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In this paper, we consider the Newton-iteration scheme based on Galerkin and multi-Galerkin operators to solve non-linear integral equations of the Fredholm-Hammerstein type for both smooth and weakly singular algebraic and logarithmic type kernels. By choosing X n as space of piecewise polynomials subspace of degree ≤ r − 1 , we show that for a smooth kernel, Galerkin and multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders O (h (k + 1) r) and O (h (2 k + 3) r) , respectively, where h is the norm of the partition. For weakly singular kernels, we show that the Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders O (h r + k (1 − α)) , 0 < α < 1 for algebraic kernels and O (h r + k (log ⁡ h) k) for logarithmic kernels. Also the multi-Galerkin approximate solution in the kth Newton-iteration scheme converges with the orders O (h r + (k + 1) (1 − α)) for algebraic kernels and O (h r + k + 1 (log ⁡ h) k + 1) for logarithmic kernels. Numerical results are given for the justification of our theoretical results. [ABSTRACT FROM AUTHOR]

In this paper we design a family of cell-average nonlinear prediction operators that make use of the generalized harmonic means and we apply the resulting schemes to image processing. The new family of nonlinear schemes conserve the numerical properties of the linear schemes, such as the L 1 -stability, the order of accuracy or compression rate but avoiding Gibbs phenomenon close to the discontinuities. The generalized harmonic mean was introduced in the framework of point-values in [A. Guessab, M. Moncayo, and G. Schmeisser, A class of nonlinear four-point subdivision schemes. Properties in terms of conditions, Adv. Comput. Math. 37 (2012), pp. 151–190] in order to improve the results of the harmonic mean. However, in the cell-average setting our conclusion is that, from a numerical point of view, the advantage of using the new mean is not clear. [ABSTRACT FROM AUTHOR]

In this paper, we study fixed point proximity algorithms for a TVL1 restoration model recovering blurred images with impulsive noise, and image inpainting. The model that minimizes the sum of a data fidelity term in theℓ¹-norm, a term in ℓ²-norm and total-variation regularization term is strictly convex. We obtain the solution of the model through finding a fixed point of a nonlinear mapping expressed in terms of the proximity operator of the ℓ¹-norm or the ℓ²-norm, each of which is explicitly given. The non-expansivity of the mapping is also analysed theoretically. This formulation naturally leads to fixed-point algorithms for numerical treatment of the model. Numerical experiments demonstrate that the proposed algorithms perform favourably. [ABSTRACT FROM AUTHOR]