arXiv subjects

Hehu Xie

Publications and source records attributed to Hehu Xie.

At least 37 records · Page 2Linked to original sources

arXiv2024

FieldTNN-based machine learning method for Maxwell eigenvalue problems

The aim of this paper is to introduce a FieldTNN-based machine learning method for solving the Maxwell eigenvalue problem in both 2D and 3D domains, including both tensor and non-tensor computational regions. First, we extend the existing TNN-based approach to address the Maxwell eigenvalue problem, a fundamental challenge in electromagnetic field theory. Second, we tackle non-tensor computational domains, which represents a novel and significant contribution of this work. Third, we incorporate the divergence-free condition into the optimization process, allowing for the automatic filtering of spurious eigenpairs. Numerical examples are presented to demonstrate the efficiency and accuracy of our algorithm, underscoring its potential for broader applications in computational electromagnetics.

math.NA↗

arXiv2024

Error estimates of finite element methods for nonlocal problems using exact or approximated interaction neighborhoods

We study the asymptotic error between the finite element solutions of nonlocal models with a bounded interaction neighborhood and the exact solution of the limiting local model. The limit corresponds to the case when the horizon parameter, the radius of the spherical nonlocal interaction neighborhood of the nonlocal model, and the mesh size simultaneously approach zero. Two important cases are discussed: one involving the original nonlocal models and the other for nonlocal models with polygonal approximations of the nonlocal interaction neighborhood. Results of numerical experiments are also reported to substantiate the theoretical studies.

math.NA↗

arXiv2024

An Efficient Approach to Regression Problems with Tensor Neural Networks

This paper introduces a tensor neural network (TNN) to address nonparametric regression problems, leveraging its distinct sub-network structure to effectively facilitate variable separation and enhance the approximation of complex, high-dimensional functions. The TNN demonstrates superior performance compared to conventional Feed-Forward Networks (FFN) and Radial Basis Function Networks (RBN) in terms of both approximation accuracy and generalization capacity, even with a comparable number of parameters. A significant innovation in our approach is the integration of statistical regression and numerical integration within the TNN framework. This allows for efficient computation of high-dimensional integrals associated with the regression function and provides detailed insights into the underlying data structure. Furthermore, we employ gradient and Laplacian analysis on the regression outputs to identify key dimensions influencing the predictions, thereby guiding the design of subsequent experiments. These advancements make TNN a powerful tool for applications requiring precise high-dimensional data analysis and predictive modeling.

stat.ML↗

arXiv2024

The weak Galerkin finite element method for the Steklov eigenvalue problem

This paper introduces the application of the weak Galerkin (WG) finite element method to solve the Steklov eigenvalue problem, focusing on obtaining lower bounds of the eigenvalues. The noncomforming finite element space of the weak Galerkin finite element method is the key to obtain lower bounds of the eigenvalues. The arbitary high order lower bound estimates are given and the guaranteed lower bounds of the eigenvalues are also discussed. Numerical results demonstrate the accuracy and lower bound property of the numerical scheme.

math.NA↗

arXiv2024

Solving High Dimensional Partial Differential Equations Using Tensor Neural Network and A Posteriori Error Estimators

In this paper, based on the combination of tensor neural network and a posteriori error estimator, a novel type of machine learning method is proposed to solve high-dimensional boundary value problems with homogeneous and non-homogeneous Dirichlet or Neumann type of boundary conditions and eigenvalue problems of the second-order elliptic operator. The most important advantage of the tensor neural network is that the high dimensional integrations of tensor neural networks can be computed with high accuracy and high efficiency. Based on this advantage and the theory of a posteriori error estimation, the a posteriori error estimator is adopted to design the loss function to optimize the network parameters adaptively. The applications of tensor neural network and the a posteriori error estimator improve the accuracy of the corresponding machine learning method. The theoretical analysis and numerical examples are provided to validate the proposed methods.

math.NA↗

arXiv2024

Enhanced Error Estimates for Augmented Subspace Method with Crouzeix-Raviart Element

In this paper, we present some enhanced error estimates for augmented subspace methods with the nonconforming Crouzeix-Raviart (CR) element. Before the novel estimates, we derive the explicit error estimates for the case of single eigenpair and multiple eigenpairs based on our defined spectral projection operators, respectively. Then we first strictly prove that the CR element based augmented subspace method exhibits the second-order convergence rate between the two steps of the augmented subspace iteration, which coincides with the practical experimental results. The algebraic error estimates of second order for the augmented subspace method explicitly elucidate the dependence of the convergence rate of the algebraic error on the coarse space, which provides new insights into the performance of the augmented subspace method. Numerical experiments are finally supplied to verify these new estimate results and the efficiency of our algorithms.

