scholarly journals An inverse problem of synthesis of nanooptical security elements for visual and automated authenticity verification

2020 ◽  
pp. 56-63
Author(s):  
А.А. Гончарский ◽  
С.Р. Дурлевич
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
White Light ◽  
Fredholm Integral Equation ◽  
Phase Function ◽  
Focal Plane ◽  
Optical Element ◽  
Coherent Radiation ◽  
Nonlinear Fredholm Integral Equation ◽  
Elementary Region

Статья посвящена решению обратных задач синтеза нанооптических защитных элементов. Синтез нанооптического элемента включает в себя как решение обратной задачи расчета его фазовой функции, так и прецизионное формирование микрорельефа. При освещении микрорельефа в любой точке нанооптического элемента когерентным излучением в фокальной плоскости, параллельной плоскости оптического элемента, формируется изображение, используемое для автоматизированного контроля. Область оптического элемента разбивается на элементарные области. Изображение в элементарных областях формируется с помощью бинарных киноформов, фазовая функция которых рассчитывается с помощью решения нелинейного интегрального уравнения Фредгольма первого рода. Глубина микрорельефа в каждой элементарной области постоянна и определяет цвет элементарной области при освещении оптического элемента белым светом. Разработанные элементы могут быть использованы для защиты документов, акцизных марок, брендов и др. This paper is concerned with solving inverse problems of the synthesis of nanooptical security elements. The synthesis of a nanooptical element involves calculating its phase function via solving an inverse problem and fabricating the microrelief with high precision. The microrelief of the nanooptical element illuminated at any point with coherent radiation produces an image in the focal plane parallel to the plane of the optical element. This image is used for the automated authenticity verification. The area of the optical element is divided into elementary regions. In each elementary region, the image is formed using binary kinoforms whose phase function is calculated via solving a nonlinear Fredholm integral equation of the first kind. The depth of the microrelief is constant in each elementary region and determines the color of that region when the optical element is illuminated with white light. The developed elements can be used to protect documents, excise stamps, and brands.

scholarly journals Numerical Treatment of Fixed Point Applied to the Nonlinear Fredholm Integral Equation

2009 ◽  
Vol 2009 (1) ◽  
pp. 735638 ◽  
Author(s):  
MI Berenguer ◽  
MV Fernández Muñoz ◽  
AI Garralda Guillem ◽  
M Ruiz Galán
Keyword(s):  
Integral Equation ◽  
Fixed Point ◽  
Fredholm Integral Equation ◽  
Numerical Treatment ◽  
Nonlinear Fredholm Integral Equation

scholarly journals SOME RAPIDLY CONVERGENT METHODS FOR NONLINEAR FREDHOLM INTEGRAL EQUATION

2005 ◽  
Vol 12 (1) ◽  
pp. 63-72
Author(s):  
I. Kaldo ◽  
O. Vaarmann
Keyword(s):  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Inverse Operator ◽  
Iteration Process ◽  
Iteration Step ◽  
Nonlinear Fredholm Integral Equation ◽  
Computational Aspects ◽  
Iteration Methods ◽  
Fréchet Differentiable ◽  
Limited Accuracy

Many problems in modelling can be reduced to the solution of a nonlinear equation F(x) = 0, where F is a Frechet‐differentiable (as many times as necessary) mapping between Banach spaces X and Y. For solving this equation we consider high order iteration methods of the type xk +1 =xk ‐ Q(xk, Ai k ), i ∈ I, I = {1,…, r}, r ≥ 1, k = 0, 1, …, where Q(x, Ai k ) is an operator from X into itself and Ai k, i ∈ I, are some approximations to the inverse operator(s) occurring in the associated exact method. In particular, this set of methods contains methods with successive approximation of the inverse operator(s) and those based on the use of iterative methods to obtain a cheap solution of limited accuracy for corresponding linear equation(s) at each iteration step. A convergence theorem is proved and computational aspects of the methods under consideration are examined. The solution of nonlinear Fredholm integral equation by means of methods with convergence order p ≥ 2 are considered and possibilities of organizing parallel computation in iteration process are also briefly discussed. Daug modeliavimo problemu galima suformuluoti netiesines lygties F(x) = 0 pavidalu. Čia F yra Banacho erdves X atvaizdavimas i Banacho erdve Y, turintis visas reikalingas Freshe išvestines. Lygčiai F(x) = 0 spresti taikomas aukštosios eiles iteracinis procesas tokio tipo xk + 1 =xk ‐ Q(xk, Ai k ), i ∈ {1,…, r}, k = 0, 1, …, Čia Q(x, Ai k ) yra tam tikras operatorius X → X, Ai k , yra atvirkštinio atvaizdavimo aproksimacijos. Irodyta konvergavimo teorema ir išnagrineti metodu taikymo skaičiavimo aspektai. Aptariamos skaičiavimu lygiagretinimo galimybes, taikant si ulomus metodus netiesinei Fredholmo integralinei lygčiai.

