arXiv subjects

Xian-Ming Gu

Publications and source records attributed to Xian-Ming Gu.

At least 19 recordsLinked to original sources

arXiv2026

A preconditioned boundary value method for advection-diffusion equations with half Laplacian via spectrum doubling

In this paper, we study an evolution equation that involves a half-Laplacian operator derived from the Riesz fractional Laplacian, combined with a differential operator $\mathcal{L}$. Using the identity $(-Δ)^{1/2}=\mathcal H(\partial_x)$, we introduce a Spectrum Doubling (SD) reformulation that transforms the original half-diffusion equation into a first-order doubled system. The reformulated system exhibits stable and unstable spectral branches, and the original half-diffusion dynamics is recovered on a suitable stable invariant subspace characterized by a compatibility condition on the initial condition. The SD reformulation provides a practical numerical advantage: the half-Laplacian is applied only to the initial condition and source term, avoiding repeated evaluation of singular integrals during time marching. For the resulting integer-order system, we develop a Boundary Value Method (BVM) and study a second-order generalized midpoint scheme. We establish its stability and second-order temporal convergence. The fully discrete scheme leads to a large Kronecker-structured linear system, which is solved efficiently by GMRES with a block $ω$-circulant preconditioner. Under simultaneous diagonalizability of the spatial discretization matrices, the preconditioner can be implemented efficiently through fast discrete transforms. Numerical experiments for three evelutionary models confirm the theoretical convergence results and demonstrate the robustness and efficiency of the proposed method, including in strongly advective regimes. The experiments also show that the approach remains effective when the Hilbert transform is evaluated numerically, and illustrate the applicability of the SD framework to a nonlocal Schrödinger-type example.

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arXiv2025

Error Analysis of Krylov Subspace approximation Based on IDR($s$) Method for Matrix Function Bilinear Forms

The matrix function bilinear form, namely $\mathbf{u}^\top f(A)\mathbf{v}$, appears in many scientific computing problems, where $\mathbf{u}, \mathbf{v} \in \mathbb{R}^n$, $A \in \mathbb{R}^{n\times n}$, and $f(z)$ is a given analytic function. The Induced Dimension Reduction (IDR($s$)) method was originally proposed to solve a large system of linear equations, and effectively reduces the complexity and storage requirement by dimensionality reduction while maintaining the numerical stability of the algorithm. In fact, the IDR($s$) method can generate an interesting Hessenberg decomposition. We make use of this decomposition to establish the numerical algorithm and a practical error estimation framework for the matrix function bilinear form. Based on an error analysis of the IDR($s$) approximation, the corresponding error expansion is derived. Crucially, we prove that, under mild conditions, the remainder term in this expansion converges to zero as the truncation order increases, thereby validating the error expansion. The leading computable term is then used as a practical a posteriori error indicator for general analytic functions. We present numerical experiments to support our theoretical findings and illustrate the efficacy of our proposed method, along with its stopping criterion, over traditional Bi-Lanczos and Arnoldi-based algorithms.

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arXiv2025

Splitting-based randomized dynamical low-rank approximations for stiff matrix differential equations

In the fields of control theory and machine learning, the dynamic low-rank approximation for large-scale matrices has received substantial attention. Considering large-scale semilinear stiff matrix differential equations, we propose splitting-based randomized dynamical low-rank approximations for a low-rank solution of the stiff matrix differential equation. We first split such the equation into a stiff linear subproblem and a nonstiff nonlinear subproblem. Then, a low-rank exponential integrator is applied to the linear subproblem. Two randomized low-rank approaches are employed for the nonlinear subproblem. Furthermore, we extend the proposed methods to rank-adaptation scenarios. Through rigorous validation on canonical stiff matrix differential problems, including spatially discretized Allen-Cahn equations and differential Riccati equations, we demonstrate that our methods achieve desired convergence orders. Numerical results confirm the robustness and accuracy of the proposed methods.

