Table of Content

20 September 2021, Volume 3 Issue 3

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Weighted Nuclear Norm Minimization-Based Regularization Method for Image Restoration

Yu-Mei Huang, Hui-Yin Yan

2021, 3(3):  371-390.  doi:10.1007/s42967-020-00076-4

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Regularization methods have been substantially applied in image restoration due to the ill-posedness of the image restoration problem. Different assumptions or priors on images are applied in the construction of image regularization methods. In recent years, matrix low-rank approximation has been successfully introduced in the image denoising problem and significant denoising effects have been achieved. Low-rank matrix minimization is an NP-hard problem and it is often replaced with the matrix's weighted nuclear norm minimization (WNNM). The assumption that an image contains an extensive amount of self-similarity is the basis for the construction of the matrix low-rank approximation-based image denoising method. In this paper, we develop a model for image restoration using the sum of block matching matrices' weighted nuclear norm to be the regularization term in the cost function. An alternating iterative algorithm is designed to solve the proposed model and the convergence analyses of the algorithm are also presented. Numerical experiments show that the proposed method can recover the images much better than the existing regularization methods in terms of both recovered quantities and visual qualities.

Hermite-Discontinuous Galerkin Overset Grid Methods for the Scalar Wave Equation

Oleksii Beznosov, Daniel Appel?

2021, 3(3):  391-418.  doi:10.1007/s42967-020-00075-5

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We present high order accurate numerical methods for the wave equation that combines efficient Hermite methods with geometrically flexible discontinuous Galerkin methods by using overset grids. Near boundaries we use thin boundary fitted curvilinear grids and in the volume we use Cartesian grids so that the computational complexity of the solvers approaches a structured Cartesian Hermite method. Unlike many other overset methods we do not need to add artificial dissipation but we find that the built-in dissipation of the Hermite and discontinuous Galerkin methods is sufficient to maintain the stability. By numerical experiments we demonstrate the stability, accuracy, efficiency, and the applicability of the methods to forward and inverse problems.

Modulus-Based Multisplitting Iteration Method for a Class of Weakly Nonlinear Complementarity Problem

Guangbin Wang, Fuping Tan

2021, 3(3):  419-428.  doi:10.1007/s42967-020-00074-6

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In this paper, we present a modulus-based multisplitting iteration method based on multisplitting of the system matrix for a class of weakly nonlinear complementarity problem. And we prove the convergence of the method when the system matrix is an H₊-matrix. Finally, we give two numerical examples.

Error Estimate of a Fully Discrete Local Discontinuous Galerkin Method for Variable-Order Time-Fractional Diffusion Equations

Leilei Wei, Shuying Zhai, Xindong Zhang

2021, 3(3):  429-444.  doi:10.1007/s42967-020-00081-7

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The aim of this paper is to develop a fully discrete local discontinuous Galerkin method to solve a class of variable-order fractional diffusion problems. The scheme is discretized by a weighted-shifted Grünwald formula in the temporal discretization and a local discontinuous Galerkin method in the spatial direction. The stability and the L²-convergence of the scheme are proved for all variable-order α(t) ∈ (0, 1). The proposed method is of accuracy-order O(τ³ + h^k+1), where τ, h, and k are the temporal step size, the spatial step size, and the degree of piecewise P^k polynomials, respectively. Some numerical tests are provided to illustrate the accuracy and the capability of the scheme.

Adaptive Moving Mesh Central-Upwind Schemes for Hyperbolic System of PDEs: Applications to Compressible Euler Equations and Granular Hydrodynamics

Alexander Kurganov, Zhuolin Qu, Olga S. Rozanova, Tong Wu

2021, 3(3):  445-480.  doi:10.1007/s42967-020-00082-6

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We introduce adaptive moving mesh central-upwind schemes for one- and two-dimensional hyperbolic systems of conservation and balance laws. The proposed methods consist of three steps. First, the solution is evolved by solving the studied system by the second-order semi-discrete central-upwind scheme on either the one-dimensional nonuniform grid or the two-dimensional structured quadrilateral mesh. When the evolution step is complete, the grid points are redistributed according to the moving mesh differential equation. Finally, the evolved solution is projected onto the new mesh in a conservative manner. The resulting adaptive moving mesh methods are applied to the one- and two-dimensional Euler equations of gas dynamics and granular hydrodynamics systems. Our numerical results demonstrate that in both cases, the adaptive moving mesh central-upwind schemes outperform their uniform mesh counterparts.

