arXiv subjects

Xiangcheng Zheng

Publications and source records attributed to Xiangcheng Zheng.

At least 37 records · Page 2Linked to original sources

arXiv2025

Numerical approximations for a hyperbolic integrodifferential equation with a non-positive variable-sign kernel and nonlinear-nonlocal damping

This work considers the Galerkin approximation and analysis for a hyperbolic integrodifferential equation, where the non-positive variable-sign kernel and nonlinear-nonlocal damping with both the weak and viscous damping effects are involved. We derive the long-time stability of the solution and its finite-time uniqueness. For the semi-discrete-in-space Galerkin scheme, we derive the long-time stability of the semi-discrete numerical solution and its finite-time error estimate by technical splitting of intricate terms. Then we further apply the centering difference method and the interpolating quadrature to construct a fully discrete Galerkin scheme and prove the long-time stability of the numerical solution and its finite-time error estimate by designing a new semi-norm. Numerical experiments are performed to verify the theoretical findings.

math.NA↗

arXiv2025

Improved high-index saddle dynamics for finding saddle points and solution landscape

We present an improved high-index saddle dynamics (iHiSD) for finding saddle points and constructing solution landscapes, which is a crossover dynamics from gradient flow to traditional HiSD such that the Morse theory for gradient flow could be involved. We propose analysis for the reflection manifold in iHiSD, and then prove its stable and nonlocal convergence from outside of the region of attraction to the saddle point, which resolves the dependence of the convergence of HiSD on the initial value. We then present and analyze a discretized iHiSD that inherits these convergence properties. Furthermore, based on the Morse theory, we prove that any two saddle points could be connected by a sequence of trajectories of iHiSD. Theoretically, this implies that a solution landscape with a finite number of stationary points could be completely constructed by means of iHiSD, which partly answers the completeness issue of the solution landscape for the first time and indicates the necessity of integrating the gradient flow in HiSD. Different methods are compared by numerical experiments to substantiate the effectiveness of the iHiSD method.

math.NA↗

arXiv2025

Inverse source problem of sub-diffusion of variable exponent

This work investigates both direct and inverse problems of the variable-exponent sub-diffusion model, which attracts increasing attentions in both practical applications and theoretical aspects. Based on the perturbation method, which transfers the original model to an equivalent but more tractable form, the analytical extensibility of the solutions and the weak unique continuation principle are proved, which results in the uniqueness of the inverse space-dependent source problem from local internal observation. Then, based on the variational identity connecting the inversion input data with the unknown source function, we propose a weak norm and prove the conditional stability for the inverse problem in this norm. The iterative thresholding algorithm and Nesterov iteration scheme are employed to numerically reconstruct the smooth and non-smooth sources, respectively. Numerical experiments are performed to investigate their effectiveness.

math.NA↗

arXiv2024

A multiscale Abel kernel and application in viscoelastic problem

We consider the variable-exponent Abel kernel and demonstrate its multiscale nature in modeling crossover dynamics from the initial quasi-exponential behavior to long-term power-law behavior. Then we apply this to an integro-differential equation modeling, e.g. mechanical vibration of viscoelastic materials with changing material properties. We apply the Crank-Nicolson method and the linear interpolation quadrature to design a temporal second-order scheme, and develop a framework of exponentially weighted energy argument in error estimate to account for the non-positivity and non-monotonicity of the multiscale kernel. Numerical experiments are carried out to substantiate the theoretical findings and the crossover dynamics of the model.

math.NA↗

arXiv2024

Convergence analysis of time-splitting projection method for nonlinear quasiperiodic Schrödinger equation

This work proposes and analyzes an efficient numerical method for solving the nonlinear Schrödinger equation with quasiperiodic potential, where the projection method is applied in space to account for the quasiperiodic structure and the Strang splitting method is used in time.While the transfer between spaces of low-dimensional quasiperiodic and high-dimensional periodic functions and its coupling with the nonlinearity of the operator splitting scheme make the analysis more challenging. Meanwhile, compared to conventional numerical analysis of periodic Schrödinger systems, many of the tools and theories are not applicable to the quasiperiodic case. We address these issues to prove the spectral accuracy in space and the second-order accuracy in time. Numerical experiments are performed to substantiate the theoretical findings.

math.NA↗

arXiv2024

Weighted Sonine conditions and application

The Sonine kernel described by the classical Sonine condition of convolution form is an important class of kernels used in integral equations and nonlocal differential equations. This work extends this idea to introduce weighted Sonine conditions where the non-convolutional weight functions accommodate the inhomogeneity in practical applications. We characterize tight relations between classical Sonine condition and its weighted versions, which indicates that the non-degenerate weight functions may not introduce significant changes on the set of Sonine kernels. To demonstrate the application of weighted Sonine conditions, we employ them to derive equivalent but more feasible formulations of weighted integral equations and nonlocal differential equations to prove their well-posedness, and discuss possible application to corresponding partial differential equation models.

