Numerical Mathematics: Theory, Methods and Applications
type: channel
description: Numerical Mathematics: Theory, Methods and Applications (NMTMA) publishes high-quality papers on the construction, analysis and application of numerical methods for solving scientific and engineering problems. Research and expository papers devoted to the numerical solution of mathematical equations arising in all areas of science and technology are expected. The journal originates from Numerical Mathematics: A Journal of Chinese Universities (English Edition), and has been sponsored by Nanjing University.
publisher: Global Science Press
language: en
publication name: Numerical Mathematics: Theory, Methods and Applications
issn: 1004-8979
items: A Gradient Recovery Technique for Enhancing the Convergence of Demagnetizing Field Based on PDE Approach
items: Theoretical Guarantees of Recovery Algorithms for Ternary Sparse Signals
items: A Space-Time Extrapolation Cascadic Multigrid Method for 2D Linear Parabolic Problems
items: Author Index to Volume 18
items: Two Globally Convergent Hybrid CG Methods with Adaptive Restart Strategy for Riemannian Optimization
items: Splitting Schemes for Second-Order Backward Stochastic Differential Equations
items: A Biquadratic Characteristic Finite Volume Element Method for Time-Fractional Convection-Dominated Equations
items: Machine Learning-Based Bias Correction for the GEIM Model
items: Numerical Analysis of a Second Order Crank-Nicolson Mixed Finite Element Method for the Swift-Hohenberg Equation
items: Parallel and Scalable Two-Grid Nonlinear Preconditioning Technique for Fully Coupled Solution of Thermal Convective Flow Problems
items: A Moving Mesh Isogeometric Method Based on Harmonic Maps
items: Decoupling Analysis for Solving the Diffusive Peterlin Viscoelastic Model
items: Solving the Helmholtz Equation by Sparse Fundamental Solution Neural Network
items: A Highly Efficient Adaptive Mesh Refinement Algorithm for the 1D Schrödinger-Poisson Problem
items: Entirely Exponential-Type Scheme (E2S) for Optimization Problems with Singularly Perturbed ODE Constraints: The Model Problem
items: Uncertainty Quantification with Physics-Informed Generative Process Distributions for the Linear Diffusion Equation
items: Approximate Method for Computing Hypersingular Integrals with Oscillatory Kernels
items: A Modified Version of the PRESB Preconditioner for a Class of Non-Hermitian Complex Systems of Linear Equations
items: Sparse Wavelet Element Method for Piezoelectric Equations in an Unbounded Domain
items: Convergence Analysis on a Second-Order, Positive, and Unconditional Energy Dissipative Scheme for the Poisson-Nernst-Planck Equations
items: Solving Generalized Tensor Eigenvalue Problem via Spectral Shifted Inverse Power Methods
items: Implicit-Explicit Time Discretization Schemes for a Class of Semilinear Wave Equations with Nonautonomous Dampings
items: Variable $s$-Step Technique for New Conjugate Residual Algorithms for Solving Non-Square Linear Systems Arising in Control Problems
items: On the Criteria for Online Updating of Basis Functions for Model Order Reduction Techniques
items: New Extrapolation Modulus-Based Economical Cascadic Multigrid Method for Variational Inequality Problems with Nonlinear Source Terms
items: A Highly Efficient USOR-like Iterative Method for the Stationary Wigner Equation with Scattering
items: Noise Robust Physics-Informed Generative Adversarial Networks for Solving Stochastic Differential Equations
items: On the Recovery of Source Term for Fractional Evolution PDEs by MC-fPINNs
items: L1 Schemes for Time-Fractional Differential Equations: A Brief Survey and New Development
items: A Modularized Algorithmic Framework for Interface Related Optimization Problems Using Characteristic Functions

A Gradient Recovery Technique for Enhancing the Convergence of Demagnetizing Field Based on PDE Approach
type: item
description: The PDE approach is a popular technique for the demagnetizing field calculation due to the flexibility in handling complex domains. However, it faces a challenge on delivering desired accuracy due to the suboptimal convergence of the function gradient and singularity on the boundary. In this work, a robust gradient recovery technique is applied and analysed for fixing such an issue. An $L^2$ error estimate of the finite element approximation for the demagnetizing field is derived, which consists of two parts, i.e., of finite element discretization error and boundary approximation error. A gradient recovery method based on the polynomial preserving recovery technique is applied to enhance the accuracy of the finite element approximation, and a superconvergence result is established. An idea of locally refined surface meshes is applied to resolve the singularity in the boundary conditions, thereby reducing boundary approximation errors. Extensive numerical tests are provided to verify our theoretical findings and the efficiency of our proposed method. The results indicate that the proposed method achieves second-order convergence for the approximate demagnetizing field, positioning it as a highly competitive technique to develop optimal algorithms in computational micromagnetics.
subject: Articles
  topic: section
subject: Demagnetizing field
  topic: keywords
subject: PDE approach
  topic: keywords
subject: finite element method
  topic: keywords
subject: gradient recovery technique
  topic: keywords
subject: polynomial preserving recovery method
  topic: keywords
subject: superconvergence
  topic: keywords
creator: Jing Wu
creator: Zixuan Cui
creator: Lei Yang
creator: Guanghui Hu
rights: Copyright (c) 2025 Numerical Mathematics: Theory, Methods and Applications
date: 2025-11-10
publication date: 2025-11-10
doi: 10.4208/nmtma.OA-2025-0043

