Pub Date : 2024-06-07DOI: 10.1134/s096554252470012x
O. N. Granichin, Yu. V. Ivanskii, K. D. Kopylova
Abstract
In 1976–1977, Polyak published in the journal Avtomatica i Telemekhanika (Automation and Remote Control) two remarkable papers on how to study the properties of estimates of iterative pseudogradient algorithms. The first paper published in 1976 considered the general case based on the stochastic Lyapunov function, and the second one considered the linear case. The assumptions formulated in these papers and the estimates obtained in them can still be considered the state-of-the art. In the current paper, Polyak’s approach is applied to the study of the properties of estimates of a (randomized) stochastic approximation search algorithm for the case of unknown-but-bounded noise in observations. The obtained asymptotic estimates were already known earlier, and exact estimates for a finite number of observations are published for the first time.
摘要 1976-1977 年，波利克在《自动化与远程控制》（Avtomatica i Telemekhanika）杂志上发表了两篇关于如何研究迭代伪梯度算法估计值特性的重要论文。1976 年发表的第一篇论文考虑了基于随机 Lyapunov 函数的一般情况，而第二篇论文则考虑了线性情况。这些论文中提出的假设和获得的估计至今仍被认为是最先进的。在本文中，Polyak 的方法被应用于研究（随机）随机逼近搜索算法的估计值特性，该算法适用于观测中存在未知但有界噪声的情况。所获得的渐近估计值早先就已为人所知，本文首次公布了有限数量观测值的精确估计值。
{"title":"Polyak’s Method Based on the Stochastic Lyapunov Function for Justifying the Consistency of Estimates Produced by a Stochastic Approximation Search Algorithm under an Unknown-but-Bounded Noise","authors":"O. N. Granichin, Yu. V. Ivanskii, K. D. Kopylova","doi":"10.1134/s096554252470012x","DOIUrl":"https://doi.org/10.1134/s096554252470012x","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In 1976–1977, Polyak published in the journal Avtomatica i Telemekhanika (Automation and Remote Control) two remarkable papers on how to study the properties of estimates of iterative pseudogradient algorithms. The first paper published in 1976 considered the general case based on the stochastic Lyapunov function, and the second one considered the linear case. The assumptions formulated in these papers and the estimates obtained in them can still be considered the state-of-the art. In the current paper, Polyak’s approach is applied to the study of the properties of estimates of a (randomized) stochastic approximation search algorithm for the case of unknown-but-bounded noise in observations. The obtained asymptotic estimates were already known earlier, and exact estimates for a finite number of observations are published for the first time.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523626","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-07DOI: 10.1134/s0965542524700143
A. V. Nazin, A. S. Poznyak
Abstract
We consider a class of controlled nonlinear plants, the dynamics of which are governed by a vector system of ordinary differential equations with a right-hand side that is partially known. The study’s objective is to construct a robust tracking controller with certain constraints on the state variables, assuming that the state variables and their time derivatives can be observed. The Legendre–Fenchel transform and a chosen proxy function are utilized to develop this mathematical development using the mirror descent approach, which is frequently employed in convex optimization problems involving static objects. The Average Subgradient Method (an improved version of the Subgradient Descent Method), and the Integral Sliding Mode technique for continuous-time control systems are basically extended by the suggested unifying architecture. The primary findings include demonstrating that the “desired regime”—a non-stationary analog of the sliding surface – can be achieved from the very start of the process and getting an explicit upper bound on the cost function’s decrement.