math.NA↗

arXiv2024

A Nonnested Augmented Subspace Method for Kohn-Sham Equation

In this paper, a novel adaptive finite element method is proposed to solve the Kohn-Sham equation based on the moving mesh (nonnested mesh) adaptive technique and the augmented subspace method. Different from the classical self-consistent field iterative algorithm which requires to solve the Kohn-Sham equation directly in each adaptive finite element space, our algorithm transforms the Kohn-Sham equation into some linear boundary value problems of the same scale in each adaptive finite element space, and then the wavefunctions derived from the linear boundary value problems are corrected by solving a small-scale Kohn-Sham equation defined in a low-dimensional augmented subspace. Since the new algorithm avoids solving large-scale Kohn-Sham equation directly, a significant improvement for the solving efficiency can be obtained. In addition, the adaptive moving mesh technique is used to generate the nonnested adaptive mesh for the nonnested augmented subspace method according to the singularity of the approximate wavefunctions. The modified Hessian matrix of the approximate wavefunctions is used as the metric matrix to redistribute the mesh. Through the moving mesh adaptive technique, the redistributed mesh is almost optimal. A number of numerical experiments are carried out to verify the efficiency and the accuracy of the proposed algorithm.

math.NA↗

arXiv2024

Tensor Neural Network Interpolation and Its Applications

Based on tensor neural network, we propose an interpolation method for high dimensional non-tensor-product-type functions. This interpolation scheme is designed by using the tensor neural network based machine learning method. This means that we use a tensor neural network to approximate high dimensional functions which has no tensor product structure. In some sense, the non-tenor-product-type high dimensional function is transformed to the tensor neural network which has tensor product structure. It is well known that the tensor product structure can bring the possibility to design highly accurate and efficient numerical methods for dealing with high dimensional functions. In this paper, we will concentrate on computing the high dimensional integrations and solving high dimensional partial differential equations. The corresponding numerical methods and numerical examples will be provided to validate the proposed tensor neural network interpolation.

math.NA↗

arXiv2024

Tensor Neural Network Based Machine Learning Method for Elliptic Multiscale Problems

In this paper, we introduce a type of tensor neural network based machine learning method to solve elliptic multiscale problems. Based on the special structure, we can do the direct and highly accurate high dimensional integrations for the tensor neural network functions without Monte Carlo process. Here, with the help of homogenization techniques, the multiscale problem is first transformed to the high dimensional limit problem with reasonable accuracy. Then, based on the tensor neural network, we design a type of machine learning method to solve the derived high dimensional limit problem. The proposed method in this paper brings a new way to design numerical methods for computing more general multiscale problems with high accuracy. Several numerical examples are also provided to validate the accuracy of the proposed numerical methods.

math.NA↗

arXiv2024

Solving High-dimensional Parametric Elliptic Equation Using Tensor Neural Network

In this paper, we introduce a tensor neural network based machine learning method for solving the elliptic partial differential equations with random coefficients in a bounded physical domain. With the help of tensor product structure, we can transform the high-dimensional integrations of tensor neural network functions to one-dimensional integrations which can be computed with the classical quadrature schemes with high accuracy. The complexity of its calculation can be reduced from the exponential scale to a polynomial scale. The corresponding machine learning method is designed for solving high-dimensional parametric elliptic equations. Some numerical examples are provided to validate the accuracy and efficiency of the proposed algorithms.

math.NA↗

arXiv2024

Augmented Subspace Scheme for Eigenvalue Problem by Weak Galerkin Finite Element Method

This study proposes a class of augmented subspace schemes for the weak Galerkin (WG) finite element method used to solve eigenvalue problems. The augmented subspace is built with the conforming linear finite element space defined on the coarse mesh and the eigenfunction approximations in the WG finite element space defined on the fine mesh. Based on this augmented subspace, solving the eigenvalue problem in the fine WG finite element space can be reduced to the solution of the linear boundary value problem in the same WG finite element space and a low dimensional eigenvalue problem in the augmented subspace. The proposed augmented subspace techniques have the second order convergence rate with respect to the coarse mesh size, as demonstrated by the accompanying error estimates. Finally, a few numerical examples are provided to validate the proposed numerical techniques.

math.NA↗

arXiv2023

Tensor Neural Network and Its Numerical Integration

In this paper, we introduce a type of tensor neural network. For the first time, we propose its numerical integration scheme and prove the computational complexity to be the polynomial scale of the dimension. Based on the tensor product structure, we develop an efficient numerical integration method by using fixed quadrature points for the functions of the tensor neural network. The corresponding machine learning method is also introduced for solving high-dimensional problems. Some numerical examples are also provided to validate the theoretical results and the numerical algorithm.