scholarly journals Approximate Solution of Nonlinear Integral Equations of the Second Kind by Using Homotopy Perturbation Method

2017 ◽  
Vol 65 (2) ◽  
pp. 151-155
Author(s):  
MM Hasan ◽  
MA Matin
Keyword(s):  
Integral Equation ◽  
Exact Solution ◽  
Approximate Solution ◽  
Perturbation Method ◽  
Fredholm Integral Equation ◽  
Homotopy Perturbation Method ◽  
Nonlinear Integral Equations ◽  
Numerical Examples ◽  
Homotopy Perturbation ◽  
Nonlinear Fredholm Integral Equation

In this paper, we apply Homotopy perturbation method (HPM) for obtaining approximate solution of nonlinear Fredholm integral equation of the second kind. Finally, some numerical examples are provided, and the obtained numerical approximations are compared with the corresponding exact solution. Dhaka Univ. J. Sci. 65(2): 151-155, 2017 (July)

scholarly journals ON THE VIBRATIONS OF INHOMOGENEOUS PIEZO DISC

2019 ◽  
Vol 81 (3) ◽  
pp. 369-380
Author(s):  
A.O. Vatulyan ◽  
Yu.N. Zubkov
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Fredholm Integral Equation ◽  
Radial Displacement ◽  
Canonical System ◽  
Cauchy Problems ◽  
Resonance Frequencies ◽  
Finite Set ◽  
Inhomogeneous Properties ◽  
Set Of Points

In the framework of the model of coupled electroelasticity of inhomogeneous bodies, the problem of steady-state oscillations of a thin piezodisc with inhomogeneous properties is considered, in particular, in the presence of radial polarization. The necessary simplifications are made within the framework of traditional hypotheses, the formulated boundary-value problem is reduced to a canonical system of first-order differential equations with respect to dimensionless components of radial displacement and radial stress with corresponding boundary conditions. The direct problem of oscillations of an inhomogeneous disk is solved numerically based on the shooting method by numerically analyzing auxiliary Cauchy problems. The analysis of the amplitude-frequency characteristics and resonance frequencies depending on various laws of variation of the inhomogeneous properties of the piezodisc is performed, which in the presented model are characterized by two functions, one of which characterizes the change in the elastic modulus, the second changes in the piezomodule. The inverse problem is formulated in the first statement, in which the laws of variation of the piezodisc heterogeneity (two functions) are restored from the values of the functions characterizing the radial displacement and stress, known in a finite set of points. The results of computational experiments on solving the inverse problem in the first formulation are presented, various aspects of reconstruction are discussed. The second formulation of the inverse problem is formulated to determine the piezoelectric characteristics of the disk, where a function that describes the laws of change in the elastic characteristics of the disk and the amplitude-frequency characteristic is considered known. To solve the inverse problem, in this formulation, the Fredholm integral equation of the first kind with a smooth kernel is formulated. The results of numerical experiments on solving the Fredholm integral equation of the first kind using the Tikhonov regularizing method are presented, various aspects of reconstruction are discussed.

AN ACOUSTIC INVERSE PROBLEM: NUMERICAL EXPERIMENT

1995 ◽  
Vol 03 (03) ◽  
pp. 229-240 ◽  
Author(s):  
R. P. GILBERT ◽  
ZHONGYAN LIN
Keyword(s):  
Inverse Problem ◽  
Integral Equation ◽  
Numerical Experiment ◽  
Fredholm Integral Equation ◽  
Numerical Experiments ◽  
Goursat Problem ◽  
Index Of Refraction ◽  
Numerical Treatment ◽  
Moments Problem

As a sequel to Refs. 1 and 2, this paper gives a numerical treatment of the inverse problem associated with the determination of the index of refraction. We show that the problem can be solved in two steps. First we must recover a function from its moments, problem (IM), which we may reformulate as a Fredholm integral equation of the first kind, problem (IE). Second we solve an inverse Goursat problem, (IG). Numerical schemes for both steps are given along with the results of some numerical experiments.

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