scholarly journals On the convergence of the wavelet-Galerkin method for nonlinear filtering

2010 ◽  
Vol 20 (1) ◽  
pp. 93-108 ◽  
Author(s):  
Łukasz Nowak ◽  
Monika Pasławska-Południak ◽  
Krystyna Twardowska
Keyword(s):  
Galerkin Method ◽  
Compact Support ◽  
Nonlinear Filtering ◽  
Euler Scheme ◽  
Zakai Equation ◽  
Implicit Euler Scheme ◽  
Galerkin Scheme ◽  
Wavelet Galerkin Method ◽  
Biorthogonal Basis ◽  
Filtering Problem

On the convergence of the wavelet-Galerkin method for nonlinear filteringThe aim of the paper is to examine the wavelet-Galerkin method for the solution of filtering equations. We use a wavelet biorthogonal basis with compact support for approximations of the solution. Then we compute the Zakai equation for our filtering problem and consider the implicit Euler scheme in time and the Galerkin scheme in space for the solution of the Zakai equation. We give theorems on convergence and its rate. The method is numerically much more efficient than the classical Galerkin method.

Fractional generalizations of filtering problems and their associated fractional Zakai equations

2014 ◽  
Vol 17 (3) ◽  
Author(s):  
Sabir Umarov ◽  
Frederick Daum ◽  
Kenric Nelson
Keyword(s):  
Brownian Motion ◽  
Fractional Derivative ◽  
Lévy Process ◽  
Nonlinear Filtering ◽  
Levy Process ◽  
Zakai Equation ◽  
Filtering Problem

AbstractIn this paper we discuss fractional generalizations of the filtering problem. The ”fractional” nature comes from time-changed state or observation processes, basic ingredients of the filtering problem. The mathematical feature of the fractional filtering problem emerges as the Riemann-Liouville or Caputo-Djrbashian fractional derivative in the associated Zakai equation. We discuss fractional generalizations of the nonlinear filtering problem whose state and observation processes are driven by time-changed Brownian motion or/and Lévy process.

Uniqueness for measure-valued equations of nonlinear filtering for stochastic dynamical systems with Lévy noise

2018 ◽  
Vol 50 (2) ◽  
pp. 396-413
Author(s):  
Huijie Qiao
Keyword(s):  
Dynamical Systems ◽  
Nonlinear Filtering ◽  
General Assumption ◽  
Observation System ◽  
Pathwise Uniqueness ◽  
Zakai Equation ◽  
Stochastic Dynamical Systems ◽  
Gaussian Signal ◽  
Non Gaussian ◽  
Filtering Problem

Abstract In the paper we study the Zakai and Kushner–Stratonovich equations of the nonlinear filtering problem for a non-Gaussian signal-observation system. Moreover, we prove that under some general assumption, the Zakai equation has pathwise uniqueness and uniqueness in joint law, and the Kushner–Stratonovich equation is unique in joint law.

scholarly journals Nonlinear filtering of a system of logistic equations

1997 ◽  
Vol 55 (2) ◽  
pp. 219-238
Author(s):  
Rehez Ahlip ◽  
Vo Anh
Keyword(s):  
Conditional Distribution ◽  
Nonlinear Filtering ◽  
Perturbation Technique ◽  
Nonlinear Filter ◽  
Conditional Density ◽  
Nonlinear Stochastic System ◽  
Stochastic Integrals ◽  
Zakai Equation ◽  
Logistic Equations ◽  
Filtering Problem

This paper is concerned with the filtering problem for a nonlinear stochastic system of prey-predator logistic equations. Based on the innovations approach, we establish the Zakai equation for the unnormalised conditional distribution and the adjoint Zakai equation for the unnormalised conditional density of the nonlinear filter. Using a perturbation technique, we obtain the appropriate expressions for the unnormalised conditional distribution and density of stochastic integrals with respect to the observation processes.

scholarly journals Unbiased estimation of the solution to Zakai’s equation

2020 ◽  
Vol 26 (2) ◽  
pp. 113-129
Author(s):  
Hamza M. Ruzayqat ◽  
Ajay Jasra
Keyword(s):  
Continuous Time ◽  
Unbiased Estimation ◽  
Finite Variance ◽  
Linear Filtering ◽  
Multilevel Monte Carlo ◽  
Zakai Equation ◽  
Unbiased Estimators ◽  
First Order ◽  
Multilevel Monte Carlo Method ◽  
Filtering Problem

AbstractIn the following article, we consider the non-linear filtering problem in continuous time and in particular the solution to Zakai’s equation or the normalizing constant. We develop a methodology to produce finite variance, almost surely unbiased estimators of the solution to Zakai’s equation. That is, given access to only a first-order discretization of solution to the Zakai equation, we present a method which can remove this discretization bias. The approach, under assumptions, is proved to have finite variance and is numerically compared to using a particular multilevel Monte Carlo method.

Time and space meshes adaptivity for the resolution of a semi-linear heat equation

2021 ◽  
pp. 1-10
Author(s):  
Nejmeddine Chorfi
Keyword(s):  
Parabolic Equation ◽  
Heat Equation ◽  
Order Of Convergence ◽  
Linear Parabolic Equation ◽  
Spectral Elements ◽  
Euler Scheme ◽  
Time Step ◽  
Implicit Euler Scheme ◽  
Linear Heat ◽  
Error Indicators

The aim of this work is to highlight that the adaptivity of the time step when combined with the adaptivity of the spectral mesh is optimal for a semi-linear parabolic equation discretized by an implicit Euler scheme in time and spectral elements method in space. The numerical results confirm the optimality of the order of convergence. The later is similar to the order of the error indicators.

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