arXiv subjects

Liwei Xu

Publications and source records attributed to Liwei Xu.

At least 19 recordsLinked to original sources

arXiv2026

High-order complete flux schemes for convection-diffusion equations on arbitrary subdivisions

We develop a novel complete flux finite volume method for convection-diffusion equations on arbitrary subdivisions in two and three dimensions. Unlike standard finite volume discretizations, where the numerical flux is directly approximated from the flux definition, we derive the exact normal flux across each control volume edge/face from the underlying PDE. This exact flux splits naturally into a homogeneous part (the classical Scharfetter--Gummel flux) and an inhomogeneous part based on a Green's function that incorporates the tangential flux and the source term. The resulting formulation is exactly equivalent to the continuous equation and, once the discrete space is chosen, yields high-order schemes without using correction or stabilization strategies. From this framework, we develop concrete numerical schemes on arbitrary grids using Lagrange finite element spaces and B-spline spaces, together with their companion dual meshes (control volume partitions). Numerical experiments in two and three dimensions confirm the optimal convergence and positivity preservation of the proposed schemes.

math.NA↗

arXiv2026

Numerical solutions of an accurate diffuse interface model of the incompressible resistive MHD free surface flow

In this paper, we derive a new model to simulate the incompressible resistive magnetohydrodynamic (MHD) free surface flow. A thermodynamically consistent diffuse interface method is adopted to characterize the moving interface in the modeling process. The formal convergence of the proposed MHD free surface flow model to the sharp interface model is established via a matched asymptotic argument, and the model can be solved without the need for sophisticated free surface capturing schemes. We design a fully decoupled linear finite element scheme that preserves the divergence-free constraint of the magnetic field at a discrete level. The reliability and robustness of the proposed model and algorithm are validated through numerical investigations of the magnetic damping effect on bubble dynamics. In particular, we provide a quantitative numerical comparison of the present results with those obtained from an inductionless MHD model and a sharp interface arbitrary Lagrangian--Eulerian model.

math.NA↗

arXiv2025

Data-driven optimized high-order WENO schemes with low-dissipation and low-dispersion

Classical high-order weighted essentially non-oscillatory (WENO) schemes are designed to achieve optimal convergence order for smooth solutions and to maintain non-oscillatory behaviors for discontinuities. However, their spectral properties are not optimal, which limits the ability to capture high-frequency waves and small-scale features. In this paper, we propose a data-driven optimized method to improve the spectral properties of the WENO schemes. By analyzing the approximate dispersion relation (ADR), the spectral error of the schemes can be bounded by the reconstructed errors of a series of trigonometric functions with different wavenumbers. Therefore, we propose the new schemes WENO5-JS/Z-NN that introduce a compensation term parameterized by a neural network to the weight function of the WENO5-JS/Z schemes. The neural network is trained such that the generated weights can minimize the reconstructed errors over a large number of spatial stencils, and furthermore, improve the spectral accuracy. Meanwhile, the Total Variation Diminishing (TVD) constraint and anti-dissipation penalization are incorporated into the loss function to enhance the shock-capturing capability and preserve stability in simulating high-frequency waves. Compared to WENO5-JS/Z, our schemes maintain the ability to capture discontinuities while providing higher resolution for fine-scale flow features. The ADR indicates that the new schemes can match the exact spectrum more accurately over a broader range of wavenumbers.

math.NA↗

arXiv2025

Modeling and simulation of inductionless magnetohydrodynamic free surface problems with unmatched densities

We propose a new diffuse interface model for simulating an inductionless magnetohydrodynamic (MHD) free surface problem. By using the Onsager's variational principle and the laws of thermodynamics, we derive a thermodynamically consistent system that couples the Cahn--Hilliard equation modeling phase separation, the Navier--Stokes equations governing fluid motion, and a generalized Darcy's law accounting for electromagnetic effects. In contrast to existing diffuse interface MHD models, the proposed model can handle general material properties in practical engineering applications. Furthermore, through asymptotic arguments, we investigate the sharp interface limit, and then demonstrate that the classical sharp interface model can be recovered as the interface thickness approaches zero, theoretically validating the proposed diffuse interface model as an approximate approach. An efficient decoupled, linear, and charge-conservative finite element scheme is designed, and it significantly facilitates the large-scale and accurate numerical simulations involving large parameter ratios. Finally, we present several three-dimensional numerical experiments of magnetic damping effects on bubble dynamics for the demonstration of the capability of the proposed model and method in capturing complex MHD phenomena.

