The primal-dual hybrid gradient (PDHG) algorithm has been applied for solving linearly constrained convex problems. However, it ‎was ‎shown ‎that ‎without ‎some additional ‎assumptions, convergence may fail. ‎In ‎this ‎work, ‎we propose a new competitive prediction-correction primal-dual hybrid gradient algorithm to solve this kind of problem. Under some conditions, we prove the global convergence for the proposed algorithm with the rate of ‎$‎O(1/N)‎$ ‎in a nonergodic sense‎.‎ Comparative performance analysis of our proposed approach with other related methods on some matrix completion and wavelet-based image inpainting test problems shows the outperformance of our approach, in terms of iteration number and CPU time.

‎A linear discrete ill-posed problem has a perturbed right-hand side vector and an ill-conditioned‎ ‎coefficient matrix‎. ‎The solution to such a problem is very sensitive to perturbation‎. ‎Replacement of the coefficient matrix by a nearby one that has less condition‎ ‎number is one of the well-known approaches for decreasing the sensitivity of the problem to perturbation. ‎In this paper‎, ‎we suggest some new regularization matrix to the Tikhonov regularization‎. ‎These new ones are based on fractional derivatives such as Grunwald-Letnikov and Caputo and can cause to have more exact solutions‎.

In this paper, we deal with the numerical approximation of the coupled wave equation of Kirchhoff type with nonlinear boundary damping and memory term. Since the equation is a nonlinear equation, the Raviart-Thomas mixed finite element method is one of the most suitable techniques to obtain the approximated solution. In this paper, we will show that using the Raviart-Thomas method the optimal convergence order of the scheme can be achieved. To that end, we prove the necessary lemmas and the main theorem. Finally, the efficiency of the method is certified by numerical examples.