{"title":"Non-Quadratic Proxy Functions in Mirror Descent Method Applied to Designing of Robust Controllers for Nonlinear Dynamic Systems with Uncertainty","authors":"A. V. Nazin, A. S. Poznyak","doi":"10.1134/s0965542524700143","DOIUrl":"https://doi.org/10.1134/s0965542524700143","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>We consider a class of controlled nonlinear plants, the dynamics of which are governed by a vector system of ordinary differential equations with a right-hand side that is partially known. The study’s objective is to construct a robust tracking controller with certain constraints on the state variables, assuming that the state variables and their time derivatives can be observed. The Legendre–Fenchel transform and a chosen proxy function are utilized to develop this mathematical development using the mirror descent approach, which is frequently employed in convex optimization problems involving static objects. The Average Subgradient Method (an improved version of the Subgradient Descent Method), and the Integral Sliding Mode technique for continuous-time control systems are basically extended by the suggested unifying architecture. The primary findings include demonstrating that the “desired regime”—a non-stationary analog of the sliding surface – can be achieved from the very start of the process and getting an explicit upper bound on the cost function’s decrement.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141523631","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-06-07DOI: 10.1134/s0965542524700027
N. Mijajlović, M. Jaćimović
Abstract
The objective of this manuscript is to study the convergence of three-step approximation methods for quasi-variational inequalities in the general case. First, we propose the three-step dynamical system and carry out an asymptotic analysis for the generated trajectories. The explicit time discretization of this system results into a three-step iterative method, which we prove to converge also when it is applied to strongly-monotone quasi-variational inequalities. In addition, we show that linear convergence is guaranteed under strong-monotonicity.
{"title":"Three-Step Approximation Methods from Continuous and Discrete Perspective for Quasi-Variational Inequalities","authors":"N. Mijajlović, M. Jaćimović","doi":"10.1134/s0965542524700027","DOIUrl":"https://doi.org/10.1134/s0965542524700027","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The objective of this manuscript is to study the convergence of three-step approximation methods for quasi-variational inequalities in the general case. First, we propose the three-step dynamical system and carry out an asymptotic analysis for the generated trajectories. The explicit time discretization of this system results into a three-step iterative method, which we prove to converge also when it is applied to strongly-monotone quasi-variational inequalities. In addition, we show that linear convergence is guaranteed under strong-monotonicity.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141550084","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.1134/s0965542524030059
O. V. Belai, L. L. Frumin, A. E. Chernyavsky
Abstract
The paper considers algorithms for solving inverse scattering problems based on the discretization of the Gelfand–Levitan–Marchenko integral equations, associated with the system of nonlinear Schrödinger equations of the Manakov model. The numerical algorithm of the first order approximation for solving the scattering problem is reduced to the inversion of a series of nested block Toeplitz matrices using the Levinson-type bordering method. Increasing the approximation accuracy violates the Toeplitz structure of block matrices. Two algorithms are described that solve this problem for second order accuracy. One algorithm uses a block version of the Levinson bordering algorithm, which recovers the Toeplitz structure of the matrix by moving some terms of the systems of equations to the right-hand side. Another algorithm is based on the Toeplitz decomposition of an almost block-Toeplitz matrix and the Tyrtyshnikov bordering algorithm. The speed and accuracy of calculations using the presented algorithms are compared on an exact solution (the Manakov vector soliton).
{"title":"Algorithms for Solving the Inverse Scattering Problem for the Manakov Model","authors":"O. V. Belai, L. L. Frumin, A. E. Chernyavsky","doi":"10.1134/s0965542524030059","DOIUrl":"https://doi.org/10.1134/s0965542524030059","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper considers algorithms for solving inverse scattering problems based on the discretization of the Gelfand–Levitan–Marchenko integral equations, associated with the system of nonlinear Schrödinger equations of the Manakov model. The numerical algorithm of the first order approximation for solving the scattering problem is reduced to the inversion of a series of nested block Toeplitz matrices using the Levinson-type bordering method. Increasing the approximation accuracy violates the Toeplitz structure of block matrices. Two algorithms are described that solve this problem for second order accuracy. One algorithm uses a block version of the Levinson bordering algorithm, which recovers the Toeplitz structure of the matrix by moving some terms of the systems of equations to the right-hand side. Another algorithm is based on the Toeplitz decomposition of an almost block-Toeplitz matrix and the Tyrtyshnikov bordering algorithm. The speed and accuracy of calculations using the presented algorithms are compared on an exact solution (the Manakov vector soliton).</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140797806","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.1134/s0965542524030072
Yu. G. Evtushenko, A. A. Tret’yakov
Abstract
The paper is devoted to the solution of a nonlinear system of equations (F(x{{) = 0}_{n}}), where (F) is a quadratic mapping acting from ({{mathbb{R}}^{n}}) to ({{mathbb{R}}^{n}}). The derivative (F{kern 1pt} ') is assumed to be degenerate at the solution point, which is a major characteristic property of nonlinearity of the mapping. Based on constructions of the p-regularity theory, a 2-factor method is proposed for solving the system of equations, which converges at a quadratic rate. Moreover, an exact formula is obtained for solving this quadratic system of equations in the case of a 2-regular mapping (F(x)).