math.CA↗

arXiv2024

Two methods addressing variable-exponent fractional initial and boundary value problems and Abel integral equation

Variable-exponent fractional models attract increasing attentions in various applications, while the rigorous analysis is far from well developed. This work provides general tools to address these models. Specifically, we first develop a convolution method to study the well-posedness, regularity, an inverse problem and numerical approximation for the sundiffusion of variable exponent. For models such as the variable-exponent two-sided space-fractional boundary value problem (including the variable-exponent fractional Laplacian equation as a special case) and the distributed variable-exponent model, for which the convolution method does not apply, we develop a perturbation method to prove their well-posedness. The relation between the convolution method and the perturbation method is discussed, and we further apply the latter to prove the well-posedness of the variable-exponent Abel integral equation and discuss the constraint on the data under different initial values of variable exponent.

math.NA↗

arXiv2024

A generalized fractional Laplacian

In this article we show that the fractional Laplacian in $R^{2}$ can be factored into a product of the divergence operator, a Riesz potential operator, and the gradient operator. Using this factored form we introduce a generalization of the fractional Laplacian, involving a matrix $K(x)$, suitable when the fractional Laplacian is applied in a non homogeneous medium. For the case of $K(x)$ a constant, symmetric positive definite matrix we show that the fractional Poisson equation is well posed, and determine the regularity of the solution in terms of the regularity of the right hand side function.

math.AP↗

arXiv2024

Addressing complex boundary conditions of miscible flow and transport in two and three dimensions with application to optimal control

We investigate complex boundary conditions of the miscible displacement system in two and three space dimensions with the commonly-used Bear-Scheidegger diffusion-dispersion tensor, which describes, e.g., the porous medium flow processes in petroleum reservoir simulation or groundwater contaminant transport. Specifically, we incorporate the no-flux boundary condition for the Darcy velocity to prove that the general no-flux boundary condition for the transport equation is equivalent to the normal derivative boundary condition of the concentration, based on which we further prove several complex boundary conditions by the Bear-Scheidegger tensor and its derivative. The derived boundary conditions not only provide new insights and distinct properties of the Bear-Scheidegger diffusion-dispersion tensor, but accommodate the coupling and the nonlinearity of the miscible displacement system and the Bear-Scheidegger tensor in deriving the first-order optimality condition of the corresponding optimal control problem for practical application.

math.OC↗

arXiv2024

An equivalent formulation of Sonine condition

Sonine kernel is characterized by the Sonine condition (denoted by SC) and is an important class of kernels in nonlocal differential equations and integral equations. This work proposes a SC with a more general form (denoted by gSC), which is more convenient than SC to accommodate complex kernels and equations. A typical kernel is given, and the first-kind Volterra integral equation under gSC is accordingly transformed and then analyzed. Based on these results, it is finally proved that the gSC is indeed equivalent to the original SC, which indicates that the Sonine kernel may be essentially characterized by the behavior of its convolution with the associated kernel at the starting point.

math.CA↗

arXiv2024

Understanding high-index saddle dynamics via numerical analysis

High-index saddle dynamics (HiSD) serves as a competitive instrument in searching the any-index saddle points and constructing the solution landscape of complex systems. The Lagrangian multiplier terms in HiSD ensure the Stiefel manifold constraint, which, however, are dropped in the commonly-used discrete HiSD scheme and are replaced by an additional Gram-Schmidt orthonormalization. Though this scheme has been successfully applied in various fields, it is still unclear why the above modification does not affect its effectiveness. We recover the same form as HiSD from this scheme, which not only leads to error estimates naturally, but indicates that the mechanism of Stiefel manifold preservation by Lagrangian multiplier terms in HiSD is nearly a Gram-Schmidt process (such that the above modification is appropriate). The developed methods are further extended to analyze the more complicated constrained HiSD on high-dimensional sphere, which reveals more mechanisms of the constrained HiSD in preserving several manifold properties.

math.NA↗

arXiv2024

Local modification of subdiffusion by initial Fickian diffusion: Multiscale modeling, analysis and computation

We propose a local modification of the standard subdiffusion model by introducing the initial Fickian diffusion, which results in a multiscale diffusion model. The developed model resolves the incompatibility between the nonlocal operators in subdiffusion and the local initial conditions and thus eliminates the initial singularity of the solutions of the subdiffusion, while retaining its heavy tail behavior away from the initial time. The well-posedness of the model and high-order regularity estimates of its solutions are analyzed by resolvent estimates, based on which the numerical discretization and analysis are performed. Numerical experiments are carried out to substantiate the theoretical findings.

math.NA↗

arXiv2023

Accelerated high-index saddle dynamics method for searching high-index saddle points

