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    A TWO-GRID METHOD FOR THE C 0 INTERIOR PENALTY DISCRETIZATION OF THE MONGE-AMPERE EQUATION
    Gerard Awanou, Hengguang Li, Eric Malitz
    Journal of Computational Mathematics    2020, 38 (4): 547-564.   DOI: 10.4208/jcm.1901-m2018-0039
    Abstract117)      PDF      
    The purpose of this paper is to analyze an efficient method for the solution of the nonlinear system resulting from the discretization of the elliptic Monge-Ampère equation by a C 0 interior penalty method with Lagrange finite elements. We consider the two-grid method for nonlinear equations which consists in solving the discrete nonlinear system on a coarse mesh and using that solution as initial guess for one iteration of Newton's method on a finer mesh. Thus both steps are inexpensive. We give quasi-optimal W 1,∞ error estimates for the discretization and estimate the difference between the interior penalty solution and the two-grid numerical solution. Numerical experiments confirm the computational efficiency of the approach compared to Newton's method on the fine mesh.
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    ORDER REDUCED METHODS FOR QUAD-CURL EQUATIONS WITH NAVIER TYPE BOUNDARY CONDITIONS
    Weifeng Zhang, Shuo Zhang
    Journal of Computational Mathematics    2020, 38 (4): 565-579.   DOI: 10.4208/jcm.1901-m2018-0150
    Abstract77)      PDF      
    Quad-curl equations with Navier type boundary conditions are studied in this paper. Stable order reduced formulations equivalent to the model problems are presented, and finite element discretizations are designed. Optimal convergence rates are proved.
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    ON NEW STRATEGIES TO CONTROL THE ACCURACY OF WENO ALGORITHM CLOSE TO DISCONTINUITIES II: CELL AVERAGES AND MULTIRESOLUTION
    Sergio Amat, Juan Ruiz, Chi-Wang Shu
    Journal of Computational Mathematics    2020, 38 (4): 661-682.   DOI: 10.4208/jcm.1903-m2019-0125
    Abstract77)      PDF      
    This paper is the second part of the article and is devoted to the construction and analysis of new non-linear optimal weights for WENO interpolation capable of rising the order of accuracy close to discontinuities for data discretized in the cell averages. Thus, now we are interested in analyze the capabilities of the new algorithm when working with functions belonging to the subspace L 1L 2 and that, consequently, are piecewise smooth and can present jump discontinuities. The new non-linear optimal weights are redesigned in a way that leads to optimal theoretical accuracy close to the discontinuities and at smooth zones. We will present the new algorithm for the approximation case and we will analyze its accuracy. Then we will explain how to use the new algorithm in multiresolution applications for univariate and bivariate functions. The numerical results confirm the theoretical proofs presented.
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    IMPLICITY LINEAR COLLOCATION METHOD AND ITERATED IMPLICITY LINEAR COLLOCATION METHOD FOR THE NUMERICAL SOLUTION OF HAMMERSTEIN FREDHOLM INTEGRAL EQUATIONS ON 2D IRREGULAR DOMAINS
    H. Laeli Dastjerdi, M. Nili Ahmadabadi
    Journal of Computational Mathematics    2020, 38 (4): 624-637.   DOI: 10.4208/jcm.1903-m2017-0206
    Abstract75)      PDF      
    In this work, we adapt and compare implicity linear collocation method and iterated implicity linear collocation method for solving nonlinear two dimensional Fredholm integral equations of Hammerstein type using IMQ-RBFs on a non-rectangular domain. IMQs show to be the most promising RBFs for this kind of equations. The proposed methods are mesh-free and they are independent of the geometry of domain. Convergence analysis of the proposed methods together with some benchmark examples are provided which support their reliability and numerical stability.
