We propose a new two-phase method for reconstruction of blurred images corrupted by impulse noise. In the first phase, we use a noise detector to identify the pixels that are contaminated by noise, and then, in the second phase, we reconstruct the noisy pixels by solving an equality constrained total variation minimization problem that preserves the exact values of the noise-free pixels. For images that are only corrupted by impulse noise (i.e., not blurred) we apply the semismooth Newton's method to a reduced problem, and if the images are also blurred, we solve the equality constrained reconstruction problem using a first-order primal-dual algorithm. The proposed model improves the computational efficiency (in the denoising case) and has the advantage of being regularization parameter-free. Our numerical results suggest that the method is competitive in terms of its restoration capabilities with respect to the other two-phase methods.

In this paper, an inverse source problem for the time-fractional diffusion equation is investigated. The observational data is on the final time and the source term is assumed to be temporally independent and with a sparse structure. Here the sparsity is understood with respect to the pixel basis, i.e., the source has a small support. By an elastic-net regularization method, this inverse source problem is formulated into an optimization problem and a semismooth Newton (SSN) algorithm is developed to solve it. A discretization strategy is applied in the numerical realization. Several one and two dimensional numerical examples illustrate the efficiency of the proposed method.

In this paper, we propose a regularized version of the generalized NCP-function proposed by Hu, Huang and Chen [J. Comput. Appl. Math., 230 (2009), pp. 69-82]. Based on this regularized function, we propose a semismooth Newton method for solving nonlinear complementarity problems, where a non-monotone line search scheme is used. In particular, we show that the proposed non-monotone method is globally and locally superlinearly convergent under suitable assumptions. We test the proposed method by solving the test problems from MCPLIB. Numerical experiments indicate that this algorithm has better numerical performance in the case of p = 5 and Θ ∈ [0.25,075] than other cases.

Optimal control problems for the heat equation with pointwisebilateral control-state constraints are considered. A locallysuperlinearly convergent numerical solution algorithm is proposedand its mesh independence is established. Further, for theefficient numerical solution reduced space and Schur complementbased preconditioners are proposed which take into account theactive and inactive set structure of the problem. The paper endsby numerical tests illustrating our theoretical findings andcomparing the efficiency of the proposed preconditioners.