{"title":"Exact Formula for Solving a Degenerate System of Quadratic Equations","authors":"Yu. G. Evtushenko, A. A. Tret’yakov","doi":"10.1134/s0965542524030072","DOIUrl":"https://doi.org/10.1134/s0965542524030072","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The paper is devoted to the solution of a nonlinear system of equations <span>(F(x{{) = 0}_{n}})</span>, where <span>(F)</span> is a quadratic mapping acting from <span>({{mathbb{R}}^{n}})</span> to <span>({{mathbb{R}}^{n}})</span>. The derivative <span>(F{kern 1pt} ')</span> is assumed to be degenerate at the solution point, which is a major characteristic property of nonlinearity of the mapping. Based on constructions of the <i>p</i>-regularity theory, a 2-factor method is proposed for solving the system of equations, which converges at a quadratic rate. Moreover, an exact formula is obtained for solving this quadratic system of equations in the case of a 2-regular mapping <span>(F(x))</span>.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140797877","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.1134/s0965542524030060
Chandra Sekhar Mahato, Siddhartha Biswas
Abstract
The present article focuses on the Rayleigh surface wave propagation in homogeneous isotropic thermoelastic medium. This works aims to develop a new nonlocal elasticity model based on three-phase-lag model of hyperbolic thermoelasticity under double porosity structure. New constitutive relations and equations are derived using nonlocal continuum mechanics. State space approach is employed to solve the problem. Frequency equation is derived using appropriate boundary conditions. Path of surface particles is found elliptical at the time of Rayleigh surface wave propagation. Different characteristics of Rayleigh waves are calculated numerically and presented graphically for different wave number. Various characteristics of wave for different thermoelastic models are also presented graphically. The effect of nonlocal parameters and void parameters on phase velocity, attenuation coefficient, penetration depth, and specific loss of Rayleigh waves are shown graphically. Some special cases are deduced from this study which agree with the existing literatures.
{"title":"State Space Approach to Characterize Rayleigh Waves in Nonlocal Thermoelastic Medium with Double Porosity under Three-Phase-Lag Model","authors":"Chandra Sekhar Mahato, Siddhartha Biswas","doi":"10.1134/s0965542524030060","DOIUrl":"https://doi.org/10.1134/s0965542524030060","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The present article focuses on the Rayleigh surface wave propagation in homogeneous isotropic thermoelastic medium. This works aims to develop a new nonlocal elasticity model based on three-phase-lag model of hyperbolic thermoelasticity under double porosity structure. New constitutive relations and equations are derived using nonlocal continuum mechanics. State space approach is employed to solve the problem. Frequency equation is derived using appropriate boundary conditions. Path of surface particles is found elliptical at the time of Rayleigh surface wave propagation. Different characteristics of Rayleigh waves are calculated numerically and presented graphically for different wave number. Various characteristics of wave for different thermoelastic models are also presented graphically. The effect of nonlocal parameters and void parameters on phase velocity, attenuation coefficient, penetration depth, and specific loss of Rayleigh waves are shown graphically. Some special cases are deduced from this study which agree with the existing literatures.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805045","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.1134/s0965542524030138
A. G. Petrov
Abstract
A boundary element scheme for the problem of potential flow over an axisymmetric toroidal body is considered. An integral equation for the velocity distribution on the body is derived. It is shown that the numerical scheme for solving the considered equation converges exponentially.