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    SOLVING SYSTEMS OF QUADRATIC EQUATIONS VIA EXPONENTIAL-TYPE GRADIENT DESCENT ALGORITHM
    Meng Huang, Zhiqiang Xu
    Journal of Computational Mathematics    2020, 38 (4): 638-660.   DOI: 10.4208/jcm.1902-m2018-0109
    Abstract66)      PDF      
    We consider the rank minimization problem from quadratic measurements, i.e., recovering a rank r matrix X ∈ R n×r from m scalar measurements y #em/em# =a#em/em#?XX?a#em/em#, a#em/em# ∈ Rn, #em/em#=1, …, m. Such problem arises in a variety of applications such as quadratic regression and quantum state tomography. We present a novel algorithm, which is termed exponential-type gradient descent algorithm, to minimize a non-convex objective function f(U)=1/4m Σ#em/em#=1m (y#em/em#-a#em/em#?UU?a#em/em#)2. This algorithm starts with a careful initialization, and then refines this initial guess by iteratively applying exponential-type gradient descent. Particularly, we can obtain a good initial guess of X as long as the number of Gaussian random measurements is O(nr), and our iteration algorithm can converge linearly to the true X (up to an orthogonal matrix) with m=O (nr log(cr)) Gaussian random measurements.
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    THE SHIFTED-INVERSE POWER WEAK GALERKIN METHOD FOR EIGENVALUE PROBLEMS
    Qilong Zhai, Xiaozhe Hu, Ran Zhang
    Journal of Computational Mathematics    2020, 38 (4): 606-623.   DOI: 10.4208/jcm.1903-m2018-0101
    Abstract57)      PDF      
    This paper proposes and analyzes a new weak Galerkin method for the eigenvalue problem by using the shifted-inverse power technique. A high order lower bound can be obtained at a relatively low cost via the proposed method. The error estimates for both eigenvalue and eigenfunction are provided and asymptotic lower bounds are shown as well under some conditions. Numerical examples are presented to validate the theoretical analysis.
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    RELU DEEP NEURAL NETWORKS AND LINEAR FINITE ELEMENTS
    Juncai He, Lin Li, Jinchao Xu, Chunyue Zheng
    Journal of Computational Mathematics    2020, 38 (3): 502-527.   DOI: 10.4208/jcm.1901-m2018-0160
    Abstract51)      PDF      
    In this paper, we investigate the relationship between deep neural networks (DNN) with rectified linear unit (ReLU) function as the activation function and continuous piecewise linear (CPWL) functions, especially CPWL functions from the simplicial linear finite element method (FEM). We first consider the special case of FEM. By exploring the DNN representation of its nodal basis functions, we present a ReLU DNN representation of CPWL in FEM. We theoretically establish that at least 2 hidden layers are needed in a ReLU DNN to represent any linear finite element functions in Ω⊆R d when d ≥ 2. Consequently, for d=2, 3 which are often encountered in scientific and engineering computing, the minimal number of two hidden layers are necessary and sufficient for any CPWL function to be represented by a ReLU DNN. Then we include a detailed account on how a general CPWL in R d can be represented by a ReLU DNN with at most 「log 2( d + 1)」 hidden layers and we also give an estimation of the number of neurons in DNN that are needed in such a representation. Furthermore, using the relationship between DNN and FEM, we theoretically argue that a special class of DNN models with low bit-width are still expected to have an adequate representation power in applications. Finally, as a proof of concept, we present some numerical results for using ReLU DNNs to solve a two point boundary problem to demonstrate the potential of applying DNN for numerical solution of partial differential equations.
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    HIGH ORDER FINITE DIFFERENCE/SPECTRAL METHODS TO A WATER WAVE MODEL WITH NONLOCAL VISCOSITY
    Mohammad Tanzil Hasan, Chuanju Xu
    Journal of Computational Mathematics    2020, 38 (4): 580-605.   DOI: 10.4208/jcm.1902-m2017-0280
    Abstract49)      PDF      
    In this paper, efficient numerical scheme is proposed for solving the water wave model with nonlocal viscous term that describe the propagation of surface water wave. By using the Caputo fractional derivative definition to approximate the nonlocal fractional operator, finite difference method in time and spectral method in space are constructed for the considered model. The proposed method employs known 5/2 order scheme for fractional derivative and a mixed linearization for the nonlinear term. The analysis shows that the proposed numerical scheme is unconditionally stable and error estimates are provided to predict that the second order backward differentiation plus 5/2 order scheme converges with order 2 in time, and spectral accuracy in space. Several numerical results are provided to verify the efficiency and accuracy of our theoretical claims. Finally, the decay rate of solutions are investigated.
