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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
    Abstract129)      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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    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
    Abstract94)      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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    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
    Abstract87)      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
    Abstract82)      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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    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
    Abstract81)      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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    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
    Abstract77)      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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    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
    Abstract73)      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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    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
    Abstract65)      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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    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
    Abstract60)      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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    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
    Abstract60)      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 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
    Abstract49)      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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    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
    Abstract49)      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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    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
    Abstract48)      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
    Abstract46)      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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    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
    Abstract29)      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 NEW ADAPTIVE SUBSPACE MINIMIZATION THREE-TERM CONJUGATE GRADIENT ALGORITHM FOR UNCONSTRAINED OPTIMIZATION
    Keke Zhang, Hongwei Liu, Zexian Liu
    Journal of Computational Mathematics    2021, 39 (2): 159-177.   DOI: 10.4208/jcm.1907-m2018-0173
    Abstract9)      PDF (221KB)(14)      
    A new adaptive subspace minimization three-term conjugate gradient algorithm with nonmonotone line search is introduced and analyzed in this paper. The search directions are computed by minimizing a quadratic approximation of the objective function on special subspaces, and we also proposed an adaptive rule for choosing different searching directions at each iteration. We obtain a significant conclusion that the each choice of the search directions satisfies the sufficient descent condition. With the used nonmonotone line search, we prove that the new algorithm is globally convergent for general nonlinear functions under some mild assumptions. Numerical experiments show that the proposed algorithm is promising for the given test problem set.
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    CONSTRAINT-PRESERVING ENERGY-STABLE SCHEME FOR THE 2D SIMPLIFIED ERICKSEN-LESLIE SYSTEM
    Xuelian Bao, Rui Chen, Hui Zhang
    Journal of Computational Mathematics    2021, 39 (1): 1-21.   DOI: 10.4208/jcm.1906-m2018-0144
    Abstract6)      PDF (300KB)(4)      
    Here we consider the numerical approximations of the 2D simplified Ericksen-Leslie system. We first rewrite the system and get a new system. For the new system, we propose an easy-to-implement time discretization scheme which preserves the sphere constraint at each node, enjoys a discrete energy law, and leads to linear and decoupled elliptic equations to be solved at each time step. A discrete maximum principle of the scheme in the finite element form is also proved. Some numerical simulations are performed to validate the scheme and simulate the dynamic motion of liquid crystals.
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    ACCURATE AND EFFICIENT IMAGE RECONSTRUCTION FROM MULTIPLE MEASUREMENTS OF FOURIER SAMPLES
    T. Scarnati, Anne Gelb
    Journal of Computational Mathematics    2020, 38 (5): 797-826.   DOI: 10.4208/jcm.2002-m2019-0192
    Abstract6)      PDF (2220KB)(27)      
    Several problems in imaging acquire multiple measurement vectors (MMVs) of Fourier samples for the same underlying scene. Image recovery techniques from MMVs aim to exploit the joint sparsity across the measurements in the sparse domain. This is typically accomplished by extending the use of l 1 regularization of the sparse domain in the single measurement vector (SMV) case to using l 2,1 regularization so that the "jointness" can be accounted for. Although effective, the approach is inherently coupled and therefore computationally inefficient. The method also does not consider current approaches in the SMV case that use spatially varying weighted l 1 regularization term. The recently introduced variance based joint sparsity (VBJS) recovery method uses the variance across the measurements in the sparse domain to produce a weighted MMV method that is more accurate and more efficient than the standard l 2,1 approach. The efficiency is due to the decoupling of the measurement vectors, with the increased accuracy resulting from the spatially varying weight. Motivated by these results, this paper introduces a new technique to even further reduce computational cost by eliminating the requirement to first approximate the underlying image in order to construct the weights. Eliminating this preprocessing step moreover reduces the amount of information lost from the data, so that our method is more accurate. Numerical examples provided in the paper verify these benefits.
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    EFFICIENT AND ACCURATE CHEBYSHEV DUAL-PETROV-GALERKIN METHODS FOR ODD-ORDER DIFFERENTIAL EQUATIONS
    Xuhong Yu, Lusha Jin, Zhongqing Wang
    Journal of Computational Mathematics    2021, 39 (1): 43-62.   DOI: 10.4208/jcm.1907-m2018-0285
    Abstract5)      PDF (249KB)(3)      
    Efficient and accurate Chebyshev dual-Petrov-Galerkin methods for solving first-order equation, third-order equation, third-order KdV equation and fifth-order Kawahara equation are proposed. Some Sobolev bi-orthogonal basis functions are constructed which lead to the diagonalization of discrete systems. Accordingly, both the exact solutions and the approximate solutions are expanded as an infinite and truncated Fourier-like series, respectively. Numerical experiments illustrate the effectiveness of the suggested approaches.
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    UNIFORM STABILITY AND ERROR ANALYSIS FOR SOME DISCONTINUOUS GALERKIN METHODS
    Qingguo Hong, Jinchao Xu
    Journal of Computational Mathematics    2021, 39 (2): 283-310.   DOI: 10.4208/jcm.2003-m2018-0223
    Abstract4)      PDF (310KB)(4)      
    In this paper, we provide a number of new estimates on the stability and convergence of both hybrid discontinuous Galerkin (HDG) and weak Galerkin (WG) methods. By using the standard Brezzi theory on mixed methods, we carefully define appropriate norms for the various discretization variables and then establish that the stability and error estimates hold uniformly with respect to stabilization and discretization parameters. As a result, by taking appropriate limit of the stabilization parameters, we show that the HDG method converges to a primal conforming method and the WG method converges to a mixed conforming method.
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