Arc Length-Based WENO Scheme for Hamilton-Jacobi Equations

Rathan Samala, Biswarup Biswas

2021, 3(3):  481-496.  doi:10.1007/s42967-020-00091-5

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In this article, novel smoothness indicators are presented for calculating the nonlinear weights of the weighted essentially non-oscillatory scheme to approximate the viscosity numerical solutions of Hamilton-Jacobi equations. These novel smoothness indicators are constructed from the derivatives of reconstructed polynomials over each sub-stencil. The constructed smoothness indicators measure the arc-length of the reconstructed polynomials so that the new nonlinear weights could get less absolute truncation error and give a high-resolution numerical solution. Extensive numerical tests are conducted and presented to show the performance capability and the numerical accuracy of the proposed scheme with the comparison to the classical WENO scheme.

Oscillation of Second-Order Half-Linear Neutral Advanced Differential Equations

Shan Shi, Zhenlai Han

2021, 3(3):  497-508.  doi:10.1007/s42967-020-00092-4

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The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form
(r(t)((y(t) + p(t)y(τ(t)))')^α)' + q(t)y^α(σ(t))=0, t ≥ t₀,
when ∫^∞ ${r^{-\frac{1}{\alpha }}}$(s)ds < ∞. We obtain sufficient conditions for the oscillation of the studied equations by the inequality principle and the Riccati transformation. An example is provided to illustrate the results.

A Jacobi Spectral Collocation Method for Solving Fractional Integro-Differential Equations

Qingqing Wu, Zhongshu Wu, Xiaoyan Zeng

2021, 3(3):  509-526.  doi:10.1007/s42967-020-00099-x

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The aim of this paper is to obtain the numerical solutions of fractional Volterra integrodifferential equations by the Jacobi spectral collocation method using the Jacobi-Gauss collocation points. We convert the fractional order integro-differential equation into integral equation by fractional order integral, and transfer the integro equations into a system of linear equations by the Gausssian quadrature. We furthermore perform the convergence analysis and prove the spectral accuracy of the proposed method in L^∞ norm. Two numerical examples demonstrate the high accuracy and fast convergence of the method at last.

A Weak Galerkin Harmonic Finite Element Method for Laplace Equation

Ahmed Al-Taweel, Yinlin Dong, Saqib Hussain, Xiaoshen Wang

2021, 3(3):  527-544.  doi:10.1007/s42967-020-00097-z

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In this article, a weak Galerkin finite element method for the Laplace equation using the harmonic polynomial space is proposed and analyzed. The idea of using the ${P_k}$-harmonic polynomial space instead of the full polynomial space ${P_k}$ is to use a much smaller number of basis functions to achieve the same accuracy when k ≥ 2. The optimal rate of convergence is derived in both H¹ and L² norms. Numerical experiments have been conducted to verify the theoretical error estimates. In addition, numerical comparisons of using the P₂-harmonic polynomial space and using the standard P₂ polynomial space are presented.

Approximations of the Fractional Integral and Numerical Solutions of Fractional Integral Equations

Yuri Dimitrov

2021, 3(3):  545-569.  doi:10.1007/s42967-021-00132-7

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In the present paper, we derive the asymptotic expansion formula for the trapezoidal approximation of the fractional integral. We use the expansion formula to obtain approximations for the fractional integral of orders α, 1 + α, 2 + α, 3 + α and 4 + α. The approximations are applied for computation of the numerical solutions of the ordinary fractional relaxation and the fractional oscillation equations expressed as fractional integral equations.