arXiv subjects

Xiangcheng Zheng

Publications and source records attributed to Xiangcheng Zheng.

At least 19 recordsLinked to original sources

arXiv2026

Asymptotic stability of high-index saddle dynamics

High-index saddle dynamics (HiSD) has been extensively adopted to compute saddle points of arbitrary index, with various applications such as constructing solution landscapes of complex systems. In this work, we prove the asymptotic stability of various HiSD methods, including the idealized, perturbed, and coupled HiSD methods, as well as their momentum-accelerated variants. The analysis is Jacobian-free and thus compensates for the linear stability analysis in the literature. The results not only provide specific characterizations for the convergence rate and the basin of attraction, but also reveal intrinsic correlations among dynamics in the coupled HiSD. Theoretical results are substantiated by numerical experiments.

math.DS↗

arXiv2026

Analysis and approximation for inverse potential problem of space-dependent variable-exponent sub-diffusion in heterogeneous medium

Anomalous transport in complex heterogeneous medium is often characterized by non-Markovian memory effects and spatially varying trapping properties. From the point of view of continuous-time random walks, such heterogeneity can be described by a position-dependent heavy-tailed waiting-time density $q(t \mid x)\sim C(x)t^{-1-α(x)}$, where $0<α(x)<1$ is the local anomalous exponent. This microscopic interpretation naturally leads to the sub-diffusion with a space-dependent variable exponent $α(x)$. Compared with constant-exponent models, the variable-exponent fractional operators generally do not commute with the spatial Laplacian, which makes both the forward and inverse analysis substantially more delicate. This paper investigates the inverse problem of recovering a spatially dependent radiative or potential coefficient in variable-exponent sub-diffusion from terminal observation data. We formulate the inverse problem as a nonlinear fixed-point equation and prove that the associated fixed-point operator is monotone such that the fixed-point iteration converges decreasingly to the unique solution. We further establish a Lipschitz-type conditional stability estimate for the terminal observation map, which also implies uniqueness of the inverse problem for sufficiently large terminal time. For numerical reconstruction, we develop a fully discrete finite element scheme and derive an error estimate for the inverse problem. Numerical simulations are conducted to validate the effectiveness of the method, even in the presence of noisy data.

math.NA↗

arXiv2026

Non-convex optimal control governed by nonlinear elliptic partial differential equation: Analysis, functional-based algorithm and control landscape

Non-convex optimal control arises from various applications but may contain multiple stationary points. Classical solvers usually perform a local search and therefore rely on good initial guesses to reach appropriate local optimal controls. In this work we introduce a novel solution strategy for the non-convex optimal control of an elliptic equation, the main idea of which is to construct the control landscape based on the functional high-index saddle dynamics (FHiSD) method. This method reduces the dependence on prescribed initial guesses by using the information of high-index saddle points, and depicts the macroscopic configuration of the space of the control variable. Then various minima could be systematically computed along transition pathways and the control strategy can then be selected among them. We prove the existence of the optimal control, and then propose and analyze the FHiSD in locating saddle points. Subsequently, the applicability of FHiSD in non-convex optimal control is rigorously justified. Numerical results not only indicate the effectiveness of the proposed method, but reveal unintuitive phenomena (e.g. the non-monotonicity of the values of the cost functional with respect to the Morse indices) that support the necessity of computing multiple solutions of high indices.

math.OC↗

arXiv2026

High-order smoothness and high-order collocation approximation for Volterra integral equation with multiscale and nonlinear exponent kernel

We consider a Volterra integral equation with multiscale and nonlinear exponent kernel. We propose the high-order smoothing conditions, under which the initial singularity of the solutions can be eliminated up to any prescribed order. This indicates the application of the multiscale nature of the kernel on local modification of the solutions. Then a discontinuous high-order collocation method with arbitrary polynomial degree is developed and analyzed on uniform or graded meshes based on the solution regularity. This work serves as a comprehensive extension and complement to [Zheng, Qiu and Stynes, SIAM J. Numer. Anal., to appear] in both mathematical and numerical aspects.

