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

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Nurullah Yilmaz ◽  
Ahmet Sahiner
Keyword(s):  
Image Restoration ◽  
Lipschitz Functions ◽  
Numerical Results ◽  
Smoothing Technique ◽  
Minimization Algorithm ◽  
Smooth Functions ◽  
Regularization Problem ◽  
Hyperbolic Smoothing

<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>

Optimal covering of solid bodies by spheres via the hyperbolic smoothing technique

2014 ◽  
Vol 30 (2) ◽  
pp. 391-403 ◽  
Author(s):  
Helder Manoel Venceslau ◽  
Daniela Cristina Lubke ◽  
Adilson Elias Xavier
Keyword(s):  
Smoothing Technique ◽  
Solid Bodies ◽  
Hyperbolic Smoothing

scholarly journals Estimation of Smooth Functions via Convex Programs

2016 ◽  
Vol 5 (4) ◽  
pp. 150
Author(s):  
Eunji Lim ◽  
Mina Attallah
Keyword(s):  
Stock Price ◽  
Noisy Data ◽  
Numerical Results ◽  
Convex Program ◽  
Option Prices ◽  
Convex Programs ◽  
Smooth Functions ◽  
Sensitivity Estimation

One of the numerically preferred methods for fitting a function to noisy data when the underlying function is known to be smooth is to minimize the roughness of the fit while placing a limit on the sum of squared errors. We show that the fit can be formulated as a solution to a convex program. Since convex programs can be solved by various methods with guaranteed convergence, our formulation enables one to use these methods to compute the fit numerically. Numerical results show that our formulation is successfully applied to the problem of sensitivity estimation of option prices as functions of the underlying stock price.

scholarly journals A conjugate gradient algorithm and its application in large-scale optimization problems and image restoration

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Gonglin Yuan ◽  
Tingting Li ◽  
Wujie Hu
Keyword(s):  
Image Restoration ◽  
Conjugate Gradient ◽  
Large Scale ◽  
Optimization Problems ◽  
Trust Region ◽  
Conjugate Gradient Algorithm ◽  
Gradient Algorithm ◽  
Smooth Functions ◽  
Nonsmooth Functions ◽  
Large Scale Unconstrained Optimization

Abstract To solve large-scale unconstrained optimization problems, a modified PRP conjugate gradient algorithm is proposed and is found to be interesting because it combines the steepest descent algorithm with the conjugate gradient method and successfully fully utilizes their excellent properties. For smooth functions, the objective algorithm sufficiently utilizes information about the gradient function and the previous direction to determine the next search direction. For nonsmooth functions, a Moreau–Yosida regularization is introduced into the proposed algorithm, which simplifies the process in addressing complex problems. The proposed algorithm has the following characteristics: (i) a sufficient descent feature as well as a trust region trait; (ii) the ability to achieve global convergence; (iii) numerical results for large-scale smooth/nonsmooth functions prove that the proposed algorithm is outstanding compared to other similar optimization methods; (iv) image restoration problems are done to turn out that the given algorithm is successful.

scholarly journals CLASSES OF OPERATOR-SMOOTH FUNCTIONS. III. STABLE FUNCTIONS AND FUGLEDE IDEALS

2005 ◽  
Vol 48 (1) ◽  
pp. 175-197 ◽  
Author(s):  
Edward Kissin ◽  
Victor S. Shulman
Keyword(s):  
Lipschitz Functions ◽  
Subject Classification ◽  
Primary 47A56 ◽  
Smooth Functions ◽  
Operator Stable ◽  
Operator Lipschitz Functions

AbstractThis paper continue to study the interrelation and hierarchy of the spaces of operator-Lipschitz functions and the spaces of functions closed to them: commutator bounded and operator stable. It also examines various properties of symmetrically normed ideals, introduces new classes of ideals: regular and Fuglede, and investigates them.AMS 2000 Mathematics subject classification: Primary 47A56; 47L20

scholarly journals On removing blur in images using least squares solutions

Filomat ◽  
2016 ◽  
Vol 30 (14) ◽  
pp. 3855-3866
Author(s):  
Predrag Stanimirovic ◽  
Igor Stojanovic ◽  
Dimitrios Pappas ◽  
Spiros Chountasis ◽  
Zoran Zdravev
Keyword(s):  
Statistical Analysis ◽  
Image Restoration ◽  
Least Squares ◽  
Numerical Results ◽  
Tikhonov Regularization Method ◽  
Truncated Singular Value Decomposition ◽  
Singular Value Decomposition Method ◽  
Restoration Method ◽  
Value Decomposition ◽  
Least Squares Approach

