The Generalized Viscosity Implicit Rules of Asymptotically Nonexpansive Mappings in Hilbert Spaces

The generalized viscosity implicit rules of nonexpansive asymptotically mappings in Hilbert spaces are considered. The strong convergence theorems of the rules are proved under certain assumptions imposed on the sequences of parameters. An application of it in the convex minimization problem is considered. The results presented in this paper improve and extend some recent corresponding results in the literature.

S-matrix bootstrap in 3+1 dimensions: regularization and dual convex problem

The S-matrix bootstrap maps out the space of S-matrices allowed by analyticity, crossing, unitarity, and other constraints. For the 2 → 2 scattering matrix S2→2 such space is an infinite dimensional convex space whose boundary can be determined by maximizing linear functionals. On the boundary interesting theories can be found, many times at vertices of the space. Here we consider 3 + 1 dimensional theories and focus on the equivalent dual convex minimization problem that provides strict upper bounds for the regularized primal problem and has interesting practical and physical advantages over the primal problem. Its variables are dual partial waves kℓ(s) that are free variables, namely they do not have to obey any crossing, unitarity or other constraints. Nevertheless they are directly related to the partial waves fℓ(s), for which all crossing, unitarity and symmetry properties result from the minimization. Numerically, it requires only a few dual partial waves, much as one wants to possibly match experimental results. We consider the case of scalar fields which is related to pion physics.

A fast viscosity forward-backward algorithm for convex minimization problems with an application in image recovery

The purpose of this paper is to invent an accelerated algorithm for the convex minimization problem which can be applied to the image restoration problem. Theoretically, we first introduce an algorithm based on viscosity approximation method with the inertial technique for finding a common fixed point of a countable family of nonexpansive operators. Under some suitable assumptions, a strong convergence theorem of the proposed algorithm is established. Subsequently, we utilize our proposed algorithm to solving a convex minimization problem of the sum of two convex functions. As an application, we apply and analyze our algorithm to image restoration problems. Moreover, we compare convergence behavior and efficiency of our algorithm with other well-known methods such as the forward-backward splitting algorithm and the fast iterative shrinkage-thresholding algorithm. By using image quality metrics, numerical experiments show that our algorithm has a higher efficiency than the mentioned algorithms.

A Projected Forward-Backward Algorithm for Constrained Minimization with Applications to Image Inpainting

In this research, we study the convex minimization problem in the form of the sum of two proper, lower-semicontinuous, and convex functions. We introduce a new projected forward-backward algorithm using linesearch and inertial techniques. We then establish a weak convergence theorem under mild conditions. It is known that image processing such as inpainting problems can be modeled as the constrained minimization problem of the sum of convex functions. In this connection, we aim to apply the suggested method for solving image inpainting. We also give some comparisons to other methods in the literature. It is shown that the proposed algorithm outperforms others in terms of iterations. Finally, we give an analysis on parameters that are assumed in our hypothesis.

An algorithm for approximating a common solution of some nonlinear problems in Banach spaces with an application

In this paper, we construct a new Halpern-type subgradient extragradient iterative algorithm. The sequence generated by this algorithm converges strongly to a common solution of a variational inequality, an equilibrium problem, and a J-fixed point of a continuous J-pseudo-contractive map in a uniformly smooth and two-uniformly convex real Banach space. Also, the theorem is applied to approximate a common solution of a variational inequality, an equilibrium problem, and a convex minimization problem. Moreover, a numerical example is given to illustrate the implementability of our algorithm. Finally, the theorem proved complements, improves, and unifies some related recent results in the literature.

Convex duality for principal frequencies

<abstract><p>We consider the sharp Sobolev-Poincaré constant for the embedding of $ W^{1, 2}_0(\Omega) $ into $ L^q(\Omega) $. We show that such a constant exhibits an unexpected dual variational formulation, in the range $ 1 &lt; q &lt; 2 $. Namely, this can be written as a convex minimization problem, under a divergence–type constraint. This is particularly useful in order to prove lower bounds. The result generalizes what happens for the torsional rigidity (corresponding to $ q = 1 $) and extends up to the case of the first eigenvalue of the Dirichlet-Laplacian (i.e., to $ q = 2 $).</p></abstract>

Superdirective Robust Algorithms’ Comparison for Linear Arrays

Frequency-invariant beam patterns are often required by systems using an array of sensors to process broadband signals. In some experimental conditions (small devices for underwater acoustic communication), the array spatial aperture is shorter than the involved wavelengths. In these conditions, superdirective beamforming is essential for an efficient system. We present a comparison between two methods that deal with a data-independent beamformer based on a filter-and-sum structure. Both methods (the first one numerical, the second one analytic) formulate a mathematical convex minimization problem, in which the variables to be optimized are the filters coefficients or frequency responses. The goal of the optimization is to obtain a frequency invariant superdirective beamforming with a tunable tradeoff between directivity and frequency-invariance. We compare pros and cons of both methods measured through quantitative metrics to wrap up conclusions and further proposed investigations.