The high-index saddle dynamics (HiSD) method [J. Yin, L. Zhang, and P. Zhang, {\it SIAM J. Sci. Comput., }41 (2019), pp.A3576-A3595] serves as an efficient tool for computing index-$k$ saddle points and constructing solution landscapes. Nevertheless, the conventional HiSD method often encounters slow convergence rates on ill-conditioned problems. To address this challenge, we propose an accelerated high-index saddle dynamics (A-HiSD) by incorporating the heavy ball method. We prove the linear stability theory of the continuous A-HiSD, and subsequently estimate the local convergence rate for the discrete A-HiSD. Our analysis demonstrates that the A-HiSD method exhibits a faster convergence rate compared to the conventional HiSD method, especially when dealing with ill-conditioned problems. We also perform various numerical experiments including the loss function of neural network to substantiate the effectiveness and acceleration of the A-HiSD method.

math.NA↗

arXiv2023

Semi-implicit method of high-index saddle dynamics and application to construct solution landscape

We analyze the semi-implicit scheme of high-index saddle dynamics, which provides a powerful numerical method for finding the any-index saddle points and constructing the solution landscape. Compared with the explicit schemes of saddle dynamics, the semi-implicit discretization relaxes the step size and accelerates the convergence, but the corresponding numerical analysis encounters new difficulties compared to the explicit scheme. Specifically, the orthonormal property of the eigenvectors at each time step could not be fully employed due to the semi-implicit treatment, and computations of the eigenvectors are coupled with the orthonormalization procedure, which further complicates the numerical analysis. We address these issues to prove error estimates of the semi-implicit scheme via, e.g. technical splittings and multi-variable circulating induction procedure. We further analyze the convergence rate of the generalized minimum residual solver for solving the semi-implicit system. Extensive numerical experiments are carried out to substantiate the efficiency and accuracy of the semi-implicit scheme in constructing solution landscapes of complex systems.

math.NA↗

arXiv2023

Error estimate for semi-implicit method of sphere-constrained high-index saddle dynamics

We prove error estimates for the semi-implicit numerical scheme of sphere-constrained high-index saddle dynamics, which serves as a powerful instrument in finding saddle points and constructing the solution landscapes of constrained systems on the high-dimensional sphere. Due to the semi-implicit treatment and the novel computational procedure, the orthonormality of numerical solutions at each time step could not be fully employed to simplify the derivations, and the computations of the state variable and directional vectors are coupled with the retraction, the vector transport and the orthonormalization procedure, which significantly complicates the analysis. We address these issues to prove error estimates for the proposed semi-implicit scheme and then carry out numerical experiments to substantiate the theoretical findings.

math.NA↗

arXiv2022

A model-free shrinking-dimer saddle dynamics for finding saddle point and solution landscape

We propose a model-free shrinking-dimer saddle dynamics for finding any-index saddle points and constructing the solution landscapes, in which the force in the standard saddle dynamics is replaced by a surrogate model trained by the Gassian process learning. By this means, the exact form of the model is no longer necessary such that the saddle dynamics could be implemented based only on some observations of the force. This data-driven approach not only avoids the modeling procedure that could be difficult or inaccurate, but also significantly reduces the number of queries of the force that may be expensive or time-consuming. We accordingly develop a sequential learning saddle dynamics algorithm to perform a sequence of local saddle dynamics, in which the queries of the training samples and the update or retraining of the surrogate force are performed online and around the latent trajectory in order to improve the accuracy of the surrogate model and the value of each sampling. Numerical experiments are performed to demonstrate the effectiveness and efficiency of the proposed algorithm.

math.NA↗

arXiv2022

Convergence analysis of discrete high-index saddle dynamics

Saddle dynamics is a time continuous dynamics to efficiently compute the any-index saddle points and construct the solution landscape. In practice, the saddle dynamics needs to be discretized for numerical computations, while the corresponding numerical analysis are rarely studied in the literature, especially for the high-index cases. In this paper we propose the convergence analysis of discrete high-index saddle dynamics. To be specific, we prove the local linear convergence rates of numerical schemes of high-index saddle dynamics, which indicates that the local curvature in the neighborhood of the saddle point and the accuracy of computing the eigenfunctions are main factors that affect the convergence of discrete saddle dynamics. The proved results serve as compensations for the convergence analysis of high-index saddle dynamics and are substantiated by numerical experiments.

math.NA↗

arXiv2022

Error estimates for Euler discretization of high-index saddle dynamics

High-index saddle dynamics provides an effective means to compute the any-index saddle points and construct the solution landscape. In this paper we prove error estimates for Euler discretization of high-index saddle dynamics with respect to the time step size, which remains untreated in the literature. We overcome the main difficulties that lie in the strong nonlinearity of the saddle dynamics and the orthonormalization procedure in the numerical scheme that is uncommon in standard discretization of differential equations. The derived methods are further extended to study the generalized high-index saddle dynamics for non-gradient systems and provide theoretical support for the accuracy of numerical implementations.

math.NA↗