CSIAM Transactions on Applied Mathematics
The CSIAM Transactions on Applied Mathematics (CSIAM-AM) is the flagship journal of the Chinese Society for Industrial and Applied Mathematics. It publishes original research articles of high quality on a topic in applied mathematics or computational mathematics or scientific computing. Papers should address mathematically and computationally interesting and challenging problems. Manuscripts applying known methods to previously solved problems will not be accepted.
Structure-Aware Indoor RGB-D SLAM via Manhattan-Constrained 2D Gaussian Splatting
Accurate and layout-consistent reconstruction remains a key challenge in indoor simultaneous localization and mapping (SLAM) due to the prevalence of planar and axis-aligned structures. Traditional visual and RGB-D SLAM methods often suffer from incomplete geometry and weak structural reasoning, while NeRF-based SLAM improves fidelity but is computationally expensive and unsuitable for real-time use. 3D Gaussian splatting offers improved efficiency but lacks structural priors, often resulting in distortions in structured scenes. To address these issues, we propose a structure-aware SLAM framework based on 2D Gaussian splatting, which provides efficient, view-consistent mapping. We introduce a lightweight regularization scheme under the Manhattan-world assumption to align Gaussian orientations and positions with dominant axes, improving layout consistency and geometric fidelity. Extensive experiments on Replica and TUM-RGBD datasets demonstrate that our method consistently outperforms existing SLAM baselines in terms of geometric accuracy and edge preservation across multiple indoor scenes.
Copyright (c) 2025 CSIAM Transactions on Applied Mathematics
Linear Maximum Bound Principle Preserving Finite Difference Schemes for the Convective Allen-Cahn Equation
The convective Allen-Cahn equation generalizes the classical Allen-Cahn equation by introducing an additional convective term associated with a solenoidal velocity field while maintaining the maximum bound principle (MBP). However, developing high-order numerical schemes that are accurate in both time and space and preserve the MBP unconditionally has remained a significant challenge. In this paper, we address this by first defining new auxiliary variables to reformulate the interaction of the velocity field with the phase field. We then transform the convective Allen-Cahn equation into a generalized Fokker-Planck form using an exponential transformation, enabling the development of MBP-preserving linear numerical schemes. Subsequently, we propose first- and second-order in time numerical schemes for the reformulated equations with a second-order quasi-symmetric finite difference discretization in space. In this approach, the auxiliary variables are replaced with known functions related to the velocity field, simplifying the numerical implementation. For the first-order in time scheme, we derive its optimal error estimate and prove its unconditional MBP-preservation. For the second-order in time scheme, we show its MBP-preservation under mild constraints on the mesh and time step sizes. Some numerical experiments in two and three dimensions are also presented to validate the theoretical findings and illustrate the accuracy and efficiency of our proposed schemes.
Generalized Product-Type Variants of RBiCG for Solving Families of Linear Systems
Families of slowly changing nonsingular large sparse linear systems arise frequently in many simulation problems in science and engineering. We consider iterative solution with recycling techniques for a general case where both left-hand sides and right-hand sides of the systems change from one family to the next. We firstly develop a generalized product-type method in the framework of recycling biconjugate gradient method (RBiCG), referred to as RGPBiCG, which can also be considered as a recycling variant of GPBiCG. However, as the same situation in RBiCG stabilized method (RBiCGSTAB), the construction of recycling spaces in RGPBiCG requires expensive computational costs due to invoking other algorithms (like RBiCG) to compute approximate eigenspaces. In order to further reduce such computational costs, we alternatively form the recycling spaces in RGPBiCG with difference vectors of approximate solutions, as employed for loose GMRES (LGMRES), resulting in a more promising algorithm termed as LR-GPBiCG. Numerical experiments on both a set of academic problems and engineering simulation problems demonstrate the efficiency of our proposed algorithms.