Theoretical Guarantees of Recovery Algorithms for Ternary Sparse Signals
type: item
description: This paper focuses on the problem of recovering ternary sparse signals with s nonzero entries of 1 and −1. We propose three novel algorithms: ternary matching pursuit (TMP), ternary generalized orthogonal matching pursuit (TGOMP), and piecewise ternary generalized orthogonal matching pursuit (PTGOMP). First, inspired by the binary matching pursuit algorithm, we introduce the TMP algorithm, which assigns values of 1 or −1 based on the most correlated residual, and provide theoretical guarantees based on the mutual coherence, denoted by $µ$ and the restricted isometry property of the measurement matrix, respectively. Second, we propose the TGOMP algorithm, by selecting multiple $(M)$ indices at each iteration in order to improve the performance of the TMP algorithm. We establish a sufficient condition $µ < 1/(2s − 1)$ that ensures the TGOMP algorithm selects $M$ correct indices and corresponding entries of $x$ in each iteration. Especially, all correct indices and entries can be selected in the first iteration. Additionally, we present a sufficient condition based on the restricted isometry property that guarantees all correct indices are selected in at most s iterations. Third, we propose the PTGOMP algorithm, which employs a piecewise selection strategy at each iteration, further improves recovery performance. Theoretical guarantees for the PTGOMP algorithm are derived based on the mutual coherence, showing its advantages in ternary sparse signal recovery. Finally, we validate the effectiveness of our algorithms through simulations and numerical experiments, demonstrating that combining appropriate matrix structures with suitable sparsity patterns can significantly improve recovery performance.
subject: Articles
  topic: section
creator: Hanbing Liu
creator: Chongjun Li
rights: Copyright (c) 2025 Numerical Mathematics: Theory, Methods and Applications
date: 2025-11-05
publication date: 2025-11-05
doi: 10.4208/nmtma.OA-2025-0014

A Space-Time Extrapolation Cascadic Multigrid Method for 2D Linear Parabolic Problems
type: item
description: We present a new space-time extrapolation cascadic multigrid method with SSOR preconditioned GMRES smoother for linear parabolic equations. This method simultaneously solves all time steps of the system using an all-at-once approach, employing Crank-Nicolson discretization in time and central difference discretization in space. The key components of the new algorithm are the Richardson extrapolation and Lagrange interpolation operators. By utilizing these techniques with numerical solutions of current and previous grids, we generate a good initial guess for the iterative solution on the next finer grid, greatly reducing the number of required iterations and computational time. Finally, we explain how to implement the new multigrid method and show its efficiency through numerical experiments.
subject: Articles
  topic: section
creator: Huaqing Wang
creator: Kejia Pan
creator: Jin Li
creator: Hongling Hu
rights: Copyright (c) 2025 Numerical Mathematics: Theory, Methods and Applications
date: 2025-11-05
publication date: 2025-11-05
doi: 10.4208/nmtma.OA-2025-0010

Author Index to Volume 18
type: item
subject: Articles
  topic: section
rights: Copyright (c) 2025 Numerical Mathematics: Theory, Methods and Applications
date: 2025-11-03
publication date: 2025-11-03
starting page: 1125
ending page: 1126
doi: 10.4208/

Two Globally Convergent Hybrid CG Methods with Adaptive Restart Strategy for Riemannian Optimization
type: item
description: Hybrid conjugate gradient methods (CGMs) with adaptive restart strategies have been well-researched in Euclidean space. In this paper, we extend two methods of this type to solve optimization problems on Riemannian manifolds. Firstly, we present two Riemannian hybrid CGMs, and their hybrid conjugate parameters are yielded by projection or convex combination of the classical parameters. The first is a Riemannian hybrid CGM that projects between the Dai-Yuan method and another flexible conjugate parameter. The second is a combination of projection and convex combination of the modified Riemannian Liu-Storey method and the modified Riemannian Hestenes-Stiefel method. In the framework of Riemannian CGMs, we apply a uniform adaptive restart strategy to both methods based on the hybrid conjugate parameters we proposed. The search direction of the presented methods satisfies the sufficient descent condition. Under the usual assumption and the weak Wolfe line search, we prove the global convergence of the two proposed methods. Finally, preliminary numerical results are reported and compared with several existing Riemannian CGMs, showing that our methods are effective.
subject: Articles
  topic: section
subject: Riemannian optimization
  topic: keywords
subject: hybrid conjugate gradient method
  topic: keywords
subject: adaptive restart strategy
  topic: keywords
subject: global convergence
  topic: keywords
creator: Yun Wang
creator: Meixuan Jiang
creator: Hu Shao
creator: Xiaoquan Wang
creator: Ting Wu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-10-22
publication date: 2025-10-22
starting page: 1100
ending page: 1124
doi: 10.4208/nmtma.OA-2024-0030

Splitting Schemes for Second-Order Backward Stochastic Differential Equations
type: item
description: In this work, we focus on splitting methods for solving second-order backward stochastic differential equations (2BSDEs). By splitting the original $d$-dimensional 2BSDEs into $d$ 2BSDEs and approximating these split 2BSDEs, we derive the splitting schemes that only require one-dimensional approximations to evaluate the conditional mathematical expectations. Combining the Sinc quadrature rule to approximate the conditional expectations, we further propose the first-order fully discrete splitting schemes. Numerical tests are presented to show the capacity of the schemes.
subject: Articles
  topic: section
subject: Second-order backward stochastic differential equations
  topic: keywords
subject: splitting method
  topic: keywords
subject: splitting scheme
  topic: keywords
subject: Sinc quadrature rule
  topic: keywords
creator: Bo Li
creator: Weidong Zhao
creator: Luying Zheng
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-10-22
publication date: 2025-10-22
starting page: 1003
ending page: 1026
doi: 10.4208/nmtma.OA-2025-0023