math.NA↗

arXiv2026

Analysis and Approximation of Stochastic Multiscale Subdiffusion Driven by Fractional Gaussian Noise

This paper investigates a stochastic multiscale subdiffusion model driven by fractional Gaussian noise, where the multiscale Abel kernel with variable exponent $α(t)\in(0,1)$ is used to capture multiscale and crossover behavior in anomalous diffusion. The main difficulties of this model lie in the complexity of the multiscale Abel kernel (e.g. non-monotonicity and non-coercivity) and the low regularity caused by the noise. Concerning these issues, we prove the well-posedness and regularity of the mild solutions by means of solution operator approach and a perturbation technique for multiscale Abel kernel. Then both the semidiscrete-in-time and fully-discrete numerical schemes are proposed and analyzed under the low-regularity numerical analysis framework, with proved temporal and spatial convergence rates. Numerical experiments are presented to substantiate the theoretical results.

math.NA↗

arXiv2026

Numerical analysis of spatiotemporal high-index saddle dynamics for finding multiple solutions of semilinear elliptic problems

This paper presents a rigorous numerical framework for computing multiple solutions of semilinear elliptic problems by spatiotemporal high-index saddle dynamics (HiSD), which extends the traditional HiSD to the continuous-in-space setting, explicitly incorporating spatial differential operators. To enforce the Stiefel manifold constraint without introducing the analytical complications of retraction-based updates, we design a fully discrete retraction-free orthonormality-preserving scheme for spatiotemporal HiSD. This scheme exhibits favorable structural properties that substantially reduce the difficulties arising from coupling and gradient nonlinearities in spatiotemporal HiSD. Exploiting these properties, we establish gradient stability and error estimates, which consequently ensure the preservation of the Morse index for the computed saddle points. The framework is further extended to the semilinear advection-reaction-diffusion equation. Numerical experiments demonstrate the efficiency of the proposed method in finding multiple solutions and constructing the solution landscape of semilinear elliptic problems. To the best of our knowledge, this work presents the first rigorous full space--time accuracy analysis of the HiSD system. It reveals intrinsic connections between saddle-search algorithms and numerical methods for PDEs, enhancing their mutual compatibility for a broad range of problems.

math.NA↗

arXiv2026

A Derivative-Free Saddle-search Algorithm With Linear Convergence Rate

We propose a derivative-free saddle-search algorithm designed to locate transition states using only function evaluations. The algorithm employs a nested architecture consisting of an inner eigenvector search and an outer saddle-point search. Through rigorous numerical analysis, we prove the almost sure convergence of the inner step under suitable assumptions. Furthermore, we establish the convergence of the outer search using a decaying step size, while demonstrating linear convergence under constant step size and boundedness conditions. Numerical experiments are provided to validate our theoretical results and demonstrate the algorithm's practical applicability.

math.NA↗

arXiv2025

Determining superconvergence points for $L2-1_σ$ scheme of variable-exponent subdiffusion and error estimate

We develop a numerical scheme for subdiffusion of variable exponent by combining the $L2-1_σ$ temporal discretization with finite element spatial approximation. In existing works, determining the superconvergence points requires solving a nonlinear equation related to the variable exponent at each time step. This work relaxes the selection criterion of superconvergence points without affecting the numerical accuracy, which may reduce the cost of determining superconvergence points. To handle the initial singularity of the solution, we employ a graded temporal mesh. Then we prove the stability and error estimates with a convergence rate $O\left(N^{-\min\{rδ,2\}}+h^μ\right)$ for the $L2-1_σ$ scheme of variable-exponent subdiffusion. Numerical results are performed to substantiate the theoretical findings.