difference vectors of approximate solutions
An Accelerated Spectral Perry-Type Conjugate Gradient Projection Method for Unconstrained Nonlinear Equations with Applications
In this paper, we propose an accelerated spectral Perry-type conjugate gradient projection method for solving systems of unconstrained nonlinear equations. The spectral parameter is generated using an accelerated gradient-descent method, which ensures that the proposed search direction always satisfies the sufficient descent condition. By incorporating a self-adaptive strategy into the optimal Perry conjugate parameter, the proposed search direction possesses the trust region property independent of any line search. Additionally, the proposed method incorporates an inertial extrapolation step to enhance computational performance. The global convergence of the proposed method is theoretically established without requiring monotonicity or pseudo-monotonicity of the underlying mapping. Numerical experiments on systems of unconstrained nonlinear equations and image de-blurring problems demonstrate the effectiveness of the proposed method.
Biharmonic Distance Driven Voronoi Diagrams Restricted on Curved Surfaces
Computing surface-based Voronoi diagrams is a fundamental operation in geometry processing, typically relying on either geodesic or straight-line distances as solvers. However, when the input is a 3D model containing thin-plate structures, geodesic distances incur significant computational overhead,whereas straight-line distances can result in ownerless regions. To address this issue, we propose integrating biharmonic embedding distances into the SurfaceVoronoi framework. Specifically, mesh vertices can be embedded into a high-dimensional spectral space, ensuring that the embedding distance closely approximates the straight-line distance between sufficiently close points. In contrast, when points reside on opposite sides of a thin plate, the embedding distance significantly exceeds the straight-line distance, effectively preventing dominance from penetrating through to the opposite side. Our proposed framework offers several advantages: 1) It operates efficiently, as the embedding distance can be rapidly evaluated as a straight-line distance in high-dimensional space. 2) It guarantees the “one site, one region” property, even for models consisting of thinplate structures. 3) It enables high-quality triangulation through iterative repositioning of each site to the centroid of its dominant region. Extensive experiments conducted on benchmark meshes demonstrate these benefits.
A Neural Network Framework for High-Dimensional Dynamic Unbalanced Optimal Transport
In this paper, we introduce a neural network-based method to address the high-dimensional dynamic unbalanced optimal transport (UOT) problem. Dynamic UOT focuses on the optimal transportation between two densities with unequal total mass, however, it introduces additional complexities compared to the traditional dynamic optimal transport problem. To efficiently solve the dynamic UOT problem in high-dimensional space, we first relax the original problem by using the generalized Kullback-Leibler divergence to constrain the terminal density. Next, we adopt the Lagrangian discretization to address the unbalanced continuity equation and apply the Monte Carlo method to approximate the high-dimensional spatial integrals. Moreover, a carefully designed neural network is introduced for modeling the velocity field and source function. Numerous experiments demonstrate that the proposed framework performs excellently in high-dimensional cases. Additionally, this method can be easily extended to more general applications, such as crowd motion problem.
Smooth Bijective Projection of Polygonal Meshes via a Cubic B-Spline Shell
Polygonal meshes are a fundamental surface representation, yet their resolution can vary significantly. Establishing a smooth, bijective projection between meshes of different resolutions is crucial for consistent attribute transfer but becomes challenging when handling sharp bends and complex geometric features. This paper presents a novel approach to address this challenge by implicitly representing the shell enclosing two polygonal meshes using a cubic trivariate B-spline function, where the inner and outer bounding surfaces are formulated as level sets of a single cubic B-spline function. Our method enforces the bijective projection requirements on the cubic B-spline function, ensuring that the resulting gradient field naturally defines a robust bijective projection. Leveraging the favorable properties of cubic B-spline functions – namely, 1) sufficient smoothness while maintaining expressive representation, and 2) computational efficiency and ease of implementation, our approach efficiently computes a smooth and bijective projection even for challenging cases. Compared to existing shell-based bijective projection methods, our method consistently produces valid bijective projections, even in complex scenarios, outperforming state-of-the-art techniques. We further demonstrate its effectiveness in robust attribute transfer and precision-controlled shape manipulation.
We provide a sufficient and necessary condition in terms of the stoichiometric coefficients for a bi-reaction network to admit multistability. Also, this result completely characterizes the bi-reaction networks according to if they admit multistability.
