Generalization of hyperbolic smoothing approach for non-smooth and non-Lipschitz functions

Abstract

<p style='text-indent:20px;'>In this study, we concentrate on the hyperbolic smoothing technique for some sub-classes of non-smooth functions and introduce a generalization of hyperbolic smoothing technique for non-Lipschitz functions. We present some useful properties of this generalization of hyperbolic smoothing technique. In order to illustrate the efficiency of the proposed smoothing technique, we consider the regularization problems of image restoration. The regularization problem is recast by considering the generalization of hyperbolic smoothing technique and a new algorithm is developed. Finally, the minimization algorithm is applied to image restoration problems and the numerical results are reported.</p>

Two-Field Weak Solutions for a Class of Contact Models

“…Thus, each of our models consists of the equilibrium Equation ( 10), the set-valued constitutive law (11) governed by the constitutive function ω : S 3 → R, the homogeneous displacement condition (12), the traction condition (13), and one of the boundary conditions ( 15) and (16). In (15), we have a frictionless contact condition involving the Winkler contact law, a law which describes in a simplified manner the interaction between a deformable body and the soil (see [12]).…”

“…We study successively the problems ( 10)-( 13) ( 15) and ( 10)-( 13) (16). Let us make the following assumptions: Assumption 1.…”

“…Next, we turn our attention to the question of how the weak solutions of the problems ( 10)-( 13) ( 15) and ( 10)-( 13) (16) depend on the data. With this end in view, we firstly prove the following lemma.…”

“…It is worth mentioning that the subdifferential of convex maps is present not only in solid mechanics but also in many other fields of mathematical physics. The subdifferential of convex maps is widely used especially in those cases in which the materials have an extremely complex intrinsic structure, such as complex fluids and/or magnetorheological fluids; see, e.g., [14][15][16][17].…”

Double stranded smoothing technique in energy minimization problems

“…The comprehensive overview on different smoothing techniques can be found in [8,13]. There are many interesting studies on solving regularization problems by using smoothing techniques [4,14,28]. It is revealed that using a nonsmooth regularization term in image restoration problems has remarkable advantages [4,9,16].…”