Nonlinear Filtering based on Log-homotopy Particle Flow : Methodological Clarification and Numerical Evaluation

The state estimation of dynamical systems based on measurements is an ubiquitous problem. This is relevant in applications like robotics, industrial manufacturing, computer vision, target tracking etc. Recursive Bayesian methodology can then be used to estimate the hidden states of a dynamical system. The procedure consists of two steps: a process update based on solving the equations modelling the state evolution, and a measurement update in which the prior knowledge about the system is improved based on the measurements. For most real world systems, both the evolution and the measurement models are nonlinear functions of the system states. Additionally, both models can also be perturbed by random noise sources, which could be non-Gaussian in their nature. Unlike linear Gaussian models, there does not exist any optimal estimation scheme for nonlinear/non-Gaussian scenarios. This thesis investigates a particular method for nonlinear and non-Gaussian data assimilation, termed as the log-homotopy based particle flow. Practical filters based on such flows have been known in the literature as Daum Huang filters (DHF), named after the developers. The key concept behind such filters is the gradual inclusion of measurements to counter a major drawback of single step update schemes like the particle filters i.e. namely the degeneracy. This could refer to a situation where the likelihood function has its probability mass well seperated from the prior density, and/or is peaked in comparison. Conventional sampling or grid based techniques do not perform well under such circumstances and in order to achieve a reasonable accuracy, could incur a high processing cost. DHF is a sampling based scheme, which provides a unique way to tackle this challenge thereby lowering the processing cost. This is achieved by dividing the single measurement update step into multiple sub steps, such that particles originating from their prior locations are graduated incrementally until they reach their final locations. The motion is controlled by a differential equation, which is numerically solved to yield the updated states. DH filters, even though not new in the literature, have not been fully explored in the detail yet. They lack the in-depth analysis that the other contemporary filters have gone through. Especially, the implementation details for the DHF are very application specific. In this work, we have pursued four main objectives. The first objective is the exploration of theoretical concepts behind DHF. Secondly, we build an understanding of the existing implementation framework and highlight its potential shortcomings. As a sub task to this, we carry out a detailed study of important factors that affect the performance of a DHF, and suggest possible improvements for each of those factors. The third objective is to use the improved implementation to derive new filtering algorithms. Finally, we have extended the DHF theory and derived new flow equations and filters to cater for more general scenarios. Improvements in the implementation architecture of a standard DHF is one of the key contributions of this thesis. The scope of the applicability of DHF is expanded by combining it with other schemes like the Sequential Markov chain Monte Carlo and the tensor decomposition based solution of the Fokker Planck equation, resulting in the development of new nonlinear filtering algorithms. The standard DHF, using improved implementation and the newly derived algorithms are tested in challenging simulated test scenarios. Detailed analysis have been carried out, together with the comparison against more established filtering schemes. Estimation error and the processing time are used as important performance parameters. We show that our new filtering algorithms exhibit marked performance improvements over the traditional schemes

Fast global spectral methods for three-dimensional partial differential equations

Global spectral methods offer the potential to compute solutions of partial differential equations numerically to very high accuracy. In this work, we develop a novel global spectral method for linear partial differential equations on cubes by extending ideas of Chebop2 [Townsend and Olver, J. Comput. Phys., 299 (2015)] to the three-dimensional setting utilizing expansions in tensorized polynomial bases. Solving the discretized PDE involves a linear system that can be recast as a linear tensor equation. Under suitable additional assumptions, the structure of these equations admits for an efficient solution via the blocked recursive solver [Chen and Kressner, Numer. Algorithms, 84 (2020)]. In the general case, when these assumptions are not satisfied, this solver is used as a preconditioner to speed up computations

A Review of Computational Stochastic Elastoplasticity

Heterogeneous materials at the micro-structural level are usually subjected to several uncertainties. These materials behave according to an elastoplastic model, but with uncertain parameters. The present review discusses recent developments in numerical approaches to these kinds of uncertainties, which are modelled as random elds like Young's modulus, yield stress etc. To give full description of random phenomena of elastoplastic materials one needs adequate mathematical framework. The probability theory and theory of random elds fully cover that need. Therefore, they are together with the theory of stochastic nite element approach a subject of this review. The whole group of di erent numerical stochastic methods for the elastoplastic problem has roots in the classical theory of these materials. Therefore, we give here the classical formulation of plasticity in very concise form as well as some of often used methods for solving this kind of problems. The main issues of stochastic elastoplasticity as well as stochastic problems in general are stochastic partial di erential equations. In order to solve them we must discretise them. Methods of solving and discretisation are called stochastic methods. These methods like Monte Carlo, Perturbation method, Neumann series method, stochastic Galerkin method as well as some other very known methods are reviewed and discussed here

GPU computing for accelerating the numerical Path Integration approach

The paper discusses a novel approach of accelerating the numerical Path Integration method, used for generating a stationary joint response probability density function of a dynamic system subjected to a random excitation, by the GPU computing. The paper proposes the parallelization of nested loops technique and demonstrates the advantages of GPU computing. Two, three and four dimensional in space problems are investigated as a part of the pilot project and the achieved maximum accelerations are reported. Three degree-of-freedom system (6D) is approached by the Path Integration technique for the first time. The application of the proposed GPU methodology for problems of stochastic dynamics and reliability are discussed