Grid methods for Bayes-optimal continuous-discrete filtering and utilizing a functional tensor train representation

Colin Fox, Sergey Dolgov, Malcolm E.K. Morrison, Timothy C.A. Molteno

Research output: Contribution to journalArticlepeer-review


Optimal continuous-discrete 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 grid-methods that represent density functions on a mesh, or grid. For low-dimensional, smooth systems the finite-volume method is an effective solver that gives estimates that converge to the optimal continuous-time values. We give numerical examples to show that this finite-volume filter is able to handle multi-modal filtering distributions that result from rank-deficient observations, and that Bayes-optimal parameter estimation may be performed within the filtering process. The naïve discretization of density functions used in the finite-volume filter leads to an exponential increase of computational cost and storage with increasing dimension, that makes the finite-volume filter unfeasible for higher-dimensional 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 languageEnglish
JournalInverse Problems in Science and Engineering
Early online date6 Jan 2021
Publication statusE-pub ahead of print - 6 Jan 2021


  • 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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