math.NA↗

arXiv2025

Numerical analysis for saddle dynamics of some semilinear elliptic problems

This work presents a numerical analysis of computing transition states of semilinear elliptic partial differential equations (PDEs) via the index-1 saddle dynamics, or equivalently, the gentlest ascent dynamics. To establish clear connections between saddle dynamics and numerical methods of PDEs, as well as improving their compatibility, we first propose the continuous-in-space formulation of saddle dynamics for semilinear elliptic problems. This formulation yields a parabolic system that converges to saddle points. We then analyze the well-posedness, $H^1$ stability and error estimates of semi- and fully-discrete finite element schemes. Significant efforts are devoted to addressing the coupling, gradient nonlinearity, nonlocality of the proposed parabolic system, and the impacts of retraction due to the norm constraint. The error estimate results demonstrate the accuracy and index-preservation of the discrete schemes.

math.NA↗

arXiv2025

Nullspace-preserving high-index saddle dynamics method for degenerate multiple solution problems

We propose the nullspace-preserving high-index saddle dynamics (NPHiSD) method for degenerating multiple solution systems in constrained and unconstrained settings. The NPHiSD efficiently locates high-index saddle points and provides parent states for downward searches of lower-index saddles, thereby constructing the solution landscape systematically. The NPHiSD method searches along multiple efficient ascent directions by excluding the nullspace, which is the key for upward searches in degenerate problems. To reduce the cost of frequent nullspace updates, the search is divided into segments, within which the ascent directions remain orthogonal to the nullspace of the initial state of each segment. A sufficient and necessary condition for characterizing the segment that admits efficient ascent directions is proved. Extensive numerical experiments for typical problems such as Lifshitz-Petrich, Gross-Pitaevskii, and Lennard-Jones models are performed to show the universality and effectiveness of the NPHiSD method.

math.NA↗

arXiv2025

Generalizing subdiffusive Black-Scholes model by variable exponent: Model transformation and numerical approximation

This work generalizes the subdiffusive Black-Scholes model by introducing the variable exponent in order to provide adequate descriptions for the option pricing, where the variable exponent may account for the variation of the memory property. In addition to standard nonlinear-to-linear transformation, we apply a further spatial-temporal transformation to convert the model to a more tractable form in order to circumvent the difficulties caused by the ``non-positive, non-monotonic'' variable-exponent memory kernel. An interesting phenomenon is that the spatial transformation not only eliminates the advection term but naturally turns the original noncoercive spatial operator into a coercive one due to the specific structure of the Black-Scholes model, which thus avoids imposing constraints on coefficients. Then we perform numerical analysis for both the semi-discrete and fully discrete schemes to support numerical simulation. Numerical experiments are carried out to substantiate the theoretical results.

math.NA↗

arXiv2025

Reconstruction of a potential parameter in subdiffusion via a Kohn--Vogelius type functional: Theory and computation

This work considers the reconstruction of a space-dependent potential from boundary observations in subdiffusion by a stable and robust recovery method. Specifically, we develop an algorithm to minimize the Kohn-Vogelius cost function, which measures the difference between the solutions of two excitations. The inverse potential problem is recast into an optimization problem, where the objective is to minimize a Kohn-Vogelius-type functional within a set of admissible potentials. We establish the well-posedness of this optimization problem by proving the existence and uniqueness of a minimizer and demonstrating its stability with respect to perturbations in the boundary data. Furthermore, we analyze the Fréchet differentiability of the Kohn-Vogelius functional and prove the Lipschitz continuity of its gradient. These theoretical results enable the development of a convergent conjugate gradient algorithm for numerical reconstruction. The effectiveness and robustness of the proposed method are confirmed through several numerical examples in both one and two dimensions, including cases with noisy data.

math.NA↗

arXiv2025

Numerical analysis of high-order methods for variable-exponent fractional diffusion-wave equation

This work considers the variable-exponent fractional diffusion-wave equation, which describes, e.g. the propagation of mechanical diffusive waves in viscoelastic media with varying material properties. Rigorous numerical analysis for this model is not available in the literature, partly because the variable-exponent Abel kernel in the leading term may not be positive definite or monotonic. We adopt the idea of model reformulation to obtain a more tractable form, which, however, still involves an ``indefinite-sign, nonpositive-definite, nonmonotonic'' convolution kernel that introduces difficulties in numerical analysis. We address this issue to design two high-order schemes and derive their stability and error estimate based on the proved solution regularity, with $α(0)$-order and second-order accuracy in time, respectively. Numerical experiments are presented to substantiate the theoretical findings and to show the transition behavior of the mechanical (diffusive) wave modeled by variable exponent.

