Numerical Mathematics: Theory, Methods and Applications
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.
A Gradient Recovery Technique for Enhancing the Convergence of Demagnetizing Field Based on PDE Approach
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.
Copyright (c) 2025 Numerical Mathematics: Theory, Methods and Applications
Theoretical Guarantees of Recovery Algorithms for Ternary Sparse Signals
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.
A Space-Time Extrapolation Cascadic Multigrid Method for 2D Linear Parabolic Problems
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.
Two Globally Convergent Hybrid CG Methods with Adaptive Restart Strategy for Riemannian Optimization
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.
Copyright (c) 2025 COPYRIGHT: © Global Science Press
Splitting Schemes for Second-Order Backward Stochastic Differential Equations
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.
Second-order backward stochastic differential equations
A Biquadratic Characteristic Finite Volume Element Method for Time-Fractional Convection-Dominated Equations
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.
Characteristic finite volume element method
Machine Learning-Based Bias Correction for the GEIM Model
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.
generalized empirical interpolation method
Numerical Analysis of a Second Order Crank-Nicolson Mixed Finite Element Method for the Swift-Hohenberg Equation
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.
Crank-Nicolson unconditionally energy stable
Parallel and Scalable Two-Grid Nonlinear Preconditioning Technique for Fully Coupled Solution of Thermal Convective Flow Problems
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.
A Moving Mesh Isogeometric Method Based on Harmonic Maps
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.
Decoupling Analysis for Solving the Diffusive Peterlin Viscoelastic Model
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.
Solving the Helmholtz Equation by Sparse Fundamental Solution Neural Network
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.
A Highly Efficient Adaptive Mesh Refinement Algorithm for the 1D Schrödinger-Poisson Problem
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.
Entirely Exponential-Type Scheme (E2S) for Optimization Problems with Singularly Perturbed ODE Constraints: The Model Problem
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.
Singularly perturbed equation-constrained optimization problem
exponential-type finite difference scheme
Uncertainty Quantification with Physics-Informed Generative Process Distributions for the Linear Diffusion Equation
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.
Approximate Method for Computing Hypersingular Integrals with Oscillatory Kernels
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.
A Modified Version of the PRESB Preconditioner for a Class of Non-Hermitian Complex Systems of Linear Equations
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.
Sparse Wavelet Element Method for Piezoelectric Equations in an Unbounded Domain
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.
Convergence Analysis on a Second-Order, Positive, and Unconditional Energy Dissipative Scheme for the Poisson-Nernst-Planck Equations
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.
Solving Generalized Tensor Eigenvalue Problem via Spectral Shifted Inverse Power Methods
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.
shifted symmetric higher-order power method
Implicit-Explicit Time Discretization Schemes for a Class of Semilinear Wave Equations with Nonautonomous Dampings
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.
Variable $s$-Step Technique for New Conjugate Residual Algorithms for Solving Non-Square Linear Systems Arising in Control Problems
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.
On the Criteria for Online Updating of Basis Functions for Model Order Reduction Techniques
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.
New Extrapolation Modulus-Based Economical Cascadic Multigrid Method for Variational Inequality Problems with Nonlinear Source Terms
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.
new extrapolation modulus-based cascadic multigrid method
new extrapolation modulus-based economical cascadic multigrid method
A Highly Efficient USOR-like Iterative Method for the Stationary Wigner Equation with Scattering
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.
Noise Robust Physics-Informed Generative Adversarial Networks for Solving Stochastic Differential Equations
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.
On the Recovery of Source Term for Fractional Evolution PDEs by MC-fPINNs
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.
L1 Schemes for Time-Fractional Differential Equations: A Brief Survey and New Development
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.
A Modularized Algorithmic Framework for Interface Related Optimization Problems Using Characteristic Functions
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.