A Biquadratic Characteristic Finite Volume Element Method for Time-Fractional Convection-Dominated Equations
type: item
description: We propose a biquadratic characteristic finite volume element (CFVE) method for two classes of time-fractional convection-dominated equations on quadrilateral meshes. For the mixed time-fractional problem (i.e., involving both time-fractional and integer-order derivatives), optimal convergence rate in $L^2$-norm is established for the CFVE method with temporal discretization based on the $L1$-formula. For the pure time-fractional problem, we decompose the equation by separating an integer-order time derivative term, thereby enabling the application of the method of characteristics. Furthermore, the influence of the decomposition parameter is investigated through numerical experiments, providing practical insights into parameter selection. Extensive numerical experiments validate the efficacy of the proposed CFVE method in simulating both classes of time-fractional convection-dominated equations.
subject: Articles
  topic: section
subject: Characteristic finite volume element method
  topic: keywords
subject: time-fractional
  topic: keywords
subject: convection-dominated
  topic: keywords
subject: finite volume element method
  topic: keywords
creator: Xinyue Zhao
creator: Ziqi Tu
creator: Xiang Wang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-10-22
publication date: 2025-10-22
starting page: 1027
ending page: 1065
doi: 10.4208/nmtma.OA-2025-0018

Machine Learning-Based Bias Correction for the GEIM Model
type: item
description: This study focuses on real-time reconstruction of the spatial distribution of nuclear power using limited measurement observations. While physical models, such as the generalized empirical interpolation method (GEIM), can reconstruct the spatial field, they often cause bias if the model used to construct it is biased. The parametrized background data weak (PBDW) method attempts to mitigate this model bias, but its effectiveness is limited. To improve model bias correction, this paper proposes leveraging machine learning techniques-specifically, support vector regression, ${\rm K}$-nearest neighbors, and decision trees to enhance the GEIM method. These techniques predict model bias distributions across the entire field based on observed model bias at measurement points. The results demonstrate that Gaussian process based correction performs comparably to PBDW, both offering superior accuracy and robustness against noise, while other machine learning methods exhibit instability under varying parameter settings.
subject: Articles
  topic: section
subject: Model order reduction
  topic: keywords
subject: model bias correction
  topic: keywords
subject: Gaussian processes
  topic: keywords
subject: generalized empirical interpolation method
  topic: keywords
subject: physical field reconstruction
  topic: keywords
creator: Taidong Niu
creator: Mingzhe Lyu
creator: Helin Gong
creator: Han Zhang
creator: Zhang Chen
creator: Zhiyong Wang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-10-22
publication date: 2025-10-22
starting page: 1066
ending page: 1099
doi: 10.4208/nmtma.OA-2025-0025

Numerical Analysis of a Second Order Crank-Nicolson Mixed Finite Element Method for the Swift-Hohenberg Equation
type: item
description: In this paper, we conduct an error analysis for a mixed finite element method combined with Crank-Nicolson time-stepping for the Swift-Hohenberg (SH) equation. The original fourth-order differential equation from the SH equation is reformulated as a coupled system comprising a nonlinear parabolic equation and an elliptic equation. This transformation allows us to propose a mixed finite element method that utilizes only continuous elements for the approximation of the system. We rigorously demonstrate that our proposed scheme is unconditionally uniquely solvable and maintains unconditional nonlinear energy stability. Additionally, we provide a thorough analysis to determine the convergence rate of the method. To substantiate our theoretical findings, we present numerical tests that confirm both the accuracy and energy stability of the scheme.
subject: Articles
  topic: section
subject: Swift-Hohenberg equation
  topic: keywords
subject: error analysis
  topic: keywords
subject: mixed finite element
  topic: keywords
subject: Crank-Nicolson unconditionally energy stable
  topic: keywords
creator: Hongling Li
creator: Manting Xie
creator: Meiling Yue
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-10-22
publication date: 2025-10-22
starting page: 947
ending page: 971
doi: 10.4208/nmtma.OA-2025-0007

Parallel and Scalable Two-Grid Nonlinear Preconditioning Technique for Fully Coupled Solution of Thermal Convective Flow Problems
type: item
description: The focus of this paper is on the numerical solution of large sparse nonlinear systems of algebraic equations, which arise from the discretization of the incompressible Navier-Stokes equations for fluid flows. The inexact Newton algorithm is an efficient and popular technique for solving this kind of nonlinear systems. However, the method may not be suitable for large scale calculations because the number of Newton iterations is not scalable with respect to the number of processors, the grid size, and robust with respect to some physical parameters such as the Reynolds number and the Grashof number. In this paper, we investigate some fully coupled parallel two-grid nonlinear preconditioning algorithm, including a grid sequencing method for the Newton iteration and a two-level overlapping Schwarz preconditioner for the linear iteration. We show numerically that it performs well for solving the thermal convective flow problems on parallel computers.
subject: Articles
  topic: section
subject: Fluid flow
  topic: keywords
subject: heat transfer
  topic: keywords
subject: nonlinear preconditioning
  topic: keywords
subject: grid sequencing
  topic: keywords
subject: domain decomposition
  topic: keywords
subject: parallel computing
  topic: keywords
creator: Xiahui He
creator: Haijian Yang
creator: Jiali Tu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-10-22
publication date: 2025-10-22
starting page: 927
ending page: 946
doi: 10.4208/nmtma.OA-2025-0012