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arXiv2025

A note on the growth factor in Gaussian elimination for Higham matrices

The Higham matrix is a complex symmetric matrix A=B+iC, where both B and C are real, symmetric and positive definite and $\mathrm{i}=\sqrt{-1}$ is the imaginary unit. For any Higham matrix A, Ikramov et al. showed that the growth factor in Gaussian elimination is less than 3. In this paper, based on the previous results, a new bound of the growth factor is obtained by using the maximum of the condition numbers of matrixes B and C for the generalized Higham matrix A, which strengthens this bound to 2 and proves the Higham's conjecture.

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arXiv2024

Parallel-in-Time Iterative Methods for Pricing American Options

For pricing American options, %after suitable discretization in space and time, a sequence of discrete linear complementarity problems (LCPs) or equivalently Hamilton-Jacobi-Bellman (HJB) equations need to be solved in a sequential time-stepping manner. In each time step, the policy iteration or its penalty variant is often applied due to their fast convergence rates. In this paper, we aim to solve for all time steps simultaneously, by applying the policy iteration to an ``all-at-once form" of the HJB equations, where two different parallel-in-time preconditioners are proposed to accelerate the solution of the linear systems within the policy iteration. Our proposed methods are generally applicable for such all-at-once forms of the HJB equation, arising from option pricing problems with optimal stopping and nontrivial underlying asset models. Numerical examples are presented to show the feasibility and robust convergence behavior of the proposed methodology.

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arXiv2024

On $\tau$-preconditioners for a quasi-compact difference scheme to Riesz fractional diffusion equations with variable coefficients

In the present study, we consider the preconditioned generalized minimal residual (GMRES) method for the asymmetric linear systems arising from the $d$-dimensional Riesz space fractional diffusion equations (RSFDEs). The Crank-Nicolson scheme and a quasi-compact finite difference method are used to discretize the temporal derivative and Riesz space fractional derivatives in such RSFDEs, respectively. For the $d$-dimensional discretized RSFDEs, the corresponding coefficient matrix is the sum of a product of a $d$-level block tridiagonal matrix multiplying a diagonal matrix and a $d$-level Toeplitz matrix. We develop a sine transform based preconditioner (namely $\tau$ preconditioner) to accelerate the convergence of the GMRES method. Theoretical analysis shows that the upper bound of relative residual norm of the GMRES method with the proposed preconditioner is mesh-independent, which leads to a linear convergence rate. Numerical results are presented to confirm the theoretical results regarding the preconditioned matrix and to illustrate the efficiency of the proposed preconditioner.

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arXiv2024

A Bernoulli-barycentric rational matrix collocation method with preconditioning for a class of evolutionary PDEs

We propose a Bernoulli-barycentric rational matrix collocation method for two-dimensional evolutionary partial differential equations (PDEs) with variable coefficients that combines Bernoulli polynomials with barycentric rational interpolations in time and space, respectively. The theoretical accuracy $O\left((2π)^{-N}+h_x^{d_x-1}+h_y^{d_y-1}\right)$ of our numerical scheme is proven, where $N$ is the number of basis functions in time, $h_x$ and $h_y$ are the grid sizes in the $x$, $y$-directions, respectively, and $0\leq d_x\leq \frac{b-a}{h_x},~0\leq d_y\leq\frac{d-c}{h_y}$. For the efficient solution of the relevant linear system arising from the discretizations, we introduce a class of dimension expanded preconditioners that take the advantage of structural properties of the coefficient matrices, and we present a theoretical analysis of eigenvalue distributions of the preconditioned matrices. The effectiveness of our proposed method and preconditioners are studied for solving some real-world examples represented by the heat conduction equation, the advection-diffusion equation, the wave equation and telegraph equations.