math.NA↗

arXiv2025

Learning high-accuracy numerical schemes for hyperbolic equations on coarse meshes

When solving partial differential equations using classical schemes such as finite difference or finite volume methods, sufficiently fine meshes and carefully designed schemes are required to achieve high-order accuracy of numerical solutions, leading to a significant increase in computational costs, especially for three-dimensional (3D) time-dependent problems. Recently, machine learning-assisted numerical methods have been proposed to enhance accuracy or efficiency. In this paper, we propose a data-driven finite difference numerical method to solve the hyperbolic equations with smooth solutions on coarse grids, which can achieve higher accuracy than classical numerical schemes based on the same mesh size. In addition, the data-driven schemes have better spectrum properties than the classical schemes, although the spectrum properties are not explicitly optimized during the training process. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed method, as well as its good performance on dispersion and dissipation.

math.NA↗

arXiv2024

Conjugate-Gradient-like Based Adaptive Moment Estimation Optimization Algorithm for Deep Learning

Training deep neural networks is a challenging task. In order to speed up training and enhance the performance of deep neural networks, we rectify the vanilla conjugate gradient as conjugate-gradient-like and incorporate it into the generic Adam, and thus propose a new optimization algorithm named CG-like-Adam for deep learning. Specifically, both the first-order and the second-order moment estimation of generic Adam are replaced by the conjugate-gradient-like. Convergence analysis handles the cases where the exponential moving average coefficient of the first-order moment estimation is constant and the first-order moment estimation is unbiased. Numerical experiments show the superiority of the proposed algorithm based on the CIFAR10/100 dataset.

cs.LG↗

arXiv2024

Macroscopic auxiliary asymptotic preserving neural networks for the linear radiative transfer equations

We develop a Macroscopic Auxiliary Asymptotic-Preserving Neural Network (MA-APNN) method to solve the time-dependent linear radiative transfer equations (LRTEs), which have a multi-scale nature and high dimensionality. To achieve this, we utilize the Physics-Informed Neural Networks (PINNs) framework and design a new adaptive exponentially weighted Asymptotic-Preserving (AP) loss function, which incorporates the macroscopic auxiliary equation that is derived from the original transfer equation directly and explicitly contains the information of the diffusion limit equation. Thus, as the scale parameter tends to zero, the loss function gradually transitions from the transport state to the diffusion limit state. In addition, the initial data, boundary conditions, and conservation laws serve as the regularization terms for the loss. We present several numerical examples to demonstrate the effectiveness of MA-APNNs.

math.NA↗

arXiv2023

Learning More Discriminative Local Descriptors for Few-shot Learning

Few-shot learning for image classification comes up as a hot topic in computer vision, which aims at fast learning from a limited number of labeled images and generalize over the new tasks. In this paper, motivated by the idea of Fisher Score, we propose a Discriminative Local Descriptors Attention (DLDA) model that adaptively selects the representative local descriptors and does not introduce any additional parameters, while most of the existing local descriptors based methods utilize the neural networks that inevitably involve the tedious parameter tuning. Moreover, we modify the traditional $k$-NN classification model by adjusting the weights of the $k$ nearest neighbors according to their distances from the query point. Experiments on four benchmark datasets show that our method not only achieves higher accuracy compared with the state-of-art approaches for few-shot learning, but also possesses lower sensitivity to the choices of $k$.

cs.CV↗

arXiv2022

A model-data asymptotic-preserving neural network method based on micro-macro decomposition for gray radiative transfer equations

We propose a model-data asymptotic-preserving neural network(MD-APNN) method to solve the nonlinear gray radiative transfer equations(GRTEs). The system is challenging to be simulated with both the traditional numerical schemes and the vanilla physics-informed neural networks(PINNs) due to the multiscale characteristics. Under the framework of PINNs, we employ a micro-macro decomposition technique to construct a new asymptotic-preserving(AP) loss function, which includes the residual of the governing equations in the micro-macro coupled form, the initial and boundary conditions with additional diffusion limit information, the conservation laws, and a few labeled data. A convergence analysis is performed for the proposed method, and a number of numerical examples are presented to illustrate the efficiency of MD-APNNs, and particularly, the importance of the AP property in the neural networks for the diffusion dominating problems. The numerical results indicate that MD-APNNs lead to a better performance than APNNs or pure data-driven networks in the simulation of the nonlinear non-stationary GRTEs.