A Moving Mesh Isogeometric Method Based on Harmonic Maps
type: item
description: Although isogeometric analysis has shown great potential in achieving highly accurate numerical solutions of partial differential equations, how to efficiently implement the method is one of the challenges that makes it more competitive in practical simulations. In this paper, an integration of isogeometric analysis and a moving mesh method is proposed, providing a competitive approach to resolve the efficiency issue. Focusing on the Poisson equation, the implementation of the algorithm is presented in detail, including the numerical discretization of the governing equation using isogeometric analysis, and a mesh redistribution technique developed via harmonic maps. It is found that the isogeometric analysis brings attractive features in the realization of moving mesh method, such as providing an accurate expression for moving direction of mesh nodes, and allowing for more choices to construct monitor functions. Through a series of numerical experiments, the efficiency and effectiveness of the proposed method are successfully verified. Moreover, the potential of the method towards the practical applications is also well presented with the simulation of a helium atom in Kohn-Sham density functional theory.
subject: Articles
  topic: section
subject: Moving mesh method
  topic: keywords
subject: isogeometric analysis
  topic: keywords
subject: mesh redistribution
  topic: keywords
subject: harmonic map
  topic: keywords
creator: Tao Wang
creator: Xucheng Meng
creator: Ran Zhang
creator: Guanghui Hu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-10-22
publication date: 2025-10-22
starting page: 903
ending page: 926
doi: 10.4208/nmtma.OA-2025-0005

Decoupling Analysis for Solving the Diffusive Peterlin Viscoelastic Model
type: item
description: The paper analyzes a system decoupling approach over the entire time domain to solve a diffusive Peterlin viscoelastic model which is a type of non-Newtonian fluids. The main idea is decoupling this non-Newtonian fluid equations into Newtonian fluid equations and second-order quasilinear parabolic equations entirely in the time domain, which can be solved iteratively. Under this approach, existing excellent computational method interfaces for Newtonian fluid equations and parabolic equations can be directly utilized. By employing the weak form of the Peterlin viscoelastic model, a specific decoupling scheme for the continuous-time case is developed. We prove the convergence of this decoupling scheme. Then, for the discrete-time case, a first-order scheme and its iterative error analysis are provided. Numerical experiments validate the theoretical convergence of the algorithm and its dependence on relative parameters.
subject: Articles
  topic: section
subject: Non-Newtonian fluid
  topic: keywords
subject: decoupling approach
  topic: keywords
subject: convergence
  topic: keywords
subject: error analysis
  topic: keywords
creator: Zhen Miao
creator: Yaolin Jiang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-10-22
publication date: 2025-10-22
starting page: 972
ending page: 1002
doi: 10.4208/nmtma.OA-2024-0130

Solving the Helmholtz Equation by Sparse Fundamental Solution Neural Network
type: item
description: A sparse fundamental solution neural network (SFSNN) for solving the Helmholtz equation with constant coefficients and relatively large wave numbers $k$ is proposed. The method combines the strengths of fundamental solution techniques and neural networks by employing a radial basis function neural network, where fundamental solution functions serve as activation functions. Since these functions inherently satisfy the homogeneous Helmholtz equation, SFSNN only requires boundary sampling, significantly accelerating training. To enhance sparsity and generalization, an $ℓ_1$ regularization term of the weights is introduced into the loss function, reformulating the weight optimization as a least absolute shrinkage and selection operator (Lasso) problem. This not only reduces the number of basis functions but also improves the network’s generalization capability. Numerical experiments validate the method’s effectiveness for high-wavenumber isotropic Helmholtz equations in two dimensions and three dimensions. The results reveal that when the analytical solution is a linear combination of fundamental solutions, SFSNN accurately identifies their centers. Otherwise, the number of required basis functions scales as $N =\mathcal{O}(k^{(τ(d−1))}),$ where $τ < 1$ and $d$ is the problem dimension. Moreover, SFSNN has been successfully extended to non-homogeneous and semi-infinite Helmholtz equations, achieving high accuracy. Codes of the examples in this paper are available at https://github.com/wangzhiwensuda/SFSNN-Helmholtz-problem.
subject: Articles
  topic: section
subject: Fundamental solution function
  topic: keywords
subject: radial basis function neural network
  topic: keywords
subject: Helmholtz equation
  topic: keywords
subject: $ℓ_1$ regularization
  topic: keywords
creator: Zhiwen Wang
creator: Minxin Chen
creator: Jingrun Chen
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-10-22
publication date: 2025-10-22
starting page: 845
ending page: 880
doi: 10.4208/nmtma.OA-2025-0033

A Highly Efficient Adaptive Mesh Refinement Algorithm for the 1D Schrödinger-Poisson Problem
type: item
description: The one-dimensional stationary Schrödinger equation, coupled with the transparent boundary conditions and self-consistently linked to the Poisson equation, is a well-established model for describing quantum effects. In this paper, we introduce a general framework for constructing arbitrarily high-order finite difference schemes on arbitrary grids, whether they are uniform or nonuniform, inspired by the analytic discrete transparent boundary conditions [M. Guo et al., arXiv:2411.13175]. To enhance the accuracy of approximations while keeping computational costs low, we develop an optimal mesh refinement strategy that balances the need to resolve intervals with large gradient and high curvature of the potential function. We further propose an adaptive mesh refinement algorithm to solve the 1D Schrödinger-Poisson problem, incorporating a third-order compact finite difference discretization of the Schrödinger-Poisson system based on nonuniform grids, and a given mesh refinement strategy. Numerical experiments on a resonant tunneling diode are conducted to validate the algorithm’s high efficiency and to study the I-V characteristic curve of the device.
subject: Articles
  topic: section
subject: 1D Schrödinger-Poisson problem
  topic: keywords
subject: adaptive mesh refinement
  topic: keywords
subject: transparent boundary condition
  topic: keywords
subject: compact finite difference scheme
  topic: keywords
subject: resonant tunneling diode
  topic: keywords
creator: Meili Guo
creator: Haiyan Jiang
creator: Tiao Lu
creator: Wenqi Yao
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-10-22
publication date: 2025-10-22
starting page: 881
ending page: 902
doi: 10.4208/nmtma.OA-2024-0140

