald formula in time and a fractional centered difference formula in space. The unconditional stability and second-order convergence in t...
Article
Based on the fixed point equation of an equivalent reformulation for the absolute value equations, in this paper we construct a novel alternating iterative method and proved its R-linear convergence rate by verifying that its generated sequence is strictly contractive. Preliminary experiments are carried out to show numerical performance of the pro...
Article
In this article, a fast and efficient numerical method is constructed for solving nonlinear space-fractional multidelay reaction-diffusion equations. Firstly, we spatially discretize the equation using the weighted and shifted Grünwald–Letnikov difference (WSGD) formula. As a result, a nonlinear system of ordinary differential equations (ODEs) is o...
Article
Full-text available
In this paper, we mainly focus on the development and study of a new global GCRO-DR method that allows both the flexible preconditioning and the subspace recycling for sequences of shifted linear systems. The novel method presented here has two main advantages: firstly, it does not require the right-hand sides to be related, and, secondly, it can a...
Preprint
Full-text available
The p-step backward difference formula (BDF) for solving systems of ODEs can be formulated as all-at-once linear systems that are solved by parallel-in-time preconditioned Krylov subspace solvers (see McDonald, Pestana, andWathen [SIAM J. Sci. Comput., 40(2) (2018): A1012-A1033] and Lin and Ng [arXiv:2002.01108, 2020, 17 pages]). However, when the...
Preprint
Full-text available
In this paper, fast numerical methods are established for solving a class of time distributed-order and Riesz space fractional diffusion-wave equations. We derive new difference schemes by the weighted and shifted Gr$\ddot{\rm{u}}$nwald formula in time and the fractional centered difference formula in space. The unconditional stability and second-o...
Article
Full-text available
Anderson(m_0) extrapolation, an accelerator to a fixed-point iteration, stores m_0+1 prior evaluations of the fixed-point iteration and computes a linear combination of those evaluations as a new iteration. The computational cost of the Anderson(m_0) acceleration becomes expensive with the parameter m0 increasing, thus m0 is a common choice in most...
Article
In this paper, we present a new algorithm by combining the cheap but slow power method with the fast but expensive Arnoldi procedure periodically. The main feature of our method is that a weighted inner product is introduced when using an Arnoldi procedure to construct a basis for a Krylov subspace. Particularly, in each cycle, the weight matrix is...
Article
Full-text available
In the current paper, for the time fractional diffusion equation with an exponential tempering, we propose a numerical algorithm based on the Lagrange-quadratic spline interpolations and the optimal technique. The discretized linear systems and some properties are investigated in detail. By using these properties, the coefficient matrix and the rig...
Article
In this paper, we study a time-fractional diffusion equation with a time-invariant type variable fractional order. We propose an implicit finite difference scheme to approximate the variable-order Caputo fractional derivative, while the central difference method is employed to discretize the spatial differential operator. A novel decomposition of t...
Article
Full-text available
The space fractional Cahn–Hilliard phase-field model is more adequate and accurate in the description of the formation and phase change mechanism than the classical Cahn–Hilliard model. In this article, we propose a temporal second-order energy stable scheme for the space fractional Cahn–Hilliard model. The scheme is based on the second-order backw...
Article
The all-at-once system arising from fractional mobile/immobile advection-diffusion equations is studied. Firstly, the finite difference method with L1 formula is employed to discretize it. The resulting implicit scheme is a time-stepping scheme, which is not suitable for parallel computing. Based on this scheme, an all-at-once system is established...
Article
Full-text available
In this paper, we develop two fast implicit difference schemes for solving a class of variable-coefficient time-space fractional diffusion equations with integral fractional Laplacian (IFL). The proposed schemes utilize the graded $L1$ formula for the Caputo fractional derivative and a special finite difference discretization for IFL, where the gra...
Article
For computing PageRank problems, a Power-Arnoldi algorithm is presented by periodically knitting the power method together with the thick restarted Arnoldi algorithm. In this paper, by using the power method with the extrapolation process based on trace (PET), a variant of the Power-Arnoldi algorithm is developed for accelerating PageRank computati...
Preprint
Full-text available
The space fractional Cahn-Hilliard phase-field model is more adequate and accurate in the description of the formation and phase change mechanism than the classical Cahn-Hilliard model. In this article, we propose a temporal second-order energy stable scheme for the space fractional Cahn-Hilliard model. The scheme is based on the second-order backw...