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arXiv2024

A Generalized Block Circulant Preconditioner for Crank-Nicolson All-at-Once Systems with Applications to Option Pricing PDEs

The Crank--Nicolson (CN) method is a widely used time integration scheme for evolutionary partial differential equations (PDEs) arising in various scientific and engineering disciplines. Since the numerical solution at each time level depends on the solution at the previous time level, the resulting discretization is inherently sequential and therefore difficult to parallelize in time. In this paper, we develop an all-at-once formulation of the CN discretization together with a generalized block circulant preconditioner that enables an efficient parallel-in-time solution within a Krylov subspace framework. We establish a detailed spectral analysis of the preconditioned system, proving that most eigenvalues are equal to $1$, while the remaining eigenvalues are confined to the annulus: \begin{equation*} \left\{ z\in\mathbb{C}: \frac{1}{1+\alpha}<|z|<\frac{1}{1-\alpha}, \ \Re(z)>0 \right\}, \end{equation*} where $0<\alpha<1$ is a free parameter. Besides, the efficient implementation of the proposed preconditioner is described. Given certain conditions, we prove that the preconditioned GMRES($m$) method achieves a fast convergence rate independent of discretization stepsizes from the residual point of view. Finally, we verify both theoretical findings and the efficacy of the proposed preconditioner via numerical experiments on financial option pricing PDEs (even with variable coefficients).

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arXiv2023

A low-rank algorithm for strongly damped wave equations with visco-elastic damping and mass terms

Damped wave equations have been used in many real-world fields. In this paper, we study a low-rank solution of the strongly damped wave equation with the damping term, visco-elastic damping term and mass term. Firstly, a second-order finite difference method is employed for spatial discretization. Then, we receive a second-order matrix differential system. Next, we transform it into an equivalent first-order matrix differential system, and split the transformed system into three subproblems. Applying a Strang splitting to these subproblems and combining a dynamical low-rank approach, we obtain a low-rank algorithm. Numerical experiments are reported to demonstrate that the proposed low-rank algorithm is robust and accurate, and has second-order convergence rate in time.

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arXiv2023

A fast compact difference scheme with unequal time-steps for the tempered time-fractional Black-Scholes model

The Black-Scholes (B-S) equation has been recently extended as a kind of tempered time-fractional B-S equations, which becomes 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 overcome the weak initial singularity of exact solution, we combine the compact difference operator with L1-type approximation under nonuniform time steps to yield the numerical scheme. The convergence of the proposed difference scheme is proved to be unconditionally stable. Moreover, the kernel function in the tempered Caputo fractional derivative is approximated by sum-of-exponentials, which leads to a fast unconditionally stable compact difference method that reduces the computational cost. Finally, numerical results demonstrate the effectiveness of the proposed methods.

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arXiv2023

On the bilateral preconditioning for an L2-type all-at-once system arising from time-space fractional Bloch-Torrey equations

Time-space fractional Bloch-Torrey equations (TSFBTEs) are developed by some researchers to investigate the relationship between diffusion and fractional-order dynamics. In this paper, we first propose a second-order implicit difference scheme for TSFBTEs by employing the recently proposed L2-type formula [A.~A.~Alikhanov, C.~Huang, Appl.~Math.~Comput.~(2021) 126545]. Then, we prove the stability and the convergence of the proposed scheme. Based on such a numerical scheme, an L2-type all-at-once system is derived. In order to solve this system in a parallel-in-time pattern, a bilateral preconditioning technique is designed to accelerate the convergence of Krylov subspace solvers according to the special structure of the coefficient matrix of the system. We theoretically show that the condition number of the preconditioned matrix is uniformly bounded by a constant for the time fractional order $α\in (0,0.3624)$. Numerical results are reported to show the efficiency of our method.

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arXiv2022

An adaptive low-rank splitting approach for the extended Fisher--Kolmogorov equation

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-discrete system is split into two stiff linear parts and a nonstiff nonlinear part. This leads to our full-rank splitting scheme. The convergence and the maximum principle of the proposed scheme are proved rigorously. Based on the frame of the full-rank splitting scheme, a rank-adaptive splitting approach for obtaining a low-rank solution of the EFK equation. Numerical examples show that our methods are robust and accurate. They can also preserve energy dissipation and the discrete maximum principle.

