Crucinio, Francesca (2021) Some interacting particle methods with non-standard interactions. PhD thesis, University of Warwick.

Preview

PDF WRAP_Theses_Crucinio_2021.pdf - Submitted Version - Requires a PDF viewer. Download (3183Kb) | Preview

Request Changes to record.

Abstract

Interacting particle methods are widely used to perform inference in complex models, with applications ranging from Bayesian statistics to applied sciences. This thesis is concerned with the study of families of interacting particles which present non-standard interactions. The non-standard interactions that we study arise from the particular class of problems we are interested in, Fredholm integral equations of the first kind or from algorithmic design, as in the case of the Divide and Conquer sequential Monte Carlo algorithm.

Fredholm integral equations of the first kind are a class of inverse ill-posed problems for which finding numerical solutions remains challenging. These equations are ubiquitous in applied sciences and engineering, with applications in epidemiology, medical imaging, nonlinear regression settings and partial differential equations. We develop two interacting particle methods which provide an adaptive stochastic discretisation and do not require strong assumptions on the solution. While similar to well-studied families of interacting particle methods the two algorithms that we develop present non-standard elements and require a novel theoretical analysis. We study the theoretical properties of the two proposed algorithms, establishing a strong law of large numbers and Lp error estimates, and compare their performances with alternatives on a suite of examples, including simulated data and realistic systems.

The Divide and Conquer sequential Monte Carlo algorithm is an interacting particle method in which different sequential Monte Carlo approximations are merged together according to the topology of a given tree. We study the effect of the additional interactions due to the merging operations on the theoretical properties of the algorithm. Specifically, we show that the approximation error decays at rate N

Item Type:	Thesis (PhD)
Subjects:	Q Science > QA Mathematics Q Science > QC Physics
Library of Congress Subject Headings (LCSH):	Fredholm equations, Monte Carlo method, Particles (Nuclear physics)
Official Date:	May 2021
Dates:	Date Event May 2021 UNSPECIFIED
Institution:	University of Warwick
Theses Department:	Department of Statistics
Thesis Type:	PhD
Publication Status:	Unpublished
Supervisor(s)/Advisor:	Johansen, Adam M. ; Doucet, Arnaud
Format of File:	pdf
Extent:	xii, 213 leaves : illustrations
Language:	eng

Request changes or add full text files to a record

Repository staff actions (login required)

View Item

Downloads