Article
Full-text available
In this paper, a local coupling multi-trace domain decomposition method (LCMT-DDM) based on surface integral equation (SIE) formulations is proposed to analyze electromagnetic scattering from multilayered dielectric objects. Different from the traditional SIE-DDM, where the interactions between sub-domains are accounted for using global radiation c...
Preprint
Full-text available
In this paper, for the the time fractional diffusion equation with an exponential tempering, we give a high-order numerical algorithm based on the Lagrange and quadratic spline interpolations and a compact optimal idea. Corresponding linear systems and some properties are given. These properties show the computational cost for implementing this alg...
Article
In this paper, we propose a fast compact implicit integration factor (FcIIF) method with non-uniform time meshes for solving the two-dimensional nonlinear Riesz space-fractional reaction-diffusion equation. The weighted and shifted Grüwald-Letnikov (WSGD) approximation is employed to the spatial discretization of the equation, and a system of nonli...
Article
In this article, we first propose an unconditionally stable implicit difference scheme for solving generalized time–space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the L1‐type formula for the generalized Caputo fractional derivative in time discretization and the second‐order weighted and shi...
Preprint
Full-text available
In this paper, we first propose an unconditionally stable implicit difference scheme for solving generalized time-space fractional diffusion equations (GTSFDEs) with variable coefficients. The numerical scheme utilizes the $L1$-type formula for the generalized Caputo fractional derivative in time discretization and the second-order weighted and shi...
Article
Full-text available
In this paper, we intend to develop an effective numerical method to solve a class of two-dimensional space-fractional advection-diffusion-reaction equations. After spatially discretizing this equation using the fractional centered difference formula, it leads to a system of nonlinear ordinary differential equations. The compact implicit integratio...
Preprint
Full-text available
Volterra subdiffusion problems with weakly singular kernel describe the dynamics of subdiffusion processes well. The graded L1 scheme is often chosen to discretize such problems since it can handle the singularity of the solution near t = 0. In this paper, we propose a modification. We first split the time interval [0, T ] into [0, T_0 ] and [T_0,...
Preprint
We present a pseudo-spectral discretization method for solving second-order elliptic partial differential equations (PDEs) with continuous/discontinuous variable coefficients in one, two and three dimensions, that shares the high approximation accuracy and ease of implementation of the conventional pseudo spectral (or spectral collocation) method,...
Article
After spatially discretizing the nonlinear space Riesz fractional reaction–diffusion equations by the fractional centered difference formula, the resulting semi-discrete equations lead to a nonlinear ordinary differential equation system. In order to obtain good stability and robustness, the implicit integration factor (IIF) method that treats the...
Article
For numerical computation of three-dimensional (3-D) large-scale magnetostatic problems, iterative solver is preferable since a huge amount of memory is needed in case of using sparse direct solvers. In this paper, a recently proposed Coulomb-gauged magnetic vector potential (MVP) formulation for magnetostatic problems is adopted for finite element...
Conference Paper
In this paper, we intend to develop an effective numerical method to solve a class of two-dimensional space-fractional advection-diffusion-reaction equations. After spatially discretizing this equation using the fractional centered difference formula, it leads to a system of nonlinear ordinary differential equations. The compact implicit integratio...
Article
In this paper, we consider a fast and second-order implicit difference method for approximation of a class of time-space fractional variable coefficients advection-diffusion equation. To begin with, we construct an implicit difference scheme, based on $L2-1_{\sigma}$ formula [A. A. Alikhanov, A new difference scheme for the time fractional diffusio...
Article
Multi-shifted linear systems with non-Hermitian coefficient matrices often arise from the numerical solutions for time-dependent partial/fractional differential equations (PDEs/FDEs), control theory, PageRank problem, etc. In this paper, we derive efficient variants of the restarted CMRH (ChangingMinimal Residual method based on the Hessenberg proc...
Article
Full-text available
The block conjugate orthogonal conjugate gradient method (BCOCG) is recognized as a common method to solve complex symmetric linear systems with multiple right-hand sides. However, breakdown always occurs if the right-hand sides are rank deficient. In this paper, based on the orthogonality conditions, we present a breakdown-free BCOCG algorithm wit...
Conference Paper
Full-text available
A kind of spatial fractional diffusion equations in this paper are studied. Firstly, an L1 formula is employed for the spatial discretization of the equations. Then, a second order scheme is derived based on the resulting semi-discrete ordinary differential system by using the implicit integration factor method, which is a class of efficient semi-i...
