arXiv subjects

Xiangcheng Zheng

Publications and source records attributed to Xiangcheng Zheng.

51 records · Page 3Linked to original sources

arXiv2022

Mathematical and numerical analysis to shrinking-dimer saddle dynamics with local Lipschitz conditions

We present a mathematical and numerical investigation to the shrinkingdimer saddle dynamics for finding any-index saddle points in the solution landscape. Due to the dimer approximation of Hessian in saddle dynamics, the local Lipschitz assumptions and the strong nonlinearity for the saddle dynamics, it remains challenges for delicate analysis, such as the the boundedness of the solutions and the dimer error. We address these issues to bound the solutions under proper relaxation parameters, based on which we prove the error estimates for numerical discretization to the shrinking-dimer saddle dynamics by matching the dimer length and the time step size. Furthermore, the Richardson extrapolation is employed to obtain a high-order approximation. The inherent reason of requiring the matching of the dimer length and the time step size lies in that the former serves a different mesh size from the later, and thus the proposed numerical method is close to a fully-discrete numerical scheme of some spacetime PDE model with the Hessian in the saddle dynamics and its dimer approximation serving as a "spatial operator" and its discretization, respectively, which in turn indicates the PDE nature of the saddle dynamics.

math.NA↗

arXiv2022

Discretization and index-robust error analysis for constrained high-index saddle dynamics on high-dimensional sphere

We develop and analyze numerical discretization to the constrained high-index saddle dynamics, the dynamics searching for the high-index saddle points confined on the high-dimensional unit sphere. Compared with the saddle dynamics without constraints, the constrained high-index saddle dynamics has more complex dynamical forms, and additional operations such as the retraction and vector transport are required due to the constraint, which significantly complicate the numerical scheme and the corresponding numerical analysis. Furthermore, as the existing numerical analysis results usually depend on the index of the saddle points implicitly, the proved numerical accuracy may be reduced if the index is high in many applications, which indicates the lack of robustness with respect to the index. To address these issues, we derive the error estimates for numerical discretization of the constrained high-index saddle dynamics on high-dimensional sphere, and then improve it by providing an index-robust error analysis in an averaged norm by adjusting the relaxation parameters. The developed results provide mathematical supports for the accuracy of numerical computations.

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arXiv2022

Analysis and Petrov-Galerkin numerical approximation for variable coefficient two-sided fractional diffusion, advection, reaction equations

In this paper we investigate the variable coefficient two-sided fractional diffusion, advection, reaction equations on a bounded interval. It is known that the fractional diffusion operator may lose coercivity due to the variable coefficient, which makes both the mathematical and numerical analysis challenging. To resolve this issue, we design appropriate test and trial functions to prove the inf-sup condition of the variable coefficient fractional diffusion, advection, reaction operators in suitable function spaces. Based on this property, we prove the well-posedness and regularity of the solutions, as well as analyze the Petrov-Galerkin approximation scheme for the proposed model. Numerical experiments are presented to substantiate the theoretical findings and to compare the behaviors of different models.

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arXiv2021

Approximate inversion for Abel integral operators of variable exponent and applications to fractional Cauchy problems

We investigate the variable-exponent Abel integral equations and corresponding fractional Cauchy problems. The main contributions of the work are enumerated as follows: (i) We develop an approximate inversion technique for variable-exponent Abel integral operators, based on which we analyze the corresponding integral and differential equations; (ii) We prove that the sensitive dependence of the well-posedness of classical Riemann-Liouville fractional differential equations on the initial value could be resolved by adjusting the initial value of the variable exponent; (iii) We prove that the singularity of the solutions to the Riemann-Liouville fractional differential equations could also be eliminated by adjusting the variable exponent and its derivatives at the initial time, which, together with (ii), demonstrates the advantages of introducing the variable exponent. The proposed approximate inversion technique provides a potential means to convert the intricate variable-exponent integral and differential problems to feasible forms, and the above findings suggest that the variable-exponent fractional problems may serve as a connection between integer-order and fractional models by adjusting the variable exponent at the initial time.

