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Abstract
Optimal continuousdiscrete filtering for a nonlinear system requires evolving the forward Kolmogorov equation, that is a Fokker–Planck equation, in alternation with Bayes' conditional updating. We present two numerical gridmethods that represent density functions on a mesh, or grid. For lowdimensional, smooth systems the finitevolume method is an effective solver that gives estimates that converge to the optimal continuoustime values. We give numerical examples to show that this finitevolume filter is able to handle multimodal filtering distributions that result from rankdeficient observations, and that Bayesoptimal parameter estimation may be performed within the filtering process. The naïve discretization of density functions used in the finitevolume filter leads to an exponential increase of computational cost and storage with increasing dimension, that makes the finitevolume filter unfeasible for higherdimensional problems. We circumvent this ‘curse of dimensionality’ by using a tensor train representation (or approximation) of density functions and employ an efficient implicit PDE solver that operates on the tensor train representation. We present numerical examples of tracking n weakly coupled pendulums in continuous time to demonstrate filtering with complex density functions in up to 80 dimensions.
Original language  English 

Pages (fromto)  11991217 
Number of pages  19 
Journal  Inverse Problems in Science and Engineering 
Volume  29 
Issue number  8 
Early online date  6 Jan 2021 
DOIs  
Publication status  Published  31 Dec 2021 
Keywords
 Bayesian sequential inference
 dynamical system
 Filtering
 numerical solution of partial differential equations
 tensor train
ASJC Scopus subject areas
 Engineering(all)
 Computer Science Applications
 Applied Mathematics
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Tensor decomposition sampling algorithms for Bayesian inverse problems
Engineering and Physical Sciences Research Council
1/03/21 → 28/02/24
Project: Research council