The further investigation of the image restoration method introduced in [19, 20] is presented in this paper. Continuing investigations in that area, two additional applications of the method are investigated. More precisely, we consider the possibility to replace the available matrix in the method by the restoration obtained applying the Tikhonov regularization method or the Truncated Singular Value decomposition method. Additionally, statistical analysis of numerical results generated by applying the proposed improvement of image restoration methods is presented. Previously performed numerical experiments as well as new numerical results and the statistical analysis confirm that the least squares approach can be used as a useful tool for improving restored images obtained by other image restoration methods.

Space-Time and Motion to Advection-Diffusion Equation

2021 ◽  
Vol 3 (1) ◽  
pp. 34-43
Author(s):  
Mohammad Ghani
Keyword(s):  
Diffusion Equation ◽  
Space Time ◽  
Numerical Results ◽  
Smooth Functions ◽  
Equation Problem ◽  
Time Motion ◽  
Advection Diffusion Equation ◽  
Time And Motion ◽  
Advection Diffusion ◽  
Finite Differences Method

AbstractWe are concerned with the study the differential equation problem of space-time and motion for the case of advection-diffusion equation. We derive the advection-diffusion equation from the conservation of mass, where this can be represented by the substance flow in and flow out through the medium. In this case, the concentration of substance and rate of flow of substance in a medium are smooth functions which is useful to generate advection-diffusion equation. A special case of the advection-diffusion equation and numerical results are also given in this paper. We use explicit and implicit finite differences method for numerical results implemented in MATLAB.Keywords: advection-diffusion; space-time; motion; finite difference method. AbstrakKami tertarik untuk mempelajari masalah persamaan diferensial ruang-waktu, dan gerak untuk kasus persamaan adveksi-difusi. Kita menurunkan persamaan adveksi-difusi dari kekekalan massa, di mana hal ini dapat diwakili oleh aliran zat yang masuk dan keluar melalui media. Dalam hal ini konsentrasi zat dan laju aliran zat dalam suatu medium merupakan fungsi halus yang berguna untuk menghasilkan persamaan adveksi-difusi. Sebuah kasus khusus persamaan adveksi-difusi dan hasil numerik juga diberikan dalam makalah ini. Kami menggunakan metode beda hingga explisit dan implisit untuk hasil numerik yang diimplementasikan dalam MATLAB.Kata kunci: adveksi-difusi; ruang-waktu; gerak; metode beda hingga.

Point derivations and cohomologies of Lipschitz algebras

2019 ◽  
Vol 62 (4) ◽  
pp. 1173-1187
Author(s):  
Kazuhiro Kawamura
Keyword(s):  
Metric Spaces ◽  
Banach Algebras ◽  
Studia Math ◽  
Lipschitz Functions ◽  
Homological Dimension ◽  
Smooth Functions ◽  
Infinite Dimensional ◽  
Higher Dimensional ◽  
Geodesic Metric ◽  
Dual Module

AbstractFor a compact metric space (K, d), LipK denotes the Banach algebra of all complex-valued Lipschitz functions on (K, d). We show that the continuous Hochschild cohomology Hn(LipK, (LipK)*) and Hn(LipK, ℂe) are both infinite-dimensional vector spaces for each n ≥ 1 if the space K contains a certain infinite sequence which converges to a point e ∈ K. Here (LipK)* is the dual module of LipK and ℂe denotes the complex numbers with a LipK-bimodule structure defined by evaluations of LipK-functions at e. Examples of such metric spaces include all compact Riemannian manifolds, compact geodesic metric spaces and infinite compact subsets of ℝ. In particular, the (small) global homological dimension of LipK is infinite for every such space. Our proof uses the description of point derivations by Sherbert [‘The structure of ideals and point derivations in Banach algebras of Lipschitz functions’, Trans. Amer. Math. Soc.111 (1964), 240–272] and directly constructs non-trivial cocycles with the help of alternating cocycles of Johnson [‘Higher-dimensional weak amenability’, Studia Math.123 (1997), 117–134]. An alternating construction of cocycles on the basis of the idea of Kleshchev [‘Homological dimension of Banach algebras of smooth functions is equal to infinity’, Vest. Math. Mosk. Univ. Ser. 1. Mat. Mech.6 (1988), 57–60] is also discussed.

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