# Xian-Ming GuSouthwestern University of Finance and Economics · School of Mathematics

Xian-Ming Gu

Doctor of Philosophy

(tenured) Associate Professor; Visiting Scholar at the Mathematical Institute of Utrecht University (2023/04~2024/04)

## About

119

Publications

38,702

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1,553

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Introduction

1. My group (including 9 MSc students) focuses on the parallel numerical methods (maybe non-uniform meshes) with low storage requirement for evolutionary PDEs (e.g., advection-diffusion equation, Black-Scholes, Heston Option Pricing Model, etc.);
2. Recently, we contribute us to do a test matrices library for scientific computing, so if you have some interesting test matrices from realistic applications and want to share with us, please email me.

Additional affiliations

April 2023 - April 2024

**Universiteit Utrecht**

Position

- Visting Scholar

Description

- Work on computational finance with Prof. Dr. Cornelis W. Oosterlee

Education

August 2014 - August 2016

September 2011 - June 2017

September 2007 - June 2011

**Tangshan Normal Unviversity**

Field of study

- Mathematics and Applied Mathematics

## Publications

Publications (119)

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...

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...

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...

The extended Fisher--Kolmogorov (EFK) equation has been used to describe some phenomena in physical, material and biology systems. In this paper, we propose a full-rank splitting scheme and a rank-adaptive splitting approach for this equation. We first use a finite difference method to approximate the space derivatives. Then, the resulting semi-dis...

In this paper, we consider fast solvers for discrete linear systems generated by Riesz space fractional diffusion equations. We extract a scalar matrix, a compensation matrix, and a $\tau$ matrix from the coefficient matrix, and use their sum to construct a class of $\tau$ splitting iterative methods. Additionally, we design a preconditioner for th...

In this paper, the values and optimal exercise prices of American option under the CGMY model with regime-switching process are considered. For this case, the pricing mathematical model is a free boundary problem which includes d coupled fractional partial differential equations (PDEs) in one dimension with free boundary conditions, d denoting the...

PageRank can be viewed as a hyperlink-based method for estimating the importance of nodes in a network, and has attracted a lot of attention from researchers. In this paper, we first propose an extrapolation procedure for PageRank vector estimation based on the linear combination of the Ritz values computed from the Hessenberg process and eigenvect...

In this note, we revisit the result of the Toeplitz matrix inversion formula proposed by Lv and Huang (Appl. Math. Lett. 20 (2007), 1189-1193), then we give a new structured perturbation analysis, which is useful for testing the stability of practical algorithms. It has been shown that our new upper bound is much sharper than the one they have give...

In the present study, we consider the numerical method for Toeplitz-like linear systems arising from the d-dimensional Riesz space fractional diffusion equations (RSFDEs). We apply the Crank-Nicolson (CN) technique to discretize the temporal derivative and apply a quasi-compact finite difference method to discretize the Riesz space fractional deriv...

Simulation has been widely used to evaluate and optimize the stochastic complex systems such as manufacturing systems, telecommunication systems, and healthcare systems, which are known as cyber-physical systems. However, research on design ranking in simulation optimization is relatively few. This research considers the design ranking of k designs...

In this paper, for variable coefficient Riesz fractional diffusion equations in one and two dimensions, we first design a second-order implicit difference scheme by using the Crank-Nicolson method and a fractional centered difference formula for time
and space variables, respectively. With the compact operator acting on, a novel fourth-order finite...

The infinity norm bounds for the inverse of Nekrasov matrices play an important role
in scientific computing. We in this paper propose a triangulation-based approach that can easily be implemented to seek sharper infinity norm bounds for the inverse of Nekrasov matrices. With the help of such sharper bounds, new error estimates for
the linear compl...

In this paper, we integrated machine learning into the field of quantitative investment and established a set of automatic stock selection and investment timing models. Based on the validity test of factors, a multi-factor stock selection model was established to select stocks with the highest investment value to create a stock pool. By comparing t...

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...

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...

The Black-Scholes (B-S) equation has been recently extended as a kind of tempered time-fractional B-S equations, which become an interesting mathematical model in option pricing. In this study, we provide a fast numerical method to approximate the solution of the tempered time-fractional B-S model. To achieve high-order accuracy in space and overco...

The ranking and selection method has successfully been used to solve simulation optimization problems for many discrete event dynamic systems. However, little research has focused on optimizing the simulation efficiency of the subset ranking problem, which has wide applications in areas such as intelligent manufacturing, electrical engineering, and...

An optimized Schwarz domain decomposition method (DDM) for solving the local optical response model (LORM) is proposed in this paper. We introduce a hybridizable discontinuous Galerkin (HDG) scheme for the discretization of such a model problem based on a triangular mesh of the computational domain. The discretized linear system of the HDG method o...

Under the background of supply chain competition with deep integration of retail and logistics, market competition has evolved into competition between supply chains. In order to explore the coordination strategy and decision-making mechanism of competition between supply chains, we construct the game models under the coordinated, uncoordinated and...

The invertibility of a Toeplitz matrix can be assessed based on the solvability of two standard equations. The inverse of the nonsingular Toeplitz matrix can then be represented as the sum of products of circulant and skew-circulant (CS) matrices. In this note, we provide a new structured perturbation analysis for the CS representation of Toeplitz...

In this paper, we present a new block preconditioner for solving the saddle point linear systems. The proposed method is developed from an augmented reformulation of the saddle point problem into a new linear system with an almost block triangular coefficient matrix. Theoretical results are derived on the eigenvalue distribution of the precondition...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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,...

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,...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...

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...