Papers by Konstantin Rybakov
Civil Aviation High TECHNOLOGIES, Apr 28, 2018
Работа выполнена при поддержке РФФИ, грант № 17-08-00530 А В статье показана связь между фильтрам... more Работа выполнена при поддержке РФФИ, грант № 17-08-00530 А В статье показана связь между фильтрами, основанными на моделировании траекторий случайного процесса с обрывами и ветвлениями, и непрерывным фильтром частиц, которые относятся к последовательным методам Монте-Карло. Даются различные варианты вычисления весовых коэффициентов в фильтре частиц для стохастических систем с непрерывным временем, т. е. стохастических систем диффузионного типа. Наряду с представлением непрерывной функцией показана возможность представления траектории весовой функции кусочно-постоянной функцией с действительными неотрицательными значениями, а также кусочно-постоянной функцией с целыми неотрицательными значениями. В основе такого представления лежит моделирование траекторий общего пуассоновского процесса. Указана связь с дифференциальным уравнением для весовой функции. Все приведенные варианты вычисления весовых коэффициентов в фильтре частиц не требуют сложной программной реализации, они подходят при разработке программного обеспечения для фильтров частиц с применением различных технологий параллельного программирования для высокопроизводительных вычислительных систем. Рассмотренный в статье непрерывный фильтр частиц может применяться в различных прикладных задачах оценивания. Например, в задачах слежения за движущимся объектом, восстановления траектории движения по косвенным наблюдениям, выделения полезного сигнала на фоне помех, идентификации параметров динамических систем и многих других задачах. В дальнейшем планируется расширить применение фильтра частиц для стохастических систем диффузионно-скачкообразного типа. Кроме того, планируется сформировать алгоритмы прогнозирования состояний непрерывных стохастических систем как диффузионного, так и диффузионно-скачкообразного типа на основе рассмотренных вариантов вычисления весовых коэффициентов в фильтре частиц. Ключевые слова: весовой коэффициент, ветвящийся процесс, метод Монте-Карло, метод статистических испытаний, оптимальная фильтрация, случайный процесс, стохастическая система, фильтр частиц.
Symmetry
A solution to the trace convergence problem, which arises in proving the mean-square convergence ... more A solution to the trace convergence problem, which arises in proving the mean-square convergence for the approximation of iterated Stratonovich stochastic integrals, is proposed. This approximation is based on the representation of factorized Volterra-type functions as the orthogonal series. Solving the trace convergence problem involves the theory of trace class operators for symmetrized Volterra-type kernels. The main results are primarily focused on the approximation of iterated Stratonovich stochastic integrals, which are used to implement numerical methods for solving stochastic differential equations based on the Taylor–Stratonovich expansion.
Mathematics
Spectral representations of iterated Itô and Stratonovich stochastic integrals of arbitrary multi... more Spectral representations of iterated Itô and Stratonovich stochastic integrals of arbitrary multiplicity, including integrals from Taylor–Itô and Taylor–Stratonovich expansions, are obtained by the spectral method. They are required for the implementation of numerical methods for solving Itô and Stratonovich stochastic differential equations with high orders of mean-square and strong convergence. The purpose of such numerical methods is the modeling of nonlinear stochastic dynamics in many fields. This paper contains necessary theoretical results, as well as the results of numerical experiments.
MATEC Web of Conferences
The spectral representation for stochastic integration operators with respect to the Wiener proce... more The spectral representation for stochastic integration operators with respect to the Wiener process is proposed in the form of a composition of spectral characteristics used in the spectral form of mathematical description for control systems. This spectral representation can be defined relative to the various orthonormal bases. For given deterministic square-integrable kernels, the spectral characteristic of a stochastic integration operator is determined as an infinite random matrix. The main applications of such a representation suppose solving linear stochastic differential equations and modeling multiple or iterated Stratonovich stochastic integrals. Specific formulas are provided that allow to represent the spectral characteristic for the stochastic integration operator, the kernel of which is the Heaviside function, relative to Walsh functions and trigonometric functions.
MATEC Web of Conferences
The article is devoted to the use of computer technologies for forming the professional competenc... more The article is devoted to the use of computer technologies for forming the professional competences. We describe the study of basic and special courses for bachelor’s or master’s degrees in the field “Applied mathematics” at the Moscow Aviation Institute (National Research University). All kinds of practices during the study are considered. The continuous forming technology of professional competences at each stage of study with using the computer technologies is shown.
On the Orthogonal Expansion of the Iterated Stratonovich Stochastic Integrals
Herald of Dagestan State University
Maximum Cross Section Method in Optimal Filtering of Jump-Diffusion Random Processes
2019 15th International Asian School-Seminar Optimization Problems of Complex Systems (OPCS), 2019
The paper deals with the optimal filtering problem for random processes in dynamical systems whos... more The paper deals with the optimal filtering problem for random processes in dynamical systems whose mathematical models include stochastic differential equations with a compound Poisson process. The particle method and the maximum cross section method are applied for optimal filtering.