Data- and Mechanism-Driven Hybrid Computing: A New Paradigm for Scientific and Engineering Computation
Data- and mechanism-driven hybrid computing refers to the integration of traditional mechanism-based computing with data-driven methods. In this article, we present three typical patterns of this emerging paradigm: (1) mechanism-driven model optimization via data-driven refinement, (2) data-driven model construction with physical constraints, and (3) alternating optimization of mechanism-driven and data-driven models. We present several concrete examples to illustrate how hybrid computing improves accuracy, efficiency, and robustness across a variety of computational tasks.
Superconvergence Points of Several Polynomial and Nonpolynomial Hermite Spectral Interpolations
In this paper, we analyze the superconvergence properties for spectral interpolations by Hermite polynomials and mapped Hermite functions. At the superconvergence points, the $(N−k)$-th term in the Hermite spectral interpolation remainder for the $(k+1)$-th derivatives vanish. To solve multi-point weakly singular nonlocal problems, we previously introduced mapped Hermite functions (MHFs), which are constructed by applying a mapping to the Hermite polynomials. We prove that the superconvergence points of the spectral interpolations based on MHFs for the $(k+1)$-th derivatives are the zero points of the $(N−k)$-th term. Additionally, due to the rapid growth of the logarithmic function at the endpoints 0 and 1, we further propose generalized mapped Hermite functions (GMHFs). We develop basic approximation theory for these new orthogonal functions and prove the projection error and interpolation error in the $L^2$-weighted space using the pseudo-derivative. We demonstrate that the superconvergence points of the spectral interpolations based on both MHFs and GMHFs for the $(k+1)$-th derivative are the zero points of the $(N−k)$-th term. Numerical experiments confirm our theoretical results.
An Efficient Iterative Convolution-Thresholding Method for Image Inpainting
Variational methods have been developed for image inpainting, which involve minimizing an objective functional consisting of the regularization term and the fidelity term. The fidelity term controls the consistency of the restored region with the original image, while the regularization term smooths the boundary of the region. In this paper, we develop an efficient iterative convolution-thresholding method to solve variational approach-based image inpainting problems. In the proposed method, the region is represented by its indicator function, and the regularization term is approximated by the heat kernel convolution with the indicator function. Based on this approximation, we derive an efficient iterative method to update the indicator function only within the damaged region by alternating the convolution and thresholding steps, relying on a relaxation and linearization procedure. Extensive numerical experiments demonstrate the simplicity and efficiency of the proposed method.
Iterative convolution-thresholding method
Double-Domain Driven Unet for Selective Segmentation of Medical Image
Selective segmentation has been a notable increase in interest in the use of interactive deep learning-based methods. However, developing an efficient and accurate segmentation method for medical applications remains a formidable challenge, primarily due to the inherent complexity and diversity of medical image structures. To address these challenges, we introduce a novel deep learning approach, termed the double-domain driven Unet method (DDUM), designed for the selective segmentation of medical images. Our approach utilizes a threshold geodesic distance in conjunction with the original images as input to construct a parallel Unet architecture that captures information from both the image domain and the geodesic distance domain. To further enhance the accuracy and efficacy of the segmentation process, we employ soft threshold dynamics as a replacement for the sigmoid activation function in the final layer. The efficacy of our proposed DDUM is substantiated through extensive experiments conducted on multiple medical image datasets. In particular, the DDUM method exhibits exceptional performance in terms of both segmentation accuracy and robustness.
Interface Laplace Learning: Learnable Interface Term Helps Semi-Supervised Learning
We introduce a novel framework, called Interface Laplace Learning, for graph-based semi-supervised learning. Motivated by the observation that an interface should exist between different classes where the function value is non-smooth, we propose a Laplace learning model that incorporates an interface term. This model challenges the long-standing assumption that functions are smooth at all unlabeled points. In the proposed approach, we add an interface term to the Laplace learning model at the interface positions. We provide a practical algorithm to approximate the interface positions using k-hop neighborhood indices, and to learn the interface term from labeled data without artificial design. Our method is efficient and effective, and we present extensive experiments demonstrating that Interface Laplace Learning achieves better performance than other recent semi-supervised learning approaches at extremely low label rates on the MNIST, FashionMNIST, and CIFAR-10 datasets.
