Numerical methods for solving the stochastic filtering problem

2002 ◽  
pp. 1-20
Author(s):  
Dan Crisan
Keyword(s):  
Numerical Methods ◽  
Stochastic Filtering ◽  
Filtering Problem

scholarly journals The stochastic filtering problem: a brief historical account

2014 ◽  
Vol 51 (A) ◽  
pp. 13-22
Author(s):  
Dan Crisan
Keyword(s):  
Risk Estimation ◽  
Historical Account ◽  
Stochastic Filtering ◽  
Research Areas ◽  
Computer Scientists ◽  
Classical Mathematics ◽  
The Subject ◽  
Mathematical Techniques ◽  
New Research ◽  
Filtering Problem

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

scholarly journals Heterogeneous population dynamical model: a filtering problem

2005 ◽  
Vol 42 (02) ◽  
pp. 346-361 ◽  
Author(s):  
A. Gerardi ◽  
P. Tardelli
Keyword(s):  
Finite Number ◽  
Conditional Distribution ◽  
Heterogeneous Population ◽  
Time Approximation ◽  
Stochastic Filtering ◽  
Discrete Time Approximation ◽  
Different Types ◽  
Number Of Classes ◽  
Filtering Problem ◽  
Over Time

We consider a heterogeneous population of identical particles divided into a finite number of classes according to their level of health. The partition can change over time, and a suitable exchangeability assumption is made to allow for having identical items of different types. The partition is not observed; we only observe the cardinality of a particular class. We discuss the problem of finding the conditional distribution of particle lifetimes, given such observations, using stochastic filtering techniques. In particular, a discrete-time approximation is given.

scholarly journals Стохастическая ячеистая модель очистки моторного масла от механических примесей объёмным фильтрованием

2020 ◽  
Author(s):  
G.P. Kicha ◽  
L.A. Semeniuk ◽  
M.I. Tarasov
Keyword(s):  
Pore Space ◽  
Planck Equation ◽  
Complex Interaction ◽  
Dispersed Phase ◽  
Generalized Coordinates ◽  
Stochastic Filtering ◽  
Physical Laws ◽  
Random Particle ◽  
Filtering Problem ◽  
Dimensionless Coordinate

Приведено описание ячеистой модели фильтрования, которая создавалась на сочетании вероятностно-статистических методов с точным описанием на основе физических законов поведения дисперсной фазы при сложном взаимодействии её с дисперсионной средой. Выделены основные силы, действующие на частицу в потоке при фильтровании. Показано, что наибольшее воздействие на отсев оказывают силы Лондона-Ван-дер-Ваальса и электрокинетические, обусловленные полярными молекулами продуктов старения масла и моюще-диспергирующими присадками, которыми оно легируется. Выяснено, что в наименьшей степени на отсев влияют силы тяжести и Архимеда. Объединение детерминированных и случайных воздействий на дисперсную фазу осуществлено на базе уравнения Колмогорова Фоккера Планка. Показаны методы его формирования так, чтобы рассматриваемому случайному переносу частиц в поровом пространстве, идентифицированному совокупностью обобщенных координат и скоростей, соответствовало уравнение для многомерной плотности вероятности, отождествляемой с концентрацией дисперсной фазы. Приведены расчётные формулы для компонентов скорости частиц, по которым можно рассчитать граничную траекторию, определить безразмерную координату и фракционный коэффициент отсева. Определены краевые условия стохастической задачи фильтрования. Записаны интегралы для расчета фракционного отсева через паток вероятности. Проанализированы возможности детерминированной и стохастической ячеистых моделей фильтрования, показана их адекватность.A description is given of a cellular filtering model that was created using a combination of probabilistic and statistical methods with an accurate description based on the physical laws of the behavior of the dispersed phase during its complex interaction with the dispersion medium. The main forces acting on a particle in a stream during filtration are identified. It has been shown that the London-Van der Waals forces and electro kinetic forces caused by polar molecules of oil aging products and detergent-dispersant additives with which it is doped have the greatest impact on screening. It has been found that gravity and Archimedes force have the leas effect on screening. The combination of deterministic and random effects on the dispersed phase was carried out based on the Kolmogorov Fokker Planck equation. The methods of its formation are shown so that the equation for the multidimensional probability density, identified with the concentration of the dispersed phase, corresponds to the random particle transport in the pore space, identified by the set of generalized coordinates and velocities. Calculation formulas are given for the particle velocity components by which one can calculate the boundary trajectory and determine the dimensionless coordinate and fractional dropout coefficient. The boundary conditions of the stochastic filtering problem are determined. The integrals for calculating the fractional dropout through the molasses of probability are recorded. The possibilities of deterministic and stochastic cellular filtration models are analyzed, and their adequacy is shown.