Entirely Exponential-Type Scheme (E2S) for Optimization Problems with Singularly Perturbed ODE Constraints: The Model Problem
type: item
description: We found that no convergence to the correct solution can happen when a popular method is applied to discretize the derivative appearing in the objective function for optimization problems with singularly perturbed ODE constraints. The non-convergence mentioned above can occur even if the error bound of the numerical solution of the state equation has nothing to do with the small parameter. We disclose that the underlying reason for non-convergence to the correct solution is an inaccurate derivative calculation in the objective function for a model problem, which is solvable mathematically. To ensure correct convergence regardless of the small parameter, we propose an entirely exponential-type scheme for solving the optimization problem, in which an exponential-type scheme is used for the derivative in the objective function, together with an exponential-type finite difference scheme for the state equation. Both theoretical analysis and numerical experiments can verify the correct convergence of E2S in solving the singularly perturbed equation-constrained optimization problem.
subject: Articles
  topic: section
subject: Singularly perturbed equation-constrained optimization problem
  topic: keywords
subject: exponential-type finite difference scheme
  topic: keywords
subject: Il’in-Allen-Southwell scheme
  topic: keywords
subject: entirely exponential-type scheme.
  topic: keywords
subject: 34E15
  topic: subjects
subject: 49M41
  topic: subjects
subject: 65L09
  topic: subjects
creator: Mengyu Li
creator: Tiegang Liu
creator: Kui Cao
creator: Chengliang Feng
creator: Bin Zhang
creator: Weixiong Yuan
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-01
publication date: 2025-09-01
starting page: 817
ending page: 844
doi: 10.4208/nmtma.OA-2024-0135

Uncertainty Quantification with Physics-Informed Generative Process Distributions for the Linear Diffusion Equation
type: item
description: We propose a novel diffusion-based generative model for solving linear stochastic diffusion equations, while enabling uncertainty quantification (UQ). By embedding the governing physical laws into the generative process, our approach establishes a mapping between control parameters and solutions across the latent space of the diffusion model. This ensures that the generated solutions satisfy the underlying physical constraints. Additionally, our method overcomes the limitation of conventional diffusion models, which struggle to generate accurate solutions for new control terms, and achieves superior accuracy compared to traditional data-driven operator learning techniques. Furthermore, by sampling different noise realizations and analyzing variations in the generated solutions, we efficiently capture solution diversity, enabling simultaneous prediction and comprehensive UQ. Experimental results demonstrate that our method outperforms deep operator networks and variational inference-based deep operator networks in both accuracy and confidence estimation.
subject: Articles
  topic: section
subject: Diffusion model
  topic: keywords
subject: uncertainty quantification
  topic: keywords
subject: linear diffusion equation
  topic: keywords
subject: physics-informed.
  topic: keywords
subject: 68T07
  topic: subjects
subject: 65C20
  topic: subjects
creator: Wei Zhang
creator: Hui Xie
creator: Tengchao Yu
creator: Heng Yong
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-01
publication date: 2025-09-01
starting page: 794
ending page: 816
doi: 10.4208/nmtma.OA-2025-0024

Approximate Method for Computing Hypersingular Integrals with Oscillatory Kernels
type: item
description: An extrapolation method is proposed for the numerical computation of hypersingular integrals with oscillatory kernels. The oscillatory integral is reformulated as the weighted integral of a Hadamard finite part, which is subsequently approximated using the weighted trapezoidal rule. The asymptotic expansion of the error function is derived, and both the convergence order and the posterior error of the algorithm are analyzed. Numerical examples verify the theoretical results and demonstrate the validity of the proposed method.
subject: Articles
  topic: section
subject: Oscillatory kernel
  topic: keywords
subject: composite rectangle rule
  topic: keywords
subject: hypersingular integral
  topic: keywords
subject: extrapolation method.
  topic: keywords
subject: 65R10
  topic: subjects
subject: 65D30
  topic: subjects
subject: 65D32
  topic: subjects
subject: 42A50
  topic: subjects
subject: 42B20
  topic: subjects
subject: 65B05
  topic: subjects
creator: Yuxin Zhang
creator: Jin Li
creator: Mengyao Yu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-01
publication date: 2025-09-01
starting page: 771
ending page: 793
doi: 10.4208/nmtma.OA-2024-0115

A Modified Version of the PRESB Preconditioner for a Class of Non-Hermitian Complex Systems of Linear Equations
type: item
description: We present a modified version of the PRESB preconditioner for two-by-two block systems of linear equations with the coefficient matrix where $F∈\mathbb{C}^{n×n}$ is Hermitian positive definite and $G ∈\mathbb{C}^{n×n}$ is positive semidefinite. Spectral analysis of the preconditioned matrix is analyzed. In each iteration of a Krylov subspace method, like GMRES, for solving the preconditioned system in conjunction with proposed preconditioner two subsystems with Hermitian positive definite coefficient matrix should be solved which can be accomplished exactly using the Cholesky factorization or inexactly utilizing the conjugate gradient method. Application of the proposed preconditioner to the systems arising from finite element discretization of PDE-constrained optimization problems is presented. Numerical results are given to demonstrate the efficiency of the preconditioner. Our theoretical and numerical results show that the proposed preconditioner is efficient when the norm of the skew-Hermitian part of $G$ is small.
subject: Articles
  topic: section
subject: Complex
  topic: keywords
subject: preconditioning
  topic: keywords
subject: PRESB
  topic: keywords
subject: modified PRESB
  topic: keywords
subject: SPD
  topic: keywords
subject: GMRES
  topic: keywords
subject: CG
  topic: keywords
subject: non-Hermitian.
  topic: keywords
subject: 65F10
  topic: subjects
subject: 65F50
  topic: subjects
creator: Owe Axelsson
creator: Davod Khojasteh Salkuyeh
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-01
publication date: 2025-09-01
starting page: 756
ending page: 770
doi: 10.4208/nmtma.OA-2024-0116