Error Estimates of Operator Splitting Spectral Method for Semiclassical Sub-Diffusive Gross-Pitaevskii Equation
Nonlinear Gross-Pitaevskii-type models are frequently seen in the fields of Bose-Einstein condensation and quantum mechanics. We derive error estimates for the Lie-Trotter operator splitting spectral method for semiclassical sub-diffusive Gross-Pitaevskii equation in the unbounded domain or with the periodic boundary condition. After establishing a priori estimates for the analytic solution in fractional Sobolev space, the local error estimates for the Lie-Trotter splitting operator method are derived. The related estimates for the Lie commutator of nonlocal linear operator and nonlinear operator play key roles in deriving the local error estimates. We then obtain the global error bounds for the fully discrete scheme based on the space approximation with mapped Chebyshev spectral-Galerkin methods in the case of the unbounded domain and with Fourier spectral methods in the case of the periodic boundary condition. Especially, their convergence orders with respect to the small (scaled) Planck constant $ε$ are obtained for the first time under the framework of Wentzel-Kramers-Brillouin analysis. Numerical experiments verify and complement our theoretical results.
mapped Chebyshev spectral-Galerkin methods
pETNNs: Partial Evolutionary Tensor Neural Networks for Solving Time-Dependent Partial Differential Equations
We present partial evolutionary tensor neural networks (pETNNs), a novel approach for solving time-dependent partial differential equations with high accuracy and capable of handling high-dimensional problems. Our architecture incorporates tensor neural networks and evolutionary parametric approximation. A posteriori error bound is proposed to support the extrapolation capabilities. In numerical implementations, we adopt a partial update strategy to achieve a significant reduction in computational cost while maintaining precision and robustness. Notably, as a low-rank approximation method of complex dynamical systems, pETNNs enhance the accuracy of evolutionary deep neural networks and empower computational abilities to address high-dimensional problems. Numerical experiments demonstrate the superior performance of the pETNNs in solving complex time-dependent equations, including the incompressible Navier-Stokes equations, high-dimensional heat equations, high-dimensional transport equations, and dispersive equations of higher-order derivatives.
Time-dependent partial differential equations
Copyright (c) 2025 COPYRIGHT: © Global Science Press
A Mathematical Analysis for the Dynamics of Multiple Languages
In many regions of the world, languages coexist in daily life, but often one tongue increases its use at the expense of another. In the present paper, we build a large compartmental system of differential equations that meets the situation of two “prestigious” tongues and many local languages, whose use is reduced by social interaction. The focus is on the preferred language in social relationships for communicating, rather than mere knowledge. We aim at stating and proving theorems on the qualitative behavior of the system. Numerical simulations illustrate the results, giving rise to distinct dynamics.
Global Solvability in a Two-Species Keller-Segel-Navier-Stokes System with Sub-Logistic Source
This paper is concerned with a two-species Keller-Segel-Navier-Stokes model with sub-logistic source in a bounded domain with smooth boundary under noflux/no-flux/no-flux/Dirichlet boundary conditions. For a large class of cell kinetics including sub-logistic degradation, it is shown that under an explicit condition involving the chemotactic strength and initial mass of cells, the two-dimensional Keller-Segel-Navier-Stokes problem possesses a global and bounded classical solution. In the case with arbitrary superlinear logistic degradation, it is proved that for all suitably regular initial data, the two-dimensional Keller-Segel-Navier-Stokes problem has at least one globally defined solution in an appropriate generalized sense. These results improves and extends the previously known ones.