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    ENHANCED BLOCK-SPARSE SIGNAL RECOVERY PERFORMANCE VIA TRUNCATED ? 2/ ? 1-2 MINIMIZATION
    Weichao Kong, Jianjun Wang, Wendong Wang, Feng Zhang
    Journal of Computational Mathematics    2020, 38 (3): 437-451.   DOI: 10.4208/jcm.1811-m2017-0275
    Abstract45)      PDF      
    In this paper, we investigate truncated ? 2/ ? 1-2 minimization and its associated alternating direction method of multipliers (ADMM) algorithm for recovering the block sparse signals. Based on the block restricted isometry property (Block-RIP), a theoretical analysis is presented to guarantee the validity of proposed method. Our theoretical results not only show a less error upper bound, but also promote the former recovery condition of truncated ? 1-2 method for sparse signal recovery. Besides, the algorithm has been compared with some state-of-the-art algorithms and numerical experiments have shown excellent performances on recovering the block sparse signals.
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    IMAGE DENOISING VIA TIME-DELAY REGULARIZATION COUPLED NONLINEAR DIFFUSION EQUATIONS
    Qianting Ma
    Journal of Computational Mathematics    2020, 38 (3): 417-436.   DOI: 10.4208/jcm.1811-m2016-0763
    Abstract43)      PDF      
    A novel nonlinear anisotropic diffusion model is proposed for image denoising which can be viewed as a novel regularized model that preserves the cherished features of PeronaMalik to some extent. It is characterized by a local dependence in the diffusivity which manifests itself through the presence of p( x)-Laplacian and time-delay regularization. The proposed model offers a new nonlinear anisotropic diffusion which makes it possible to effectively enhance the denoising capability and preserve the details while avoiding artifacts. Accordingly, the restored image is very clear and becomes more distinguishable. By Galerkin’s method, we establish the well-posedness in the weak setting. Numerical experiments illustrate that the proposed model appears to be overwhelmingly competitive in restoring the images corrupted by Gaussian noise.
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    A NEW STABILIZED FINITE ELEMENT METHOD FOR SOLVING TRANSIENT NAVIER-STOKES EQUATIONS WITH HIGH REYNOLDS NUMBER
    Chunmei Xie, Minfu Feng
    Journal of Computational Mathematics    2020, 38 (3): 395-416.   DOI: 10.4208/jcm.1810-m2018-0096
    Abstract42)      PDF      
    In this paper, we present a new stabilized finite element method for transient NavierStokes equations with high Reynolds number based on the projection of the velocity and pressure. We use Taylor-Hood elements and the equal order elements in space and second order difference in time to get the fully discrete scheme. The scheme is proven to possess the absolute stability and the optimal error estimates. Numerical experiments show that our method is effective for transient Navier-Stokes equations with high Reynolds number and the results are in good agreement with the value of subgrid-scale eddy viscosity methods, Petro-Galerkin finite element method and streamline diffusion method.
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    EFFICIENT LINEAR SCHEMES WITH UNCONDITIONAL ENERGY STABILITY FOR THE PHASE FIELD MODEL OF SOLID-STATE DEWETTING PROBLEMS
    Jie Chen, Zhengkang He, Shuyu Sun, Shimin Guo, Zhangxin Chen
    Journal of Computational Mathematics    2020, 38 (3): 452-468.   DOI: 10.4208/jcm.1812-m2018-0058
    Abstract36)      PDF      
    In this paper, we study linearly first and second order in time, uniquely solvable and unconditionally energy stable numerical schemes to approximate the phase field model of solid-state dewetting problems based on the novel “scalar auxiliary variable” (SAV) approach, a new developed efficient and accurate method for a large class of gradient flows. The schemes are based on the first order Euler method and the second order backward differential formulas (BDF2) for time discretization, and finite element methods for space discretization. The proposed schemes are proved to be unconditionally stable and the discrete equations are uniquely solvable for all time steps. Various numerical experiments are presented to validate the stability and accuracy of the proposed schemes.