Sparse Wavelet Element Method for Piezoelectric Equations in an Unbounded Domain
type: item
description: This paper is concerned with devising an efficient numerical method for the piezoelectric equations in an unbounded domain, which plays a fundamental role in design and analysis of microacoustic devices with piezoelectric substrate. We make use of the perfectly matched layer method to transform the underlying problem as a surrogate in a bounded domain, which is further solved by a sparse wavelet element method. The latter method can be viewed as a combination of a wavelet element method and a sparse grid method. The numerical results are performed to show the proposed method is very efficient and outperforms the usual finite element method. It can be naturally extended to two/three dimensional problems in an unbounded domain whose boundary consists of line segments or rectangles parallel to coordinate lines or planes.
subject: Articles
  topic: section
subject: Piezoelectric equation
  topic: keywords
subject: sparse grid method
  topic: keywords
subject: sparse tensor product method
  topic: keywords
subject: wavelet element method.
  topic: keywords
subject: 65D40
  topic: subjects
subject: 65T60
  topic: subjects
subject: 65N22
  topic: subjects
creator: Jianguo Huang
creator: Likun Qiu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-01
publication date: 2025-09-01
starting page: 733
ending page: 755
doi: 10.4208/nmtma.OA-2024-0137

Convergence Analysis on a Second-Order, Positive, and Unconditional Energy Dissipative Scheme for the Poisson-Nernst-Planck Equations
type: item
description: In this paper, we propose and analyze a second-order accurate numerical scheme for the Poisson-Nernst-Planck equations. The proposed scheme combines a novel second-order temporal discretization with the centered finite difference method in space. By using a Taylor expansion for the logarithmic term in the chemical potential, this second-order accurate numerical scheme is able to preserve original energy dissipation. Based on the gradient flow formulation, the resulting scheme guarantees several crucial physical properties: mass conservation, positivity of ionic concentration, preservation of original energy dissipation, and steady states preservation. Remarkably, these properties are ensured without any restriction on the time step size. Furthermore, an optimal rate convergence estimate is provided for the proposed numerical scheme. Due to the non-constant mobility and the nonlinear and singular properties of the logarithm terms, a higher-order asymptotic expansion and a combination of rough and refined error estimation techniques are introduced to accomplish this analysis. A few numerical tests are provided to validate our theoretical claims.
subject: Articles
  topic: section
subject: Poisson-Nernst-Planck equation
  topic: keywords
subject: second-order accurate
  topic: keywords
subject: original energy dissipation
  topic: keywords
subject: optimal rate convergence analysis.
  topic: keywords
subject: 35K35
  topic: subjects
subject: 35K61
  topic: subjects
subject: 65M06
  topic: subjects
subject: 65M12
  topic: subjects
subject: 65Z05
  topic: subjects
creator: Jie Ding
creator: Xiang Ji
creator: Ziheng Xu
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-01
publication date: 2025-09-01
starting page: 703
ending page: 732
doi: 10.4208/nmtma.OA-2025-0002

Solving Generalized Tensor Eigenvalue Problem via Spectral Shifted Inverse Power Methods
type: item
description: In this paper, we propose a generalized eigenproblem algorithm, called spectral shifted inverse power method (SIPM) for computing tensor generalized eigenvalue. The SIPM method is developed to overcome the limitations of existing approaches, such as the generalized eigenproblem adaptive power, tensor Noda iteration, modified tensor Noda iteration, and generalized Newton-Noda iteration. These methods often suffer from slow convergence, sensitivity to initial conditions, and computational inefficiency. First, we express SIPM as a fixed point iteration form and establish the connection between the fixed points and generalized eigenvectors of symmetric tensors. Moreover, we introduce a shift power method that further enhances SIPM. Next, we provide a technique for selecting the optimal starting point. Finally, we present numerical results that confirm the effectiveness of our method in solving the tensor generalized eigenvalue problem more efficiently than other methods.
subject: Articles
  topic: section
subject: Spectral shifted inverse power method
  topic: keywords
subject: shifted symmetric higher-order power method
  topic: keywords
subject: $Z$-eigenpair
  topic: keywords
subject: $H$-eigenpair
  topic: keywords
subject: generalized tensor eigenvalue problem.
  topic: keywords
subject: 15A18
  topic: subjects
subject: 15A69
  topic: subjects
creator: Mehri Pakmanesh
creator: Hamidreza Afshin
creator: Masoud Hajarian
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-01
publication date: 2025-09-01
starting page: 680
ending page: 702
doi: 10.4208/nmtma.OA-2024-0139

Implicit-Explicit Time Discretization Schemes for a Class of Semilinear Wave Equations with Nonautonomous Dampings
type: item
description: This paper is concerned about the implicit-explicit (IMEX) methods for a class of dissipative wave systems with time-varying velocity feedbacks and nonlinear potential energies, equipped with different boundary conditions. Firstly, we approximate the problems by using the Hochbruck-Leibold IMEX method, which is a second-order scheme for the problems when the damping terms are time-independent. However, rigors analysis shows that the error rate declines from second to first order due to the nonautonomous dampings. To recover the convergence order, we propose a modified IMEX scheme and apply it to the nonautonomous wave equations with a kinetic boundary condition. Our numerical experiments demonstrate that the modified scheme can not only achieve second-order accuracy but also improve the computational efficiency.
subject: Articles
  topic: section
subject: Semilinear wave equations
  topic: keywords
subject: nonautonomous damping
  topic: keywords
subject: dynamic boundary condition
  topic: keywords
subject: IMEX
  topic: keywords
subject: error analysis.
  topic: keywords
subject: 65M12
  topic: subjects
subject: 65M15
  topic: subjects
subject: 65J08
  topic: subjects
creator: Zhe Jiao
creator: Yaxu Li
creator: Lijing Zhao
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-01
publication date: 2025-09-01
starting page: 650
ending page: 679
doi: 10.4208/nmtma.OA-2024-0131