A Sharp Uniform-in-Time Error Estimate for Stochastic Gradient Langevin Dynamics
Abstract. We establish a sharp uniform-in-time error estimate for the stochastic gradient Langevin dynamics (SGLD), which is a widely-used sampling algorithm. Under mild assumptions, we obtain a uniform-in-time $\mathcal{O}(η^2)$ bound for the Kullback-Leibler divergence between the SGLD iteration and the Langevin diffusion, where $η$ is the step size (or learning rate). Our analysis is also valid for varying step sizes. Consequently, we are able to derive an $\mathcal{O}(\eta)$ bound for the distance between the invariant measures of the SGLD iteration and the Langevin diffusion, in terms of Wasserstein or total variation distances. Our result can be viewed as a significant improvement compared with existing analysis for SGLD in related literature.
An Efficient Iteration Based on Reduced Basis Method for Time-Dependent Problems with Random Inputs
In this paper, we propose an efficient iterative method called RB-iteration, based on reduced basis (RB) techniques, for addressing time-dependent problems with random input parameters. This method reformulates the original model such that the left-hand side is parameter-independent, while the right-hand side remains parameterdependent, facilitating the application of fixed-point iteration for solving the system. High-fidelity simulations for time-dependent problems often demand considerable computational resources, rendering them impractical for many applications. RB-iteration enhances computational efficiency by executing iterations in a reduced order space. This approach results in significant reductions in computational costs. We conduct a rigorous convergence analysis and present detailed numerical experiments for the RB-iteration method. Our results clearly demonstrate that RB-iteration achieves superior efficiency compared to the direct fixed-point iteration method and provides enhanced accuracy relative to the classical proper orthogonal decomposition (POD) greedy method.
Two Methods Addressing Variable-Exponent Fractional Initial and Boundary Value Problems and Abel Integral Equation
Variable-exponent fractional models attract increasing attentions in various applications, while rigorous mathematical and numerical analysis for typical models remains largely untreated. This work provides general tools to address these models. Specifically, we first develop a convolution method to study the well-posedness, regularity, an inverse problem and numerical approximation for the subdiffusion of variable exponent. For models such as the variable-exponent two-sided space-fractional boundary value problem (including the variable-exponent fractional Laplacian equation as a special case) and the distributed variable-exponent model, for which the convolution method does not apply, we develop a perturbation method to prove their well-posedness. The relations between the convolution method and the perturbation method are discussed, and we further apply the latter to prove the well-posedness of the variable-exponent Abel integral equation and discuss the constraint on the data under different initial values of variable exponent.
Convergent Finite Elements on Arbitrary Meshes, the WG Method
On meshes with the maximum angle condition violated, the standard conforming, nonconforming, and discontinuous Galerkin finite elements do not converge to the true solution when the mesh size goes to zero. It is shown that one type of weak Galerkin finite element method converges on triangular and tetrahedral meshes violating the maximum angle condition, i.e. on arbitrary meshes. Numerical tests confirm the theory.
Benchmarking Density Functional Theory for Noble Metal Hydrides: A High-Fidelity PES and Vibrational Analysis of the ${\rm AgAuH}^−$ Anion
The accurate description of noble metal hydrides remains a fundamental challenge for electronic structure theory, especially in systems involving heavy elements where relativistic effects and electron correlation are significant. In this study, we present a high-accuracy potential energy surface (PES) for the ${\rm AgAuH}^−$ anion, constructed from 3,595 UCCSD(T)-F12a energy points and fitted using a feedforward neural network with a root mean square error of 0.21 meV. The PES captures the entire configuration space, including linear and bent minima, transition-state-like structures, and dissociation pathways. Quantum vibrational bound states were computed using time-independent quantum dynamics, enabling detailed mode assignments. The high-fidelity PES and vibrational dataset were used to benchmark some widely employed density functional theory (DFT) methods, B3LYP, ωB97XD, XYG3, and XYGJ-OS. Among these, XYGJ-OS provided the best agreement with the reference data in terms of equilibrium geometries and vibrational frequencies. This study provides a robust benchmark for method development and validation in metal-containing systems and highlights the importance of using high-level reference data when modeling complex coinage-metal hydrides.