Article
An implicit finite difference scheme based on the $L2$-$1_{\sigma}$ formula is presented for a class of one-dimensional time fractional reaction-diffusion equations with variable coefficients and time drift term. The unconditional stability and convergence of this scheme are proved rigorously by the discrete energy method, and the optimal convergen...
Article
In this paper, we investigate the well-posedness of the real fractional Ginzburg-Landau equation in several different function spaces, which have been used to deal with the Burgers’ equation, the semilinear heat equation, the Navier-Stokes equations, etc. The long time asymptotic behavior of the nonnegative global solutions is also studied in detai...
Article
In this paper, two efficient iterative algorithms based on the Simpler GMRES method are proposed for solving shifted linear systems. To make full use of the shifted structure, the proposed algorithms utilizing the deflated restarting strategy and flexible preconditioning can significantly reduce the number of matrix-vector products and the elapsed...
Preprint
Full-text available
PageRank is a greatly essential ranking algorithm in web information retrieval or search engine. In the current paper, we present a cost-effective Hessenberg-type method built upon the Hessenberg process for the computation of PageRank vector, which is better suited than the Arnoldi-type algorithm when the damping factor becomes high and especially...
Preprint
A kind of spatial fractional diffusion equations in this paper are studied. Firstly, an L1 formula is employed for the spatial discretization of the equations. Then, a second order scheme is derived based on the resulting semi-discrete ordinary differential system by using the implicit integration factor method, which is a class of efficient semi-i...
Article
A block lower triangular Toeplitz system arising from the time-space fractional diffusion equation is discussed. For efficient solutions of such the linear system, the preconditioned biconjugate gradient stabilized method and the flexible general minimal residual method are exploited. The main contribution of this paper has two aspects: (i) A block...
Article
Full-text available
As is well-known, the advantage of the high-order compact difference scheme (H-OCD) is unconditionally stable and convergent with the order $O(\tau^2+h^4)$ under the maximum norm. In this article, a new numerical gradient scheme based on the collocation polynomial and Hermite interpolation is presented. Moreover, the convergence order of this kind...
Article
In this paper, we introduce and prove the generalizations of Radon inequality. The proofs in the paper unify and are simpler than those in former work. Meanwhile, we also find mathematical equivalences among the Bernoulli inequality, the weighted AM-GM inequality, the Hölder inequality, the weighted power mean inequality and the Minkowski inequalit...
Article
Full-text available
In the current note, we investigate mathematical relations among the weighted arithmetic mean-geometric mean (AM-GM) inequality, the H\"{o}lder inequality and the weighted power-mean inequality. Meanwhile, the proofs of mathematical equivalence among the weighted AM-GM inequality, the weighted power-mean inequality and the H\"{o}lder inequality are...
Article
Full-text available
PageRank problem is the cornerstone of Google search engine and is usually stated as solving a huge linear system. Moreover, when the damping factor approaches 1, the spectrum properties of this system deteriorate rapidly and this system becomes difficult to solve. In this paper, we demonstrate that the coefficient matrix of this system can be tran...
Article
Stochastic Automata Networks (SANs) have a large amount of applications in modeling queueing systems and communication systems. To find the steady state probability distribution of the SANs, it often needs to solve linear systems which involve their generator matrices. However, some classical iterative methods such as the Jacobi and the Gauss-Seide...
Preprint
Full-text available
The block lower triangular Toeplitz system arising from time-space fractional diffusion equation is discussed. For the purpose of fast and economically solving the system, the preconditioned biconjugate gradient stabilized method and flexible general minimal residual method are used. The main contribution of this paper has two aspects: (i) A block...
Article
Full-text available
This study seeks to address the delay-probability-dependent stability problem for a new class of stochastic neural networks with randomly occurring uncertainties, neutral type delay, distributed delay and probability-distribution delay. The system not only includes the randomly occurring uncertainties of parameters (ROUPs) but also contains stochas...
Article
Recently, the adaptive algebraic aggregation multigrid method has been proposed for computing stationary distributions of Markov chains. This method updates aggregates on every iterative cycle to keep high accuracies of coarse-level corrections. Accordingly, its fast convergence rate is well guaranteed, but often a large proportion of time is cost...
Article
Full-text available
In this paper, employing the piecewise linear and quadratic Lagrange interpolation functions, we propose a novel numerical approximate method for the Caputo fractional derivative. For the obtained explicit recursion formula, the truncation error is investigated, which shows the involved convergence order is O(\tau^{3−\beta}) with \beta ∈ (0, 1). As...