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    A FULLY DISCRETE IMPLICIT-EXPLICIT FINITE ELEMENT METHOD FOR SOLVING THE FITZHUGH-NAGUMO MODEL
    Li Cai, Ye Sun, Feifei Jing, Yiqiang Li, Xiaoqin Shen, Yufeng Nie
    Journal of Computational Mathematics    2020, 38 (3): 469-486.   DOI: 10.4208/jcm.1901-m2017-0263
    Abstract29)      PDF      
    This work develops a fully discrete implicit-explicit finite element scheme for a parabolicordinary system with a nonlinear reaction term which is known as the FitzHugh-Nagumo model from physiology. The first-order backward Euler discretization for the time derivative, and an implicit-explicit discretization for the nonlinear reaction term are employed for the model, with a simple linearization technique used to make the process of solving equations more efficient. The stability and convergence of the fully discrete implicit-explicit finite element method are proved, which shows that the FitzHugh-Nagumo model is accurately solved and the trajectory of potential transmission is obtained. The numerical results are also reported to verify the convergence results and the stability of the proposed method.
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    A STOCHASTIC MOVING BALLS APPROXIMATION METHOD OVER A SMOOTH INEQUALITY CONSTRAINT
    Leiwu Zhang
    Journal of Computational Mathematics    2020, 38 (3): 528-546.   DOI: 10.4208/jcm.1912-m2016-0634
    Abstract28)      PDF      
    We consider the problem of minimizing the average of a large number of smooth component functions over one smooth inequality constraint. We propose and analyze a stochastic Moving Balls Approximation (SMBA) method. Like stochastic gradient (SG) methods, the SMBA method’s iteration cost is independent of the number of component functions and by exploiting the smoothness of the constraint function, our method can be easily implemented. Theoretical and computational properties of SMBA are studied, and convergence results are established. Numerical experiments indicate that our algorithm dramatically outperforms the existing Moving Balls Approximation algorithm (MBA) for the structure of our problem.
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    A HIGH-ORDER ACCURACY METHOD FOR SOLVING THE FRACTIONAL DIFFUSION EQUATIONS
    Maohua Ran, Chengjian Zhang
    Journal of Computational Mathematics    2020, 38 (2): 239-253.   DOI: 10.4208/jcm.1805-m2017-0081
    Abstract27)      PDF      
    In this paper, an efficient numerical method for solving the general fractional diffusion equations with Riesz fractional derivative is proposed by combining the fractional compact difference operator and the boundary value methods. In order to efficiently solve the generated linear large-scale system, the generalized minimal residual (GMRES) algorithm is applied. For accelerating the convergence rate of the iterative, the Strang-type, Chantype and P-type preconditioners are introduced. The suggested method can reach higher order accuracy both in space and in time than the existing methods. When the used boundary value method is A k1,k2-stable, it is proven that Strang-type preconditioner is invertible and the spectra of preconditioned matrix is clustered around 1. It implies that the iterative solution is convergent rapidly. Numerical experiments with the absorbing boundary condition and the generalized Dirichlet type further verify the efficiency.
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    ERROR ANALYSIS OF A STABILIZED FINITE ELEMENT METHOD FOR THE GENERALIZED STOKES PROBLEM
    Huoyuan Duan, Roger C. E. Tan
    Journal of Computational Mathematics    2020, 38 (2): 254-290.   DOI: 10.4208/jcm.1805-m2017-0192
    Abstract26)      PDF      
    This paper is devoted to the establishment of sharper a priori stability and error estimates of a stabilized finite element method proposed by Barrenechea and Valentin for solving the generalized Stokes problem, which involves a viscosity ν and a reaction constant σ. With the establishment of sharper stability estimates and the help of ad hoc finite element projections, we can explicitly establish the dependence of error bounds of velocity and pressure on the viscosity ν, the reaction constant σ, and the mesh size h. Our analysis reveals that the viscosity ν and the reaction constant σ respectively act in the numerator position and the denominator position in the error estimates of velocity and pressure in standard norms without any weights. Consequently, the stabilization method is indeed suitable for the generalized Stokes problem with a small viscosity ν and a large reaction constant σ. The sharper error estimates agree very well with the numerical results.