math.CA↗

arXiv2021

Numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method

We study a numerical approximation for a nonlinear variable-order fractional differential equation via an integral equation method. Due to the lack of the monotonicity of the discretization coefficients of the variable-order fractional derivative in standard approximation schemes, existing numerical analysis techniques do not apply directly. By an approximate inversion technique, the proposed model is transformed as a second kind Volterra integral equation, based on which a collocation method under uniform or graded mesh is developed and analyzed. In particular, the error estimates improve the existing results by proving a consistent and sharper mesh grading parameter and characterizing the convergence rates in terms of the initial value of the variable order, which demonstrates its critical role in determining the smoothness of the solutions and thus the numerical accuracy.

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arXiv2021

Analysis of viscoelastic flow with a generalized memory and its exponential convergence to steady state

We investigate a viscoelastic flow model with a generalized memory, in which a weak-singular component is introduced in the exponential convolution kernel of classical viscoelastic flow equations that remains untreated in the literature. We prove the well-posedness and regularity of the solutions, based on which we prove the exponential convergence of the solutions to the steady state. The proposed model serves as an extension of classical viscoelastic flow equations by adding a dimension characterized by the power of the weak-singular kernel, and the derived results provide theoretical supports for designing numerical methods for both the considered equation and its steady state.

math.AP↗

arXiv2021

Computing solution landscape of nonlinear space-fractional problems via fast approximation algorithm

The nonlinear space-fractional problems often allow multiple stationary solutions, which can be much more complicated than the corresponding integer-order problems. In this paper, we systematically compute the solution landscapes of nonlinear constant/variable-order space-fractional problems. A fast approximation algorithm is developed to deal with the variable-order spectral fractional Laplacian by approximating the variable-indexing Fourier modes, and then combined with saddle dynamics to construct the solution landscape of variable-order space-fractional phase field model. Numerical experiments are performed to substantiate the accuracy and efficiency of fast approximation algorithm and elucidate essential features of the stationary solutions of space-fractional phase field model. Furthermore, we demonstrate that the solution landscapes of spectral fractional Laplacian problems can be reconfigured by varying the diffusion coefficients in the corresponding integer-order problems.

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arXiv2020

Optimal Petrov-Galerkin spectral approximation method for the fractional diffusion, advection, reaction equation on a bounded interval

In this paper we investigate the numerical approximation of the fractional diffusion, advection, reaction equation on a bounded interval. Recently the explicit form of the solution to this equation was obtained. Using the explicit form of the boundary behavior of the solution and Jacobi polynomials, a Petrov-Galerkin approximation scheme is proposed and analyzed. Numerical experiments are presented which support the theoretical results, and demonstrate the accuracy and optimal convergence of the approximation method.

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arXiv2019

A fast method for variable-order space-fractional diffusion equations

We develop a fast divided-and-conquer indirect collocation method for the homogeneous Dirichlet boundary value problem of variable-order space-fractional diffusion equations. Due to the impact of the space-dependent variable order, the resulting stiffness matrix of the numerical approximation does not have a Toeplitz-like structure. In this paper we derive a fast approximation of the coefficient matrix by the means of a sum of Toeplitz matrices multiplied by diagonal matrices. We show that the approximation is asymptotically consistent with the original problem, which requires $O(kN\log^2 N)$ memory and $O(k N\log^3 N)$ computational complexity with $N$ and $k$ being the numbers of unknowns and the approximants, respectively. Numerical experiments are presented to demonstrate the effectiveness and the efficiency of the proposed method.

math.NA↗

arXiv2019

Uniqueness of determining the variable fractional order in variable-order time-fractional diffusion equations