$ density functional theory benchmarking
high-level ab initio potential energy surface
Construction and Analysis for Adams Explicit Discretization of High-Index Saddle Dynamics
Saddle points largely exist in complex systems and play important roles in various scientific problems. High-index saddle dynamics (HiSD) is an efficient method for computing any-index saddle points and constructing solution landscape. In this paper, we propose a two-step Adams explicit scheme for HiSD and analyze its error estimate versus time step. Through careful argumentation and overcoming the difficulties caused by nonlinear coupling and orthogonalisation, we prove that the two-step Adams explicit scheme has second-order accuracy. The theoretical results are further verified by two numerical experiments.
Well-Posedness and Regularity Analyses for Nonlocal Nonautonomous System
The time-space nonlocal evolution equations are powerful implementation for modeling anomalous diffusion. In this research, we study the nonlocal nonautonomous reaction-diffusion equation where $\chi$ is a Lusin space, $∂^w_t$ is a generalized time fractional derivative, $κ$ is a bounded reaction rate, and $\mathcal{L}$ is an infinitesimal generator in terms of semigroup induced by a symmetric Markov process $X.$ We show that the stochastic representation $u(t,x)$ defined by is the unique mild as well as weak solution. By further analysis, one can get that the above stochastic representation is also the unique strong solution, and the higher spatial and temporal regularity are obtained. In some particular cases, the corresponding dynamical behaviors are displayed.
Global-in-Time Energy Stability for a General Class of Stabilization Single-Step Schemes Applied to the Swift-Hohenberg Equation
The linear stabilization approach is well-known for facilitating the use of large time steps in solving gradient flows while maintaining stability. However, the up-to-date analysis of energy stability relies on either a global Lipschitz nonlinearity or an $ℓ^∞$ bound assumption of numerical solutions. Considering the Swift-Hohenberg equation that lacks a global Lipschitz nonlinearity, we develop a unified framework to analyze the energy stability and characterize the stabilization size for a class of single-step schemes employing spatial Fourier pseudo-spectral discretization. First, assuming that all stage solutions are bounded in the $ℓ^∞$ norm, we illustrate that the energy obtained from a single-step scheme with non-negative energy-stability-preserving coefficient is unconditionally dissipative, as long as a sufficiently large stabilization parameter is employed. To justify the $ℓ^∞$ bound assumption of solutions, we use the third-order exponential-time-differencing Runge-Kutta scheme as a case study to establish a uniform-in-time discrete $H^2$ bound for stage solutions under an $\mathcal{O}(1)$ time step constraint. This leads to a uniform $ℓ^∞$ bound of stage solutions through discrete Sobolev embedding. Consequently, we achieve a stabilization parameter of $\mathcal{O}(1),$ which is independent of the time step, thereby ensuring the energy stability. The global-in-time energy stability analysis and characterization of the stabilization parameter represent significant advancements for general single-step schemes applied to a gradient flow without the global Lipschitz continuity.
New Finite Volume Element Schemes Based on a Two-Layer Dual Strategy
A two-layer dual strategy is proposed in this work to construct a new family of high-order finite volume element (FVE-2L) schemes that can avoid main common drawbacks of the existing high-order finite volume element (FVE) schemes. The existing high-order FVE schemes are complicated to construct since the number of the dual elements in each primary element used in their construction increases with a rate $\mathcal{O}((k+1)^2),$ where $k$ is the order of the scheme. Moreover, all $k$-th-order FVE schemes require a higher regularity $H^{k+2}$ than the approximation theory for the $L^2$ theory. Furthermore, all FVE schemes lose local conservation properties over boundary dual elements when dealing with Dirichlet boundary conditions. The proposed FVE-2L schemes has a much simpler construction since they have a fixed number (four) of dual elements in each primary element. They also reduce the regularity requirement for the $L^2$ theory to $H^{k+1}$ and preserve the local conservation law on all dual elements of the second dual layer for both flux and equation forms. Their stability and $H^1$ and $L^2$ convergence are proved. Numerical results are presented to illustrate the convergence and conservation properties of the FVE-2L schemes. Moreover, the condition number of the stiffness matrix of the FVE-2L schemes for the Laplacian operator is shown to have the same growth rate as those for the existing FVE and finite element schemes.