scholarly journals The stochastic filtering problem: a brief historical account

2014 ◽  
Vol 51 (A) ◽  
pp. 13-22 ◽  
Author(s):  
Dan Crisan
Keyword(s):  
Risk Estimation ◽  
Historical Account ◽  
Stochastic Filtering ◽  
Research Areas ◽  
Computer Scientists ◽  
Classical Mathematics ◽  
The Subject ◽  
Mathematical Techniques ◽  
New Research ◽  
Filtering Problem

Onwards from the mid-twentieth century, the stochastic filtering problem has caught the attention of thousands of mathematicians, engineers, statisticians, and computer scientists. Its applications span the whole spectrum of human endeavour, including satellite tracking, credit risk estimation, human genome analysis, and speech recognition. Stochastic filtering has engendered a surprising number of mathematical techniques for its treatment and has played an important role in the development of new research areas, including stochastic partial differential equations, stochastic geometry, rough paths theory, and Malliavin calculus. It also spearheaded research in areas of classical mathematics, such as Lie algebras, control theory, and information theory. The aim of this paper is to give a brief historical account of the subject concentrating on the continuous-time framework.

scholarly journals Cellular filtration model in purifying fuels and lubricants from mechanical impurities

2020 ◽  
Vol 2020 (1) ◽  
pp. 60-71
Author(s):  
Liudmila Anatolievna Semeniuk ◽  
Gennadiy Petrovich Kicha ◽  
Maxim Igorevich Tarasov
Keyword(s):  
Numerical Methods ◽  
Complex Structure ◽  
Tangential Velocity ◽  
Constrained Motion ◽  
Dispersion Phase ◽  
Particle Position ◽  
Stochastic Filtering ◽  
Insoluble Particles ◽  
External Forces ◽  
Filtration Model

The paper describes a cellular model of fuel and lubricants cleaning developed by using a filtering material with a complex structure of cells with cylindrical and spherical collectors. It was offered to carry out the correction of moving dirty particles in cells of a filtering material subject to deviations of trajectories of insoluble particles under the external forces concerning trajectory of the dispersive medium in a cell. The constrained motion of polydisperse particles (subject to parietal effects) under perturbed flow of cylindrical and spherical collectors by the dispersion medium has been considered. Modeling efficient purification of motor fuels and oils from mechanical impurities has been carried out with a complex account of the main mechanisms of dispersion phase elimination during filtration. There has been given a diagram of the cellular filtration model with the radii of the filter cell (shell) and collector, the radial coordinates of the particle position, the angular coordinates of the particle position and the boundary screening path, the direction of gravity (acceleration), the velocity vectors of the incoming flow and a particle, the vectors of the radial and tangential velocity. According to the results of solving the differential equation by numerical methods, the boundary trajectory was constructed and the fractional dropout coefficient was found. The results of a computational experiment are illustrated. It has been inferred that considering the deviation when calculating the resistance against the particle motion using Stokes method reduces the filterability of the dispersed phase due to the proximity of the collector. The complexity of numerical methods for implementing a cellular stochastic filtering model based on the finite difference method is noted. It is recommended to consider various mechanisms of restraining the insoluble impurities in the interaction

scholarly journals Heterogeneous population dynamical model: a filtering problem

2005 ◽  
Vol 42 (2) ◽  
pp. 346-361 ◽  
Author(s):  
A. Gerardi ◽  
P. Tardelli
Keyword(s):  
Finite Number ◽  
Conditional Distribution ◽  
Heterogeneous Population ◽  
Time Approximation ◽  
Stochastic Filtering ◽  
Discrete Time Approximation ◽  
Different Types ◽  
Number Of Classes ◽  
Filtering Problem ◽  
Over Time

We consider a heterogeneous population of identical particles divided into a finite number of classes according to their level of health. The partition can change over time, and a suitable exchangeability assumption is made to allow for having identical items of different types. The partition is not observed; we only observe the cardinality of a particular class. We discuss the problem of finding the conditional distribution of particle lifetimes, given such observations, using stochastic filtering techniques. In particular, a discrete-time approximation is given.

Numerical Methods

2019 ◽  
Author(s):  
Rajesh Kumar Gupta
Keyword(s):  
Numerical Methods
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