math.NA↗

arXiv2022

A highly accurate perfectly-matched-layer boundary integral equation solver for acoustic layered-medium problems

Based on the perfectly matched layer (PML) technique, this paper develops a high-accuracy boundary integral equation (BIE) solver for acoustic scattering problems in locally defected layered media in both two and three dimensions. The original scattering problem is truncated onto a bounded domain by the PML. Assuming the vanishing of the scattered field on the PML boundary, we derive BIEs on local defects only in terms of using PML-transformed free-space Green's function, and the four standard integral operators: single-layer, double-layer, transpose of double-layer, and hyper-singular boundary integral operators. The hyper-singular integral operator is transformed into a combination of weakly-singular integral operators and tangential derivatives. We develop a high-order Chebyshev-based rectangular-polar singular-integration solver to discretize all weakly-singular integrals. Numerical experiments for both two- and three-dimensional problems are carried out to demonstrate the accuracy and efficiency of the proposed solver.

math.NA↗

arXiv2022

Stochastic Gauss-Newton Algorithms for Online PCA

In this paper, we propose a stochastic Gauss-Newton (SGN) algorithm to study the online principal component analysis (OPCA) problem, which is formulated by using the symmetric low-rank product (SLRP) model for dominant eigenspace calculation. Compared with existing OPCA solvers, SGN is of improved robustness with respect to the varying input data and algorithm parameters. In addition, turning to an evaluation of data stream based on approximated objective functions, we develop a new adaptive stepsize strategy for SGN (AdaSGN) which requires no priori knowledge of the input data, and numerically illustrate its comparable performance with SGN adopting the manaully-tuned diminishing stepsize. Without assuming the eigengap to be positive, we also establish the global and optimal convergence rate of SGN with the specified stepsize using the diffusion approximation theory.

math.OC↗

arXiv2022

Optimal error estimates of a Crank--Nicolson finite element projection method for magnetohydrodynamic equations

In this paper, we propose and analyze a fully discrete finite element projection method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method and the Galerkin finite element method are used to discretize the model in time and space, respectively, and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a second-order decoupling projection method of the Van Kan type \cite{vankan1986} in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Error estimates are proved in the discrete $L^\infty(0,T;L^2)$ norm for the proposed decoupled finite element projection scheme. Numerical examples are provided to illustrate the theoretical results.

math.NA↗

arXiv2022

On the hyper-singular boundary integral equation methods for dynamic poroelasticity: three dimensional case

In our previous work [SIAM J. Sci. Comput. 43(3) (2021) B784-B810], an accurate hyper-singular boundary integral equation method for dynamic poroelasticity in two dimensions has been developed. This work is devoted to studying the more complex and difficult three-dimensional problems with Neumann boundary condition and both the direct and indirect methods are adopted to construct combined boundary integral equations. The strongly-singular and hyper-singular integral operators are reformulated into compositions of weakly-singular integral operators and tangential-derivative operators, which allow us to prove the jump relations associated with the poroelastic layer potentials and boundary integral operators in a simple manner. Relying on both the investigated spectral properties of the strongly-singular operators, which indicate that the corresponding eigenvalues accumulate at three points whose values are only dependent on two Lamé constants, and the spectral properties of the Calderón relations of the poroelasticity, we propose low-GMRES-iteration regularized integral equations. Numerical examples are presented to demonstrate the accuracy and efficiency of the proposed methodology by means of a Chebyshev-based rectangular-polar solver.

math.NA↗

arXiv2021

On the generalized Calderón formulas for closed- and open-surface elastic scattering problems

The Calderón formulas (i.e., the combination of single-layer and hyper-singular boundary integral operators) have been widely utilized in the process of constructing valid boundary integral equation systems which could possess highly favorable spectral properties. This work is devoted to studying the theoretical properties of elastodynamic Calderón formulas which provide us with a solid basis for the design of fast boundary integral equation methods solving elastic wave problems defined on a close-surface or an open-surface in two dimensions. For the closed-surface case, it is proved that the Calderón formula is a Fredholm operator of second-kind except for certain circumstances. Regarding to the open-surface case, we investigate weighted integral operators instead of the original integral operators which are resulted from dealing with edge singularities of potentials corresponding to the elastic scattering problems by open-surfaces, and show that the Calderón formula is a compact perturbation of a bounded and invertible operator. To complete the proof, we need to use the well-posedness result of the elastic scattering problem, the analysis of the zero-frequency integral operators defined on the straight arc, the singularity decompositions of the kernels of integral operators, and a new representation formula of the hyper-singular operator. Moreover, it can be demonstrated that the accumulation point of the spectrum of the invertible operator is the same as that of the eigenvalues of the Calderón formula in the closed-surface case.