Variable $s$-Step Technique for New Conjugate Residual Algorithms for Solving Non-Square Linear Systems Arising in Control Problems
type: item
description: This paper introduces two algorithms based on the conjugate residual method and variable $s$-step for solving linear systems with non-square coefficient matrices. The introduced algorithms demonstrate their effectiveness through numerical comparisons with other methods, specifically CGNE and CGNR, to solve non-square linear systems. The approach aims to enhance the efficiency of solving problems related to control systems and imaging underground layers, particularly in the context of seismic tomography.
subject: Articles
  topic: section
subject: Non-square linear system
  topic: keywords
subject: $s$-step algorithm
  topic: keywords
subject: control system
  topic: keywords
subject: least square problem
  topic: keywords
subject: Krylov subspace method.
  topic: keywords
subject: 65F18
  topic: subjects
subject: 65F10
  topic: subjects
subject: 65J10
  topic: subjects
creator: Hojjatollah Shokri Kaveh
creator: Masoud Hajarian
creator: Anthony T. Chronopoulos
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-01
publication date: 2025-09-01
starting page: 598
ending page: 618
doi: 10.4208/nmtma.OA-2024-0109

On the Criteria for Online Updating of Basis Functions for Model Order Reduction Techniques
type: item
description: The model order reduction and data assimilation techniques have been extensively applied to construct surrogate models as core models for digital twins in different industries. However, in most applications, the online phases rely on a single offline basis construction within the parameter domain, which is computationally expensive and often overlooks local parameter variations due to anisotropic behavior. Consequently, this poses challenges for real-time monitoring where parameters vary dynamically. To address this, we introduce a relative deviation score of reduced basis coefficients, represented by $\mathcal{W},$ to assess the reconstruction capability of basis functions under current parameters. Additionally, we provide a quantitative threshold $\overline{\mathcal{W}}$ for $\mathcal{W}.$ Using the reconstruction of a two-dimensional reactor core field distribution as a case study, we demonstrate that if $\mathcal{W}$ remains below the threshold $\overline{\mathcal{W}}=1,$ the average reconstruction error can be maintained within the same order of magnitude as the fundamental error limit $ε (κ ≤ 10).$ Furthermore, we compare $\mathcal{W}$ with the traditional one-time reduced basis selection method and the metric $C_{R2F}$ [F. Bai and Y. Wang, SN Applied Sciences, 2 (2020), p. 2165], confirming the lower computational cost and enhanced robustness of our proposed algorithm.
subject: Articles
  topic: section
subject: POD
  topic: keywords
subject: model order reduction
  topic: keywords
subject: data assimilation
  topic: keywords
subject: parameterized dynamical restriction
  topic: keywords
subject: 2D LWR problem.
  topic: keywords
subject: 35Q92
  topic: subjects
subject: 93B07
  topic: subjects
creator: Han Zhang
creator: Zhiyong Wang
creator: Zihao Mou
creator: Taidong Niu
creator: Helin Gong
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-01
publication date: 2025-09-01
starting page: 619
ending page: 649
doi: 10.4208/nmtma.OA-2024-0127

New Extrapolation Modulus-Based Economical Cascadic Multigrid Method for Variational Inequality Problems with Nonlinear Source Terms
type: item
description: In this paper, we combine a new extrapolation formula with a cascadic multigrid method to solve variational inequality problems with nonlinear source terms. We construct a new extrapolation modulus-based cascadic multigrid method and a new extrapolation modulus-based economical cascadic multigrid method. Based on a new formula for controlling the number of iterations, the convergence is analyzed. Numerical experiments demonstrate the effectiveness of the proposed methods.
subject: Articles
  topic: section
subject: Variational inequality
  topic: keywords
subject: nonlinear source terms
  topic: keywords
subject: new extrapolation modulus-based cascadic multigrid method
  topic: keywords
subject: new extrapolation modulus-based economical cascadic multigrid method
  topic: keywords
subject: iterative step control formula.
  topic: keywords
creator: Xin Lu
creator: Chenliang Li
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-09-01
publication date: 2025-09-01
starting page: 575
ending page: 597
doi: 10.4208/nmtma.OA-2024-129

A Highly Efficient USOR-like Iterative Method for the Stationary Wigner Equation with Scattering
type: item
description: The stationary Wigner equation (SWE) is often used to model the quantum transport in semiconductor devices. The discrete stationary Wigner equation is derived using the first-order upwind scheme and the sinc-Galerkin method [H. Jiang, T. Lu, and W. Zhang, J. Comput. Appl. Math. 409 (2022), 114152]. Based on the successive over relaxation (SOR) iterative method, we develop the basic SOR-like (BSOR-like) and updated SOR-like (USOR-like) iterative methods for solving the discrete stationary Wigner equation efficiently. The main difference between these two iterative methods is that the USOR-like iterative method aims to make more use of the updated components of the Wigner function by splitting the pseudo-differential term. The convergence range of the relaxation parameter in the USOR-like iterative method is numerically investigated. Compared with that of the BSOR-like iterative method, this interval is significantly enlarged. Numerical results have also shown that the USOR-like iterative method is more computationally efficient than the BSOR-like iterative method. As an application, the resonant tunneling effect and the effects of scattering are investigated by simulating a resonant tunneling diode (RTD) using the USOR-like iterative method.
subject: Articles
  topic: section
creator: Yidan Wang
creator: Haiyan Jiang
creator: Tiao Lu
creator: Wenqi Yao
rights: Copyright (c) 2025 Numerical Mathematics: Theory, Methods and Applications
date: 2025-09-30
publication date: 2025-09-30
doi: 10.4208/nmtma.OA-2025-0019