{"title":"Exponentially Convergent Numerical Scheme for the Stream Function of Potential Flow over Axisymmetric Bodies","authors":"A. G. Petrov","doi":"10.1134/s0965542524030138","DOIUrl":"https://doi.org/10.1134/s0965542524030138","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A boundary element scheme for the problem of potential flow over an axisymmetric toroidal body is considered. An integral equation for the velocity distribution on the body is derived. It is shown that the numerical scheme for solving the considered equation converges exponentially.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806825","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.1134/s0965542524030126
I. A. Nosikov, A. A. Tolchennikov, M. V. Klimenko
Abstract
A direct variational method for solving the problem of finding ocean wave rays reflected from the coastline with given positions of the source and the point of observation is considered. It is shown that the original boundary value problem can be reduced to the direct search of stationary points of the functional equal to the time of wave propagation along the ray. Information about the objective function in the area of solutions to the ray tracing problem allows us to construct a systematic procedure for finding minima, saddle points, and maxima. A feature of the proposed approach is the optimization of the ray reflection point along a given coastline.
{"title":"Boundary Value Problem of Calculating Ray Characteristics of Ocean Waves Reflected from Coastline","authors":"I. A. Nosikov, A. A. Tolchennikov, M. V. Klimenko","doi":"10.1134/s0965542524030126","DOIUrl":"https://doi.org/10.1134/s0965542524030126","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>A direct variational method for solving the problem of finding ocean wave rays reflected from the coastline with given positions of the source and the point of observation is considered. It is shown that the original boundary value problem can be reduced to the direct search of stationary points of the functional equal to the time of wave propagation along the ray. Information about the objective function in the area of solutions to the ray tracing problem allows us to construct a systematic procedure for finding minima, saddle points, and maxima. A feature of the proposed approach is the optimization of the ray reflection point along a given coastline.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140806754","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.1134/s0965542524030096
A. N. Konenkov
Abstract
The Cauchy problem for the heat equation with zero right-hand side is considered. The initial function is assumed to belong to the space of tempered distributions. The problem of determining the support of the initial function from solution values at some fixed time (T > 0) is studied. Necessary and sufficient conditions for the support to lie in a given convex compact set are obtained. These conditions are formulated in terms of the solution’s decay rate at infinity. A sharp constant in the exponential for the Landis–Oleinik conjecture on the nonexistence of fast decaying solutions is found.
{"title":"Localizing the Initial Condition for Solutions of the Cauchy Problem for the Heat Equation","authors":"A. N. Konenkov","doi":"10.1134/s0965542524030096","DOIUrl":"https://doi.org/10.1134/s0965542524030096","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>The Cauchy problem for the heat equation with zero right-hand side is considered. The initial function is assumed to belong to the space of tempered distributions. The problem of determining the support of the initial function from solution values at some fixed time <span>(T > 0)</span> is studied. Necessary and sufficient conditions for the support to lie in a given convex compact set are obtained. These conditions are formulated in terms of the solution’s decay rate at infinity. A sharp constant in the exponential for the Landis–Oleinik conjecture on the nonexistence of fast decaying solutions is found.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140797799","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2024-04-22DOI: 10.1134/s0965542524030023
M. V. Abgaryan, A. M. Bishaev
Abstract
In this work, a system of kinetic equations is constructed to study the processes in a three-component plasma. It is an analogue of the Krook model, which is widely used in the dynamics of rarefied gases. The model is supposed to be used to study the processes in the channels of rocket electric thrusters.
{"title":"Kinetic Model of a Three-Component Plasma","authors":"M. V. Abgaryan, A. M. Bishaev","doi":"10.1134/s0965542524030023","DOIUrl":"https://doi.org/10.1134/s0965542524030023","url":null,"abstract":"<h3 data-test=\"abstract-sub-heading\">Abstract</h3><p>In this work, a system of kinetic equations is constructed to study the processes in a three-component plasma. It is an analogue of the Krook model, which is widely used in the dynamics of rarefied gases. The model is supposed to be used to study the processes in the channels of rocket electric thrusters.</p>","PeriodicalId":55230,"journal":{"name":"Computational Mathematics and Mathematical Physics","volume":null,"pages":null},"PeriodicalIF":0.7,"publicationDate":"2024-04-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140805156","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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