math.AP↗

arXiv2021

Optimal error estimates of a second-order projection finite element method for magnetohydrodynamic equations

In this paper, we propose and analyze a temporally second-order accurate, fully discrete finite element method for the magnetohydrodynamic (MHD) equations. A modified Crank--Nicolson method is used to discretize the model and appropriate semi-implicit treatments are applied to the fluid convection term and two coupling terms. These semi-implicit approximations result in a linear system with variable coefficients for which the unique solvability can be proved theoretically. In addition, we use a decoupling projection method of the Van Kan type \cite{vankan1986} in the Stokes solver, which computes the intermediate velocity field based on the gradient of the pressure from the previous time level, and enforces the incompressibility constraint via the Helmholtz decomposition of the intermediate velocity field. The energy stability of the scheme is theoretically proved, in which the decoupled Stokes solver needs to be analyzed in details. Optimal-order convergence of $\mathcal{O} (τ^2+h^{r+1})$ in the discrete $L^\infty(0,T;L^2)$ norm is proved for the proposed decoupled projection finite element scheme, where $τ$ and $h$ are the time stepsize and spatial mesh size, respectively, and $r$ is the degree of the finite elements. Existing error estimates of second-order projection methods of the Van Kan type \cite{vankan1986} were only established in the discrete $L^2(0,T;L^2)$ norm for the Navier--Stokes equations. Numerical examples are provided to illustrate the theoretical results.

math.NA↗

arXiv2020

An extended sampling-ensemble Kalman filter approach for partial data inverse elastic problems

Inverse problems are more challenging when only partial data are available in general. In this paper, we propose a two-step approach combining the extended sampling method and the ensemble Kalman filter to reconstruct an elastic rigid obstacle using partial data. In the first step, the approximate location of the unknown obstacle is obtained by the extended sampling method. In the second step, the ensemble Kalman filter is employed to reconstruct the shape. The location obtained in the first step guides the construction of the initial particles of the ensemble Kalman filter, which is critical to the performance of the second step. Both steps are based on the same physical model and use the same scattering data. Numerical examples are shown to illustrate the effectiveness of the proposed method.

math.NA↗

arXiv2020

An accurate hyper-singular boundary integral equation method for dynamic poroelasticity in two dimensions

This paper is concerned with the boundary integral equation method for solving the exterior Neumann boundary value problem of dynamic poroelasticity in two dimensions. The main contribution of this work consists of two aspescts: the proposal of a novel regularized boundary integral equation, and the presentation of new regularized formulations of the strongly-singular and hyper-singular boundary integral operators. Firstly, turning to the spectral properties of the double-layer operator and the corresponding Calderón relation of the poroelasticity, we propose the novel low-GMRES-iteration integral equation whose eigenvalues are bounded away from zero and infinity. Secondly, with the help of the Günter derivatives, we reformulate the strongly-singular and hyper-singular integral operators into combinations of the weakly-singular operators and the tangential derivatives. The accuracy and efficiency of the proposed methodology are demonstrated through several numerical examples.

physics.comp-ph↗

arXiv2020

Quality-Bayesian approach to inverse acoustic source problems with partial data

A quality-Bayesian approach, combining the direct sampling method and the Bayesian inversion, is proposed to reconstruct the locations and intensities of the unknown acoustic sources using partial data. First, we extend the direct sampling method by constructing a new indicator function to obtain the approximate locations of the sources. The behavior of the indicator is analyzed. Second, the inverse problem is formulated as a statistical inference problem using the Bayes' formula. The well-posedness of the posterior distribution is proved. The source locations obtained in the first step are coded in the priors. Then an Metropolis-Hastings MCMC algorithm is used to explore the posterior density. Both steps use the same physical model and measured data. Numerical experiments show that the proposed method is effective even with partial data.

math.NA↗