math.NA↗

arXiv2025

Multiscale solution decomposition of nonlocal-in-time problems with application in numerical computation

This work develops a multiscale solution decomposition (MSD) method for nonlocal-in-time problems to separate a series of known terms with multiscale singularity from the original singular solution such that the remaining unknown part becomes smoother. We demonstrate that the MSD provides a scenario where the smoothness assumption for solutions of weakly singular nonlocal-in-time problems, a commonly encountered assumption in numerous literature of numerical methods that is in general not true for original solutions, becomes appropriate such that abundant numerical analysis results therein become applicable. From computational aspect, instead of handling solution singularity, the MSD significantly reduces the numerical difficulties by separating and thus circumventing the solution singularity. We consider typical problems, including the fractional relaxation equation, Volterra integral equation, subdiffusion, integrodifferential equation and diffusion-wave equation, to demonstrate the universality of MSD and its effectiveness in improving the numerical accuracy or stability in comparison with classical methods.

math.NA↗

arXiv2025

Long-time error estimate and decay of finite element method to a generalized viscoelastic flow

This work analyzes the finite element approximation to a viscoelastic flow model, which generalizes the Navier-Stokes equation and Oldroyd's model by introducing the tempered power-law memory kernel. We prove regularity and long-time exponential decay of the solutions, as well as a long-time convolution-type Grönwall inequality to support numerical analysis. A Volterra-Stokes projection is developed and analyzed to facilitate the parabolic-type duality argument, leading to the long-time error estimates and exponential decay of velocity and pressure. A benchmark problem of planar four-to-one contraction flow is simulated to substantiate the generality of the proposed model in comparison with the Navier-Stokes equation and Oldroyd's model.

math.NA↗

arXiv2025

Numerical analysis for subdiffusion problem with non-positive memory

This work considers the subdiffusion problem with non-positive memory, which not only arises from physical laws with memory, but could be transformed from sophisticated models such as subdiffusion or subdiffusive Fokker-Planck equation with variable exponent. We apply the non-uniform L1 formula and interpolation quadrature to discretize the fractional derivative and the memory term, respectively, and then adopt the complementary discrete convolution kernel approach to prove the stability and first-order temporal accuracy of the scheme. The main difficulty in numerical analysis lies in the non-positivity of the kernel and its coupling with the complementary discrete convolution kernel (such that different model exponents are also coupled), and the results extend those in [Chen, Thomée and Wahlbin, Math. Comp. 1992] to the subdiffusive case. Numerical experiments are performed to substantiate the theoretical results.

math.NA↗

arXiv2025

Finite difference method for nonlinear damped viscoelastic Euler-Bernoulli beam model

We propose and analyze the numerical approximation for a viscoelastic Euler-Bernoulli beam model containing a nonlinear strong damping coefficient. The finite difference method is used for spatial discretization, while the backward Euler method and the averaged PI rule are applied for temporal discretization. The long-time stability and the finite-time error estimate of the numerical solutions are derived for both the semi-discrete-in-space scheme and the fully-discrete scheme. Furthermore, the Leray-Schauder theorem is used to derive the existence and uniqueness of the fully-discrete numerical solutions. Finally, the numerical results verify the theoretical analysis.

math.NA↗

arXiv2025

Analysis of evolution equation with variable-exponent memory modeling multiscale viscoelasticity

We investigate the well-posedness and solution regularity of an evolution equation with non-positive type variable-exponent memory, which describes multiscale viscoelasticity in materials with memory. The perturbation method is applied for model transformation, based on which the well-posedness is proved. Then the weighted solution regularity is derived, where the initial singularity is characterized by the initial value of variable exponent.

math.AP↗