The Hitchhiker's Guide to Nonlinear Filtering

Nonlinear filtering is the problem of online estimation of a dynamic hidden variable from incoming data and has vast applications in different fields, ranging from engineering, machine learning, economic science and natural sciences. We start our review of the theory on nonlinear filtering from the simplest `filtering' task we can think of, namely static Bayesian inference. From there we continue our journey through discrete-time models, which is usually encountered in machine learning, and generalize to and further emphasize continuous-time filtering theory. The idea of changing the probability measure connects and elucidates several aspects of the theory, such as the parallels between the discrete- and continuous-time problems and between different observation models. Furthermore, it gives insight into the construction of particle filtering algorithms. This tutorial is targeted at scientists and engineers and should serve as an introduction to the main ideas of nonlinear filtering, and as a segway to more advanced and specialized literature.Comment: 64 page

Negative-free approximation of probability density function for nonlinear projection filter

Several approaches have been developed to estimate probability density functions (pdfs). The pdf has two important properties: the integration of pdf over whole sampling space is equal to 1 and the value of pdf in the sampling space is greater than or equal to zero. The first constraint can be easily achieved by the normalisation. On the other hand, it is hard to impose the non-negativeness in the sampling space. In a pdf estimation, some areas in the sampling space might have negative pdf values. It produces unreasonable moment values such as negative probability or variance. A transformation to guarantee the negative-free pdf over a chosen sampling space is presented and it is applied to the nonlinear projection filter. The filter approximates the pdf to solve nonlinear estimation problems. For simplicity, one-dimensional nonlinear system is used as an example to show the derivations and it can be readily generalised for higher dimensional systems. The efficiency of the proposed method is demonstrated by numerical simulations. The simulations also show that, for the same level of approximation error in the filter, the required number of basis functions with the transformation is a lot smaller than the ones without transformation. This would largely benefit the computational cost reduction

Stochastic Particle Flow for Nonlinear High-Dimensional Filtering Problems

A series of novel filters for probabilistic inference that propose an alternative way of performing Bayesian updates, called particle flow filters, have been attracting recent interest. These filters provide approximate solutions to nonlinear filtering problems. They do so by defining a continuum of densities between the prior probability density and the posterior, i.e. the filtering density. Building on these methods' successes, we propose a novel filter. The new filter aims to address the shortcomings of sequential Monte Carlo methods when applied to important nonlinear high-dimensional filtering problems. The novel filter uses equally weighted samples, each of which is associated with a local solution of the Fokker-Planck equation. This hybrid of Monte Carlo and local parametric approximation gives rise to a global approximation of the filtering density of interest. We show that, when compared with state-of-the-art methods, the Gaussian-mixture implementation of the new filtering technique, which we call Stochastic Particle Flow, has utility in the context of benchmark nonlinear high-dimensional filtering problems. In addition, we extend the original particle flow filters for tackling multi-target multi-sensor tracking problems to enable a comparison with the new filter

Learning dynamics on invariant measures using PDE-constrained optimization

We extend the methodology in [Yang et al., 2023] to learn autonomous continuous-time dynamical systems from invariant measures. The highlight of our approach is to reformulate the inverse problem of learning ODEs or SDEs from data as a PDE-constrained optimization problem. This shift in perspective allows us to learn from slowly sampled inference trajectories and perform uncertainty quantification for the forecasted dynamics. Our approach also yields a forward model with better stability than direct trajectory simulation in certain situations. We present numerical results for the Van der Pol oscillator and the Lorenz-63 system, together with real-world applications to Hall-effect thruster dynamics and temperature prediction, to demonstrate the effectiveness of the proposed approach.Comment: This article may be downloaded for personal use only. Any other use requires prior permission of the author and AIP Publishing. This article appeared in Chaos: An Interdisciplinary Journal of Nonlinear Science, Volume 33, Issue 6, June 2023, and may be found at https://doi.org/10.1063/5.014967

Runaway-electron model development and validation in tokamaks

Magnetic confinement fusion (MCF), in which a hot plasma at more than 100 million kelvin is confined using magnetic fields, is the most successful fusion energy concept developed to date. After decades of research, MCF devices designed to demonstrate a positive net energy output are being constructed, completing a crucial milestone on the path to making fusion a commercially viable energy source. Several hurdles remain on this path, however, and one of the most pressing issues concerns the sudden and rapid loss of confinement of the fusion plasma, known as a disruption. An undesirable consequence of disruptions is the acceleration of a fraction of the plasma electrons to relativistic energies which---if the electrons were to strike the device wall---could deposit a significant portion of the plasma energy on a small area, causing severe and potentially irreparable damage.The aim of this thesis is to develop a robust simulation tool capable of accurately predicting the number of runaway electrons produced in different disruption scenarios. Since the evolution of the runaway electrons affects the background plasma, it is important to also allow quantities such as electron temperature, ion density, and electric field to evolve self-consistently in the simulation. This leads to a tightly coupled system of non-linear equations, and to solve it we have developed the numerical tool DREAM.The complexity of the models used to simulate runaway electrons demands that the validity of the models is carefully evaluated by comparing predictions with existing experimental data. One of the most informative techniques for studying the dynamics of runaway electrons in MCF experiments utilises synchrotron radiation, and to facilitate direct comparison of runaway electron simulations with experiments we have developed the synthetic diagnostic framework SOFT. Using SOFT, we study runaway electrons in the ASDEX Upgrade and TCV fusion devices, and develop powerful techniques for\ua0 accurately extracting information about the location and momentum of runaway electrons

Stochastic Particle Flow for Nonlinear High-Dimensional Filtering Problems

A series of novel filters for probabilistic inference that propose an alternative way of performing Bayesian updates, called particle flow filters, have been attracting recent interest. These filters provide approximate solutions to nonlinear filtering problems. They do so by defining a continuum of densities between the prior probability density and the posterior, i.e. the filtering density. Building on these methods' successes, we propose a novel filter. The new filter aims to address the shortcomings of sequential Monte Carlo methods when applied to important nonlinear high-dimensional filtering problems. The novel filter uses equally weighted samples, each of which is associated with a local solution of the Fokker-Planck equation. This hybrid of Monte Carlo and local parametric approximation gives rise to a global approximation of the filtering density of interest. We show that, when compared with state-of-the-art methods, the Gaussian-mixture implementation of the new filtering technique, which we call Stochastic Particle Flow, has utility in the context of benchmark nonlinear high-dimensional filtering problems. In addition, we extend the original particle flow filters for tackling multi-target multi-sensor tracking problems to enable a comparison with the new filter