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    A C 0-WEAK GALERKIN FINITE ELEMENT METHOD FOR THE TWO-DIMENSIONAL NAVIER-STOKES EQUATIONS IN STREAM-FUNCTION FORMULATION
    Baiju Zhang, Yan Yang, Minfu Feng
    Journal of Computational Mathematics    2020, 38 (2): 310-336.   DOI: 10.4208/jcm.1806-m2017-0287
    Abstract24)      PDF      
    We propose and analyze a C 0-weak Galerkin (WG) finite element method for the numerical solution of the Navier-Stokes equations governing 2D stationary incompressible flows. Using a stream-function formulation, the system of Navier-Stokes equations is reduced to a single fourth-order nonlinear partial differential equation and the incompressibility constraint is automatically satisfied. The proposed method uses continuous piecewisepolynomial approximations of degree k+2 for the stream-function ψ and discontinuous piecewise-polynomial approximations of degree k+1 for the trace of ∇ ψ on the interelement boundaries. The existence of a discrete solution is proved by means of a topological degree argument, while the uniqueness is obtained under a data smallness condition. An optimal error estimate is obtained in L 2-norm, H 1-norm and broken H 2-norm. Numerical tests are presented to demonstrate the theoretical results.
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    CONVERGENCE ANALYSIS OF PARAREAL ALGORITHM BASED ON MILSTEIN SCHEME FOR STOCHASTIC DIFFERENTIAL EQUATIONS
    Liying Zhang, Jing Wang, Weien Zhou, Landong Liu, Li Zhang
    Journal of Computational Mathematics    2020, 38 (3): 487-501.   DOI: 10.4208/jcm.1901-m2018-0085
    Abstract23)      PDF      
    In this paper, we propose a parareal algorithm for stochastic differential equations (SDEs), which proceeds as a two-level temporal parallelizable integrator with the Milstein scheme as the coarse propagator and the exact solution as the fine propagator. The convergence order of the proposed algorithm is analyzed under some regular assumptions. Finally, numerical experiments are dedicated to illustrate the convergence and the convergence order with respect to the iteration number k, which show the efficiency of the proposed method.
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    A BALANCED OVERSAMPLING FINITE ELEMENT METHOD FOR ELLIPTIC PROBLEMS WITH OBSERVATIONAL BOUNDARY DATA
    Zhiming Chen, Rui Tuo, Wenlong Zhang
    Journal of Computational Mathematics    2020, 38 (2): 355-374.   DOI: 10.4208/jcm.1810-m2017-0168
    Abstract22)      PDF      
    In this paper we propose a finite element method for solving elliptic equations with observational Dirichlet boundary data which may subject to random noises. The method is based on the weak formulation of Lagrangian multiplier and requires balanced oversampling of the measurements of the boundary data to control the random noises. We show the convergence of the random finite element error in expectation and, when the noise is subGaussian, in the Orlicz ψ 2-norm which implies the probability that the finite element error estimates are violated decays exponentially. Numerical examples are included.
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    PIECEWISE SPARSE RECOVERY VIA PIECEWISE INVERSE SCALE SPACE ALGORITHM WITH DELETION RULE
    Yijun Zhong, Chongjun Li
    Journal of Computational Mathematics    2020, 38 (2): 375-394.   DOI: 10.4208/jcm.1810-m2017-0233
    Abstract19)      PDF      
    In some applications, there are signals with piecewise structure to be recovered. In this paper, we propose a piecewise_ISS (P_ISS) method which aims to preserve the piecewise sparse structure (or the small-scaled entries) of piecewise signals. In order to avoid selecting redundant false small-scaled elements, we also implement the piecewise_ISS algorithm in parallel and distributed manners equipped with a deletion rule. Numerical experiments indicate that compared with aISS, the P ISS algorithm is more effective and robust for piecewise sparse recovery.
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