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.
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.
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.
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 $L^2$-convergence of the scheme are proved for all variable-order $\alpha (t)\in (0,1)$. The proposed method is of accuracy-order $O(\tau ^3+h^{k+1})$ , where $\tau$, 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.
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.
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.
The purpose of this paper is to study the oscillation of second-order half-linear neutral differential equations with advanced argument of the form $\begin{aligned} (r(t)((y(t)+p(t)y(\tau (t)))')^\alpha )'+q(t)y^\alpha (\sigma (t))=0,\ t\geqslant t_0, \end{aligned}$ when $\int _{}^\infty r^{-\frac{1}{\alpha }}(s){\text{d}}s<\infty $. 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.