scholarly journals Kalman Filter and Its Modern Extensions for the Continuous-Time Nonlinear Filtering Problem

2017 ◽  
Vol 140 (3) ◽  
Author(s):  
Amirhossein Taghvaei ◽  
Jana de Wiljes ◽  
Prashant G. Mehta ◽  
Sebastian Reich
Keyword(s):  
Continuous Time ◽  
Nonlinear Filtering ◽  
Sequential Monte Carlo ◽  
Optimal Transport ◽  
Nonlinear Problems ◽  
Improved Stability ◽  
Consistent Solution ◽  
Gaussian Case ◽  
Non Gaussian ◽  
Filtering Problem

This paper is concerned with the filtering problem in continuous time. Three algorithmic solution approaches for this problem are reviewed: (i) the classical Kalman–Bucy filter, which provides an exact solution for the linear Gaussian problem; (ii) the ensemble Kalman–Bucy filter (EnKBF), which is an approximate filter and represents an extension of the Kalman–Bucy filter to nonlinear problems; and (iii) the feedback particle filter (FPF), which represents an extension of the EnKBF and furthermore provides for a consistent solution in the general nonlinear, non-Gaussian case. The common feature of the three algorithms is the gain times error formula to implement the update step (to account for conditioning due to the observations) in the filter. In contrast to the commonly used sequential Monte Carlo methods, the EnKBF and FPF avoid the resampling of the particles in the importance sampling update step. Moreover, the feedback control structure provides for error correction potentially leading to smaller simulation variance and improved stability properties. The paper also discusses the issue of nonuniqueness of the filter update formula and formulates a novel approximation algorithm based on ideas from optimal transport and coupling of measures. Performance of this and other algorithms is illustrated for a numerical example.

scholarly journals Application of ensemble transform data assimilation methods for parameter estimation in nonlinear problems

2020 ◽  
Author(s):  
Sangeetika Ruchi ◽  
Svetlana Dubinkina ◽  
Jana de Wiljes
Keyword(s):  
Monte Carlo ◽  
Sequential Monte Carlo ◽  
Optimal Transport ◽  
Hybrid Approach ◽  
Computational Cost ◽  
Nonlinear Problems ◽  
High Dimensional ◽  
Unknown Parameters ◽  
Sequential Monte Carlo Method ◽  
Ensemble Transform

Abstract. Identification of unknown parameters on the basis of partial and noisy data is a challenging task in particular in high dimensional and nonlinear settings. Gaussian approximations to the problem, such as ensemble Kalman inversion, tend to be robust, computationally cheap and often produce astonishingly accurate estimations despite the inherently wrong underlying assumptions. Yet there is a lot of room for improvement specifically regarding the description of the associated statistics. The tempered ensemble transform particle filter is an adaptive sequential Monte Carlo method, where resampling is based on optimal transport mapping. Unlike ensemble Kalman inversion it does not require any assumptions regarding the posterior distribution and hence has shown to provide promising results for non-linear non-Gaussian inverse problems. However, the improved accuracy comes with the price of much higher computational complexity and the method is not as robust as the ensemble Kalman inversion in high dimensional problems. In this work, we add an entropy inspired regularisation factor to the underlying optimal transport problem that allows to considerably reduce the high computational cost via Sinkhorn iterations. Further, the robustness of the method is increased via an ensemble Kalman inversion proposal step before each update of the samples, which is also referred to as hybrid approach. The promising performance of the introduced method is numerically verified by testing it on a steady-state single-phase Darcy flow model with two different permeability configurations. The results are compared to the output of ensemble Kalman inversion, and Markov Chain Monte Carlo methods results are computed as a benchmark.

scholarly journals Fast hybrid tempered ensemble transform filter formulation for Bayesian elliptical problems via Sinkhorn approximation

2021 ◽  
Vol 28 (1) ◽  
pp. 23-41
Author(s):  
Sangeetika Ruchi ◽  
Svetlana Dubinkina ◽  
Jana de Wiljes
Keyword(s):  
Monte Carlo ◽  
Posterior Distribution ◽  
Sequential Monte Carlo ◽  
Optimal Transport ◽  
Hybrid Approach ◽  
Computational Cost ◽  
High Dimensional ◽  
Non Linear ◽  
Non Gaussian ◽  
Ensemble Transform