Noise Robust Physics-Informed Generative Adversarial Networks for Solving Stochastic Differential Equations
type: item
description: This paper proposes a class of physics-informed neural networks called noise robust physics-informed generative adversarial networks (NR-PIGANs) to solve stochastic differential equations in the presence of noisy measurements. In these scenarios, while the governing equations are known, only a limited number of sensor measurements of the system parameters are available, and some may contain significant measurement errors. To address this, NR-PIGAN incorporates an additional noise generator with specific distribution constraints into a physics-informed generative adversarial network framework. The noise generator is trained alongside the clean data generators in an end-to-end manner, enabling the model to effectively capture both clean and noisy data distributions under the given physical constraints. Numerical experiments demonstrate that NR-PIGAN excels in handling forward and inverse problems under diverse noise perturbations, and its advantage becomes more pronounced as the noise level increases.
subject: Articles
  topic: section
subject: Stochastic differential equation
  topic: keywords
subject: inverse problem
  topic: keywords
subject: noisy measurement
  topic: keywords
subject: physics-informed
  topic: keywords
subject: generative adversarial network.
  topic: keywords
subject: 35Q68
  topic: subjects
subject: 68T07
  topic: subjects
subject: 68W25
  topic: subjects
creator: Lin Wang
creator: Min Yang
creator: Ruisong Gao
creator: Chuanjun Chen
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-05-16
publication date: 2025-05-16
starting page: 521
ending page: 543
doi: 10.4208/nmtma.OA-2024-0123

On the Recovery of Source Term for Fractional Evolution PDEs by MC-fPINNs
type: item
description: In this paper, we solve the inverse source problem of fractional evolution PDEs by MC-fPINNs. We construct the loss function in terms of the governing equation residual, boundary residual, initial residual and measurement data with noise. Meanwhile, we present a rigorous error analysis of this method. In the experimental section, we present the reconstruction outcomes of the source term for three evolutionary fractional partial differential equations (fPDEs): the evolutionary fractional Laplacian equation, the time-space fractional diffusion equation, and the fractional advection-diffusion equation. These experiments illustrate robust performance of MC-fPINNs in both low-dimensional and high-dimensional scenarios. Our results confirm the effectiveness of MC-fPINNs in solving such inverse source problem, and also provide a theoretical foundation to choose neural networks parameters in this algorithm.
subject: Articles
  topic: section
subject: MC-fPINNs
  topic: keywords
subject: fractional evolution PDEs
  topic: keywords
subject: inverse source problem.
  topic: keywords
subject: 68T07
  topic: subjects
subject: 65M12
  topic: subjects
subject: 62G05
  topic: subjects
creator: Rui Sheng
creator: Jerry Zhijian Yang
creator: Cheng Yuan
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-05-16
publication date: 2025-05-16
starting page: 487
ending page: 520
doi: 10.4208/nmtma.OA-2024-0100

L1 Schemes for Time-Fractional Differential Equations: A Brief Survey and New Development
type: item
description: We summarize the L1 method and its variants for the discretization of the Caputo derivative and develop a weighted L1 (WL1) method. The WL1 method includes the classical L1 method as a special case but improves the L1 method in accuracy while accommodating the intrinsic weak singularity of solutions to fractional differential equations. The WL1 method inherits several properties of the L1 method and the analysis of the new method is presented in a simple way. Compared to several variants of the L1 method, the new method accommodates the intrinsic weak singularity of the time-fractional derivatives in a more flexible way, especially for variable-order fractional differential equation. We also develop the fast WL1 method and apply it to initial and boundary value problems. Numerical simulations and comparisons verify our theoretical analysis and demonstrate the flexibility and efficiency of our method.
subject: Articles
  topic: section
subject: The weighted interpolation
  topic: keywords
subject: fractional initial value problem
  topic: keywords
subject: fractional subdiffusion equation
  topic: keywords
subject: fast convolution method
  topic: keywords
subject: variable-order fractional operator.
  topic: keywords
subject: 26A33
  topic: subjects
subject: 65M06
  topic: subjects
subject: 65M12
  topic: subjects
subject: 65M15
  topic: subjects
subject: 35R11
  topic: subjects
creator: Jingjing Xiao
creator: Yanping Chen
creator: Fanhai Zeng
creator: Zhongqiang Zhang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-05-16
publication date: 2025-05-16
starting page: 544
ending page: 574
doi: 10.4208/nmtma.OA-2024-0085

A Modularized Algorithmic Framework for Interface Related Optimization Problems Using Characteristic Functions
type: item
description: In this paper, we consider the algorithms and convergence for a general optimization problem, which has a wide range of applications in image segmentation, topology optimization, flow network formulation, and surface reconstruction. In particular, the problem focuses on interface related optimization problems where the interface is implicitly described by characteristic functions of the corresponding domains. Under such representation and discretization, the problem is then formulated into a discretized optimization problem where the objective function is concave with respect to characteristic functions and convex with respect to state variables. We show that under such structure, the iterative scheme based on alternative minimization can converge to a local minimizer. Extensive numerical examples are performed to support the theory.
subject: Articles
  topic: section
subject: Interface problems
  topic: keywords
subject: thresholding
  topic: keywords
subject: characteristic function
  topic: keywords
subject: convergence analysis.
  topic: keywords
subject: 65K05
  topic: subjects
subject: 90C26
  topic: subjects
creator: Dong Wang
creator: Shangzhi Zeng
creator: Jin Zhang
creator: Ning Zhang
rights: Copyright (c) 2025 COPYRIGHT: © Global Science Press
date: 2025-05-16
publication date: 2025-05-16
starting page: 437
ending page: 462
doi: 10.4208/nmtma.OA-2024-0082