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Title: Assimilating data into mathematical models
Author: Sanz-Alonso, Daniel
ISNI:       0000 0004 5992 6983
Awarding Body: University of Warwick
Current Institution: University of Warwick
Date of Award: 2016
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Chapter 1 is a brief overview of the Bayesian approach to blending mathematical models with data. For this introductory chapter, I do not claim any originality in the material itself, but only in the presentation, and in the choice of contents. Chapters 2, 3 and 4 are transcripts of published and submitted papers, with minimal cosmetic modifications. I now detail my contributions to each of these papers. Chapter 2 is a transcript of the published paper Long-time Asymptotics of the Filtering Distribution for Partially Observed Chaotic Dynamical Systems" [Sanz-Alonso and Stuart, 2015] written in collaboration with Andrew Stuart. The idea of building a unified framework for studying filtering of chaotic dissipative dynamical systems is from Andrew. My ideas include the truncation of the 3DVAR algorithm that allows for unbounded observation noise, using the squeezing property as the unifying arch across all models, and most of the links with control theory. I stated and proved all the results of the paper. I also wrote the first version of the paper, which was subsequently much improved with Andrew's input. Chapter 3 is a transcript of the published paper \Filter Accuracy for the Lorenz 96 Model: Fixed Versus Adaptive Observation Operators" [Law et al., 2016], written in collaboration with Kody Law, Abhishek Shukla, and Andrew Stuart. My contribution to this paper was in proving most of the theoretical results. I did not contribute to the numerical experiments. The idea of using adaptive observation operators is from Abhishek. Chapter 4 is a transcript of the submitted paper\Importance Sampling: Computational Complexity and Intrinsic Dimension" [Agapiou et al., 2015], written in collaboration with Sergios Agapiou, Omiros Papaspiliopoulos, and Andrew Stuart. The idea of relating the two notions of intrinsic dimension described in the paper is from Omiros. Sergios stated and proved Theorem 4.2.3. Andrew's input was fundamental in making the paper well structured, and in the overall writing style. The paper was written very collaboratively among the four of us, and some of the results were the fruit of many discussions involving different subsets of authors. Some of my inputs include: the idea of using metrics between probability measures to study the performance of importance sampling, establishing connections to tempering, the analysis of singular limits both for inverse problems and filtering, most of the filtering section and in particular the use of the theory of inverse problems to analyze different proposals in the filtering set-up, the proof of Theorem 4.2.1, and substantial input in the proof of all the results of the paper not mentioned before. This paper aims to bring cohesion and new insights into a topic with a vast literature, and I helped towards this goal by doing most of the literature review involved.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available
Keywords: QA Mathematics