math.NA↗

arXiv2022

A New Subspace Iteration Algorithm for Solving Generalized Eigenvalue Problems

It is needed to solve generalized eigenvalue problems (GEP) in many applications, such as the numerical simulation of vibration analysis, quantum mechanics, electronic structure, etc. The subspace iteration is a kind of widely used algorithm to solve eigenvalue problems. To solve the generalized eigenvalue problem, one kind of subspace iteration method, Chebyshev-Davidson algorithm, is proposed recently. In Chebyshev-Davidson algorithm, the Chebyshev polynomial filter technique is incorporated in the subspace iteration. In this paper, based on Chebyshev-Davidson algorithm, a new subspace iteration algorithm is constructed. In the new algorithm, the Chebyshev filter and inexact Rayleigh quotient iteration techniques are combined together to enlarge the subspace in the iteration. Numerical results of a vibration analysis problem show that the number of iteration and computing time of the proposed algorithm is much less than that of the Chebyshev-Davidson algorithm and some typical GEP solution algorithms. Furthermore, the new algorithm is more stable and reliable than the Chebyshev-Davidson algorithm in the numerical results.

math.NA↗

arXiv2022

Solving High Dimensional Partial Differential Equations Using Tensor Type Discretization and Optimization Process

In this paper, we propose a tensor type of discretization and optimization process for solving high dimensional partial differential equations. First, we design the tensor type of trial function for the high dimensional partial differential equations. Based on the tensor structure of the trial functions, we can do the direct numerical integration of the approximate solution without the help of Monte-Carlo method. Then combined with the Ritz or Galerkin method, solving the high dimensional partial differential equation can be transformed to solve a concerned optimization problem. Some numerical tests are provided to validate the proposed numerical methods.

math.NA↗

arXiv2022

On the convergence to local limit of nonlocal models with approximated interaction neighborhoods

Many nonlocal models have adopted Euclidean balls as the nonlocal interaction neighborhoods. When solving them numerically, it is sometimes convenient to adopt polygonal approximations of such balls. A crucial question is, to what extent such approximations affect the nonlocal operators and the corresponding solutions. While recent works have analyzed this issue for a fixed horizon parameter, the question remains open in the case of a small or vanishing horizon parameter, which happens often in many practical applications and has significant impact on the reliability and robustness of nonlocal modeling and simulations. In this work, we are interested in addressing this issue and establishing the convergence of the nonlocal solutions associated with polygonally approximated interaction neighborhoods to the local limit of the original nonlocal solutions. Our finding reveals that the new nonlocal solution does not converge to the correct local limit when the number of sides of polygons is uniformly bounded. On the other hand, if the number of sides tends to infinity, the desired convergence can be established. These results may be used to guide future computational studies of nonlocal models.

math.NA↗

arXiv2022

An Efficient Adaptive Finite Element Method for Eigenvalue Problems

The aim of this paper is to propose an efficient adaptive finite element method for eigenvalue problems based on the multilevel correction scheme and inverse power method. This method involves solving associated boundary value problems on each adaptive partitions and very low dimensional eigenvalue problems on some special meshes which are controlled by the proposed algorithm. Since we Hence the efficiency of solving eigenvalue problems can be improved to be similar to the adaptive finite element method for the associated boundary value problems. The convergence and optimal complexity is theoretically verified and numerically demonstrated.

math.NA↗

arXiv2021

GCGE: A Package for Solving Large Scale Eigenvalue Problems by Parallel Block Damping Inverse Power Method

We propose an eigensolver and the corresponding package, GCGE, for solving large scale eigenvalue problems. This method is the combination of damping idea, subspace projection method and inverse power method with dynamic shifts. To reduce the dimensions of projection subspaces, a moving mechanism is developed when the number of desired eigenpairs is large. The numerical methods, implementing techniques and the structure of the package are presented. Plenty of numerical results are provided to demonstrate the efficiency, stability and scalability of the concerned eigensolver and the package GCGE for computing many eigenpairs of large symmetric matrices arising from applications.

math.NA↗

arXiv2021

Enhanced Error Estimates for Augmented Subspace Method

In this paper, some enhanced error estimates are derived for the augmented subspace methods which are designed for solving eigenvalue problems. We will show that the augmented subspace methods have the second order convergence rate which is better than the existing results. These sharper estimates provide a new dependence of convergence rate on the coarse spaces in augmented subspace methods. These new results are also validated by some numerical examples.

math.NA↗