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arXiv2021

A Hessenberg-type Algorithm for Computing PageRank Problems

PageRank is a widespread model for analysing the relative relevance of nodes within large graphs arising in several applications. In the current paper, we present a cost-effective Hessenberg-type method built upon the Hessenberg process for the solution of difficult PageRank problems. The new method is very competitive with other popular algorithms in this field, such as Arnoldi-type methods, especially when the damping factor is close to $1$ and the dimension of the search subspace is large. The convergence and the complexity of the proposed algorithm are investigated. Numerical experiments are reported to show the efficiency of the new solver for practical PageRank computations.

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arXiv2021

A preconditioning technique for an all-at-once system from Volterra subdiffusion equations with graded time steps

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 $[T_0, T]$, where $T_0$ ($0 < T_0 < T$) is reasonably small. Then, the graded $L1$ scheme is applied in $[0, T_0]$, while the uniform one is used in $[T_0, T]$. Our all-at-once system is derived based on this strategy. In order to solve the arising system efficiently, we split it into two subproblems and design two preconditioners. Some properties of these two preconditioners are also investigated. Moreover, we extend our method to solve semilinear subdiffusion problems. Numerical results are reported to show the efficiency of our method.

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arXiv2021

The Weighted Arithmetic Mean-Geometric Mean Inequality is Equivalent to the Hölder Inequality

In the current note, we investigate the mathematical relations among the weighted arithmetic mean-geometric mean (AM-GM) inequality, the Hölder 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 Hölder inequality are fully achieved. The new results are more generalized than those of previous studies.

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arXiv2021

A note on parallel preconditioning for the all-at-once solution of Riesz fractional diffusion equations

The $p$-step backwards difference formula (BDF) for solving the system of ODEs can result in a kind of all-at-once linear systems, which are solved via the parallel-in-time preconditioned Krylov subspace solvers (see McDonald, Pestana, and Wathen [SIAM J. Sci. Comput., 40(2) (2018): A1012-A1033] and Lin and Ng [arXiv:2002.01108, 17 pages]. However, these studies ignored that the $p$-step BDF ($p\geq 2$) is not selfstarting, when they are exploited to solve time-dependent PDEs. In this note, we focus on the 2-step BDF which is often superior to the trapezoidal rule for solving the Riesz fractional diffusion equations, but its resultant all-at-once discretized system is a block triangular Toeplitz system with a low-rank perturbation. Meanwhile, we first give an estimation of the condition number of the all-at-once systems and then adapt the previous work to construct two block circulant (BC) preconditioners. Both the invertibility of these two BC preconditioners and the eigenvalue distributions of preconditioned matrices are discussed in details. The efficient implementation of these BC preconditioners is also presented especially for handling the computation of dense structured Jacobi matrices. Finally, numerical experiments involving both the one- and two-dimensional Riesz fractional diffusion equations are reported to support our theoretical findings.

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arXiv2020

An efficient second-order energy stable BDF scheme for the space fractional Cahn-Hilliard equation

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 backward differentiation formula in time and a finite difference method in space. Energy stability and convergence of the scheme are analyzed, and the optimal convergence orders in time and space are illustrated numerically. Note that the coefficient matrix of the scheme is a $2 \times 2$ block matrix with a Toeplitz-like structure in each block. Combining the advantages of this special structure with a Krylov subspace method, a preconditioning technique is designed to solve the system efficiently. Numerical examples are reported to illustrate the performance of the preconditioned iteration.

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arXiv2020

A low-rank Lie-Trotter splitting approach for nonlinear fractional complex Ginzburg-Landau equations

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 using a fractional centered difference method. Then, the resulting matrix differential equation is split into a stiff linear part and a nonstiff (nonlinear) one. For solving these two subproblems, a dynamical low-rank approach is used. The convergence of our method is proved rigorously. Numerical examples are reported which show that the proposed method is robust and accurate.

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