We study an initial-boundary value problem of variable-order time-fractional diffusion equations in one space dimension. Based on the wellposedness of the proposed model and the smoothing properties of its solutions, which are shown to be determined by the behavior of the variable order at the initial time, a uniqueness result for an important inverse problem of determination of the variable order in the time-fractional derivative contained in the proposed model from observations of its solutions is obtained.

math.AP↗

arXiv2019

Optimal order finite element approximations for variable-order time-fractional diffusion equations

We study a fully discrete finite element method for variable-order time-fractional diffusion equations with a time-dependent variable order. Optimal convergence estimates are proved with the first-order accuracy in time (and second order accuracy in space) under the uniform or graded temporal mesh without full regularity assumptions of the solutions. Numerical experiments are presented to substantiate the analysis.

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arXiv2019

Numerical approximations for the variable coefficient fractional diffusion equations with non-smooth data

In this article we study the numerical approximation of a variable coefficient fractional diffusion equation. Using a change of variable, the variable coefficient fractional diffusion equation is transformed into a constant coefficient fractional diffusion equation of the same order. The transformed equation retains the desirable stability property of being an elliptic equation. A spectral approximation scheme is proposed and analyzed for the transformed equation, with error estimates for the approximated solution derived. An approximation to the unknown of the variable coefficient fractional diffusion equation is then obtained by post processing the computed approximation to the transformed equation. Error estimates are also presented for the approximation to the unknown of the variable coefficient equation with both smooth and non-smooth diffusivity coefficient and right-hand side. Three numerical experiments are given whose convergence results are in strong agreement with the theoretically derived estimates.

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arXiv2019

Data-driven physics informed deep learning of solute transport with anomalous diffusion

The fractional advection-dispersion equation (FADE) has attracted increased attention from researchers as it provides an accurate description for challenging phenomenas with long-range time memory and spatial interactions, such as the anomalous diffusion behavior in the solute transport in porous media. Practically, a full characterization of the model parameters, such as the fluid velocity, dispersion coefficient and the order of the fractional derivative, often implies a huge amount of experiments and measurements and thus are hard to be determined. In this paper, we employ the framework of feedforward deep neural networks (DNNs) to develop an efficient data-driven deep learning algorithm for inferring the aforementioned parameters of the FADE, such as the time-dependent space-fractional advection-dispersion equation (sFADE) and the variable-order fractional mobile/immobile equation (VoFMIE), in which the feedforward DNNs are trained to minimize the mean square error loss function formulated by means of the finite difference approximations of sFADE and VoFMIE, respectively. Several numerical experiments, in which we discover the model parameters by the feedforward DNNs for both the synthetic and field data, are presented to demonstrate the effectiveness and robustness of the proposed data-driven deep learning algorithm.

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arXiv2018

Wellposedness of the two-sided variable coefficient Caputo flux fractional diffusion equation and error estimate of its spectral approximation

In this article a two-sided variable coefficient fractional diffusion equation (FDE) is investigated, where the variable coefficient occurs outside of the fractional integral operator. Under a suitable transformation the variable coefficient equation is transformed to a constant coefficient equation. Then, using the spectral decomposition approach with Jacobi polynomials, we proved the wellposedness of the model and the regularity of its solution. A spectral approximation scheme is proposed and the accuracy of its approximation studied. Two numerical experiments are presented to demonstrate the derived error estimates.

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arXiv2018

Spectral approximation of a variable coefficient fractional diffusion equation in one space dimension

In this article we consider the approximation of a variable coefficient (two-sided) fractional diffusion equation (FDE), having unknown $u$. By introducing an intermediate unknown, $q$, the variable coefficient FDE is rewritten as a lower order, constant coefficient FDE. A spectral approximation scheme, using Jacobi polynomials, is presented for the approximation of $q$, $q_{N}$. The approximate solution to $u$, $u_{N}$, is obtained by post processing $q_{N}$. An a priori error analysis is given for $(q \, - \, q_{N})$ and $(u \, - \, u_{N})$. Two numerical experiments are presented whose results demonstrate the sharpness of the derived error estimates.

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