Abstract. Identification of unknown parameters on the basis of partial and noisy data is a challenging task, in particular in high dimensional and non-linear settings. Gaussian approximations to the problem, such as ensemble Kalman inversion, tend to be robust and computationally cheap and often produce astonishingly accurate estimations despite the simplifying underlying assumptions. Yet there is a lot of room for improvement, specifically regarding a correct approximation of a non-Gaussian posterior distribution. The tempered ensemble transform particle filter is an adaptive Sequential Monte Carlo (SMC) method, whereby resampling is based on optimal transport mapping. Unlike ensemble Kalman inversion, it does not require any assumptions regarding the posterior distribution and hence has shown to provide promising results for non-linear non-Gaussian inverse problems. However, the improved accuracy comes with the price of much higher computational complexity, and the method is not as robust as ensemble Kalman inversion in high dimensional problems. In this work, we add an entropy-inspired regularisation factor to the underlying optimal transport problem that allows the high computational cost to be considerably reduced via Sinkhorn iterations. Further, the robustness of the method is increased via an ensemble Kalman inversion proposal step before each update of the samples, which is also referred to as a hybrid approach. The promising performance of the introduced method is numerically verified by testing it on a steady-state single-phase Darcy flow model with two different permeability configurations. The results are compared to the output of ensemble Kalman inversion, and Markov chain Monte Carlo methods results are computed as a benchmark.

scholarly journals A Kusuoka–Lyons–Victoir particle filter

2013 ◽  
Vol 469 (2156) ◽  
pp. 20130076 ◽  
Author(s):  
Dan Crisan ◽  
Salvador Ortiz-Latorre
Keyword(s):  
Particle Filter ◽  
Continuous Time ◽  
Nonlinear Filtering ◽  
Computational Effort ◽  
Observation Data ◽  
Minimal Variance ◽  
Discretization Grid ◽  
Filtering Problem ◽  
Grid Mesh ◽  
Number Of Particles

The aim of this paper is to introduce a new numerical algorithm for solving the continuous time nonlinear filtering problem. In particular, we present a particle filter that combines the Kusuoka–Lyons–Victoir (KLV) cubature method on Wiener space to approximate the law of the signal with a minimal variance ‘thinning’ method, called the tree-based branching algorithm (TBBA) to keep the size of the cubature tree constant in time. The novelty of our approach resides in the adaptation of the TBBA algorithm to simultaneously control the computational effort and incorporate the observation data into the system. We provide the rate of convergence of the approximating particle filter in terms of the computational effort (number of particles) and the discretization grid mesh. Finally, we test the performance of the new algorithm on a benchmark problem (the Beneš filter).

scholarly journals Information geometric nonlinear filtering

2015 ◽  
Vol 18 (02) ◽  
pp. 1550014 ◽  
Author(s):  
Nigel J. Newton
Keyword(s):  
Continuous Time ◽  
Nonlinear Filtering ◽  
Riemannian Metric ◽  
Point Of View ◽  
Posterior Distributions ◽  
Hilbert Manifold ◽  
Information Theoretic ◽  
Finite Dimensional ◽  
Geometric Representations ◽  
Filtering Problem

This paper develops information geometric representations for nonlinear filters in continuous time. The posterior distribution associated with an abstract nonlinear filtering problem is shown to satisfy a stochastic differential equation on a Hilbert information manifold. This supports the Fisher metric as a pseudo-Riemannian metric. Flows of Shannon information are shown to be connected with the quadratic variation of the process of posterior distributions in this metric. Apart from providing a suitable setting in which to study such information-theoretic properties, the Hilbert manifold has an appropriate topology from the point of view of multi-objective filter approximations. A general class of finite-dimensional exponential filters is shown to fit within this framework, and an intrinsic evolution equation, involving Amari's -1-covariant derivative, is developed for such filters. Three example systems, one of infinite dimension, are developed in detail.

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 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.

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