Analysis of jump diffusion systems by spectral method
COMPUTATIONAL MECHANICS AND MODERN APPLIED SOFTWARE SYSTEMS (CMMASS’2019), 2019
Systems with regime switching on manifolds
2018 14th International Conference "Stability and Oscillations of Nonlinear Control Systems" (Pyatnitskiy's Conference) (STAB), 2018
We propose an extension for the stochastic dynamoical systems whose trajectories belong to a give... more We propose an extension for the stochastic dynamoical systems whose trajectories belong to a given manifold. This extension is the stochastic systems with regime switching, namely the systems with a variable and random structure (stochastic hybrid systems, switching diffusions). The description, modeling and statistical analysis problems for such systems are considered.
Modified Continuous-Time Particle Filter Algorithm Without Overflow Errors
Applied Mathematics and Computational Mechanics for Smart Applications, 2021
2017 International Multi-Conference on Engineering, Computer and Information Sciences (SIBIRCON), 2017
Problems of complex systems' optim. / IEEE Sibircon Robust DMZ eq. for non-stationary systems Pro... more Problems of complex systems' optim. / IEEE Sibircon Robust DMZ eq. for non-stationary systems Problems of complex systems' optim. / IEEE Sibircon Robust DMZ eq. for non-stationary systems 2 K. A. Rybakov, "Solving approximately an optimal nonlinear filtering problem for stochastic differential systems by statistical modeling," Numer.
2017 International Multi-Conference on Engineering, Computer and Information Sciences (SIBIRCON), 2017
Problems of complex systems' optim. / IEEE Sibircon Random processes with invariants Dynamical sy... more Problems of complex systems' optim. / IEEE Sibircon Random processes with invariants Dynamical system is defined by the following Itô SDEs (n = 2): where W (t) is a scalar Wiener process (s = 1), i = 1, 2. Corresponding Stratonovich SDEs: Problems of complex systems' optim. / IEEE Sibircon Random processes with invariants
Approximate Filtering Methods in Continuous-Time Stochastic Systems
Advances in Theory and Practice of Computational Mechanics, 2020
The goal of this chapter is to consider algorithms based on the particle method for solving the o... more The goal of this chapter is to consider algorithms based on the particle method for solving the optimal filtering problem for nonlinear continuous-time stochastic observation systems not only by the minimum mean squared error estimate, but also by the maximum a posteriori estimate. Particle filters are proposed on the basis of Duncan–Mortensen–Zakai equation, as well as, on the basis of robust Duncan–Mortensen–Zakai equation. To find the mode of the conditional distribution approximately, Edgeworth series is used for the conditional probability density expansion. This approach allows to reduce significantly the computation time in contrast to finding the mode by estimating the conditional probability density, for example, the histogram or kernel estimations.
Approximate MMSE and MAP estimation using continuous-time particle filter
COMPUTATIONAL MECHANICS AND MODERN APPLIED SOFTWARE SYSTEMS (CMMASS’2019), 2019
The algorithm is proposed for solving approximately the optimal filtering problem for nonlinear c... more The algorithm is proposed for solving approximately the optimal filtering problem for nonlinear continuous-time stochastic observation systems that provides two estimates for the state. These estimates are the minimum mean squared error estimate and the maximum a posteriori estimate. The proposed algorithm is based on the continuous-time particle filter, which corresponds to the Duncan–Mortensen–Zakai equation. To find the mode of the conditional distribution approximately, the Edgeworth series is used for the conditional probability density function expansion. This approach allows one to significantly reduce the computation time in contrast to finding the mode by estimating the conditional probability density function, for example, by the histogram or the kernel density estimation.The algorithm is proposed for solving approximately the optimal filtering problem for nonlinear continuous-time stochastic observation systems that provides two estimates for the state. These estimates are the minimum mean squared error estimate and the maximum a posteriori estimate. The proposed algorithm is based on the continuous-time particle filter, which corresponds to the Duncan–Mortensen–Zakai equation. To find the mode of the conditional distribution approximately, the Edgeworth series is used for the conditional probability density function expansion. This approach allows one to significantly reduce the computation time in contrast to finding the mode by estimating the conditional probability density function, for example, by the histogram or the kernel density estimation.
Journal of Mathematical Sciences, 2020
Using maximum cross section method for filtering jump-diffusion random processes
Russian Journal of Numerical Analysis and Mathematical Modelling, 2020
The paper is focused on problem of filtering random processes in dynamical systems whose mathemat... more The paper is focused on problem of filtering random processes in dynamical systems whose mathematical models are described by stochastic differential equations with a Poisson component. The solution of a filtering problem supposes simulation of trajectories of solutions to a stochastic differential equation. The trajectory modelling procedure includes simulation of a Poisson flow permitting application of the maximum cross section method and its modification.
Russian Journal of Numerical Analysis and Mathematical Modelling, 2018
The aim of the paper is the construction and numerical solution of stochastic differential equati... more The aim of the paper is the construction and numerical solution of stochastic differential equations whose trajectories are located on a given smooth manifold with probability 1. Second order cylindrical surfaces, i.e., elliptic, hyperbolic, and parabolic cylinders serve as examples of such manifolds for the tree-dimensional space (the phase space is two-dimensional). Classes of stochastic differential equations are constructed for these surfaces and linear equations with multiplicative noise are marked in these classes. The results of modelling were used to estimate the deviations of numerical solutions from the manifold. A comparative analysis of considered examples was carried out for accuracy of eight numerical solution methods for stochastic differential equations.
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Papers by Konstantin Rybakov