Highly Efficient Gauss’s Law-Preserving Spectral Algorithms for Maxwell’s Double-Curl Source and Eigenvalue Problems Based on Eigen-Decomposition
In this paper, we present the Gauss’s law-preserving spectral methods and their efficient solution algorithms for curl-curl source and eigenvalue problems arising from Maxwell’s equations. Arbitrary order $\mathbf{H}{\rm (curl)}$-conforming spectral basis functions in two and three dimensions are firstly proposed using compact combination of Legendre polynomials. A mixed formulation involving a Lagrange multiplier is then adopted to preserve the Gauss’s law in the weak sense. To overcome the bottleneck of computational efficiency caused by the saddle-point nature of the mixed scheme, we present highly efficient algorithms based on reordering and decoupling of the linear system and numerical eigen-decomposition of 1D mass matrix. The proposed solution algorithms are direct methods requiring only several matrix-matrix or matrix-tensor products of $N$-by-$N$ matrices, where $N$ is the highest polynomial order in each direction. Compared with other direct methods, the computational complexities are reduced from $\mathcal{O}(N^6)$ and $\mathcal{O}(N^9)$ to $\mathcal{O}(N^{log_2 7})$ and $\mathcal{O}(N^{1+log_2 7})$ with small and constant pre-factors for 2D and 3D cases, respectively. Moreover, these algorithms strictly obey the Helmholtz-Hodge decomposition, thus totally eliminate the spurious eigen-modes of non-physical zero eigenvalues for convex domains. Ample numerical examples for solving Maxwell’s source and eigenvalue problems are presented to demonstrate the accuracy and efficiency of the proposed methods.
$\mathbf{H}$(${\rm curl}^2$) Conforming Element for Maxwell’s Transmission Eigenvalue Problem Using Fixed-Point Approach
Using newly developed $\mathbf{H}$(${\rm curl}^2$) conforming elements, we solve the Maxwell’s transmission eigenvalue problem. Both real and complex eigenvalues are considered. Based on the fixed-point weak formulation with reasonable assumptions, the optimal error estimates for numerical eigenvalues and eigenfunctions (in the $\mathbf{H}$(${\rm curl}^2$)-norm and $\mathbf{H}{\rm (curl)}$-semi-norm) are established. Numerical experiments are performed to verify the theoretical assumptions and confirm our theoretical analysis.
Efficient Numerical Methods for Computing Stationary States of Spherical Landau-Brazovskii Model
In this paper, we develop a set of efficient methods to compute stationary states of the spherical Landau-Brazovskii (LB) model in a discretization-then-optimization way. First, we discretize the spherical LB energy functional into a finite-dimensional energy function by the spherical harmonic expansion. Then five optimization methods are developed to compute stationary states of the discretized energy function, including the accelerated adaptive Bregman proximal gradient, Nesterov, adaptive Nesterov, adaptive nonlinear conjugate gradient and adaptive gradient descent methods. To speed up the convergence, we propose a principal mode analysis (PMA) method to estimate good initial configurations and sphere radius. The PMA method also reveals the relationship between the optimal sphere radius and the dominant degree of spherical harmonics. Numerical experiments show that our approaches significantly reduce the number of iterations and the computational time.
A Hue-Preserving Color Image Restoration Model Using Saturation-Value Total Variation
In this paper, we propose and develop a novel hue-preserving color image restoration model. In the proposed model, saturation-value total variation is used as the regularization term. In order to reduce hue loss during regularization, the hue-preserving technique is incorporated into the proposed energy functional. We then rewrite the proposed model equivalently into a three-block separable convex version and apply an alternating direction method of multipliers based algorithm to solve the proposed equivalent model numerically. We also give the convergence analysis of the proposed algorithm. Numerical examples are presented to demonstrate that the performance of the proposed restoration model is better than that of other testing methods in terms of visual quality and some criteria such as peak signal-to-noise ratio, structure similarity, and spatial-CIELAB color error.
alternating direction method of multipliers