arXiv subjects

Zhongqin Xue

Publications and source records attributed to Zhongqin Xue.

5 recordsLinked to original sources

arXiv2026

Error Estimates for the Arnoldi Approximation of a Matrix Square Root

The Arnoldi process provides an efficient framework for approximating functions of a matrix applied to a vector, i.e., of the form $f(M)\bm{b}$, by repeated matrix-vector multiplications. In this paper, we derive error estimates for approximating the action of a matrix square root using the Arnoldi process, where the integral representation of the error is reformulated in terms of the error for solving the linear system $M\bm{x}=\bm{b}$. The results extend the error analysis of the Lanczos method for Hermitian matrices in [Chen et al., SIAM J. Matrix Anal. Appl., 2022] to non-Hermitian cases and provide an improved bound for the Hermitian case. Furthermore, in practical settings, the matrix may only be available via approximate or structured representations. Motivated by this, we extend the analysis and establish a generalized error bound for perturbed matrices. The numerical results on matrices with different structures demonstrate that our theoretical analysis yields a reliable upper bound. Finally, simulations on large-scale matrices arising in particulate suspensions, represented in hierarchical matrix form, validate the effectiveness and practicality of the approach.

math.NA↗

arXiv2026

An adaptive framework for first-order gradient methods

Gradient methods are widely used in optimization problems. In practice, while the smoothness parameter can be estimated utilizing techniques such as backtracking, estimating the strong convexity parameter remains a challenge; moreover, even with the optimal parameter choice, convergence can be slow. In this work, we propose a framework for dynamically adapting the step size and momentum parameters in first-order gradient methods for the optimization problem, without prior knowledge of the strong convexity parameter. The main idea is to use the geometric average of the ratios of successive residual norms as an empirical estimate of the upper bound on the convergence rate, which in turn allows us to adaptively update the algorithm parameters. The resulting algorithms are simple to implement, yet efficient in practice, requiring only a few additional computations on existing information. The proposed adaptive gradient methods are shown to converge at least as fast as gradient descent for quadratic optimization problems. Numerical experiments on both quadratic and nonlinear problems validate the effectiveness of the proposed adaptive algorithms. The results show that the adaptive algorithms are comparable to their counterparts using optimal parameters, and in some cases, they capture local information and exhibit improved performance.

math.OC↗

arXiv2024

Improving Greedy Algorithms for Rational Approximation

When developing robust preconditioners for multiphysics problems, fractional functions of the Laplace operator often arise and need to be inverted. Rational approximation in the uniform norm can be used to convert inverting those fractional operators into inverting a series of shifted Laplace operators. Care must be taken in the approximation so that the shifted Laplace operators remain symmetric positive definite, making them better conditioned. In this work, we study two greedy algorithms for finding rational approximations to such fractional operators. The first algorithm improves the orthogonal greedy algorithm discussed in [Li et al., SISC, 2024] by adding one minimization step in the uniform norm to the procedure. The second approach employs the weak Chebyshev greedy algorithm in the uniform norm. Both methods yield non-increasing error. Numerical results confirm the effectiveness of our proposed algorithms, which are also flexible and applicable to other approximation problems. Moreover, with effective rational approximations to the fractional operator, the resulting algorithms show good performance in preconditioning a Darcy-Stokes coupled problem.

math.NA↗

arXiv2023

Convergence analysis of variable steps BDF2 method for the space fractional Cahn-Hilliard model

An implicit variable-step BDF2 scheme is established for solving the space fractional Cahn-Hilliard equation, involving the fractional Laplacian, derived from a gradient flow in the negative order Sobolev space $H^{-α}$, $α\in(0,1)$. The Fourier pseudo-spectral method is applied for the spatial approximation. The proposed scheme inherits the energy dissipation law in the form of the modified discrete energy under the sufficient restriction of the time-step ratios. The convergence of the fully discrete scheme is rigorously provided utilizing the newly proved discrete embedding type convolution inequality dealing with the fractional Laplacian. Besides, the mass conservation and the unique solvability are also theoretically guaranteed. Numerical experiments are carried out to show the accuracy and the energy dissipation both for various interface widths. In particular, the multiple-time-scale evolution of the solution is captured by an adaptive time-stepping strategy in the short-to-long time simulation.

math.NA↗

arXiv2023

Energy stable and $L^2$ norm convergent BDF3 scheme for the Swift-Hohenberg equation

A fully discrete implicit scheme is proposed for the Swift-Hohenberg model, combining the third-order backward differentiation formula (BDF3) for the time discretization and the second-order finite difference scheme for the space discretization. Applying the Brouwer fixed-point theorem and the positive definiteness of the convolution coefficients of BDF3, the presented numerical algorithm is proved to be uniquely solvable and unconditionally energy stable, further, the numerical solution is shown to be bounded in the maximum norm. The proposed scheme is rigorously proved to be convergent in $L^2$ norm by the discrete orthogonal convolution (DOC) kernel, which transfer the four-level-solution form into the three-level-gradient form for the approximation of the temporal derivative. Consequently, the error estimate for the numerical solution is established by utilization of the discrete Gronwall inequality. Numerical examples in 2D and 3D cases are provided to support the theoretical results.

math.NA↗