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Statistics From Computations

Sigurgeirsson, Hersir and Stuart, A. M. (2001) Statistics From Computations. In: Foundations of Computational Mathematics. London Mathematical Society lecture note series. No.284. Cambridge University Press , New York, NY, pp. 323-344. ISBN 978-0-521-00349-0. https://resolver.caltech.edu/CaltechAUTHORS:20170614-073434784

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Abstract

The study of numerical methods for initial value problems by considering their approximation properties from a dynamical systems viewpoint is now a well-established field; a substantial body of knowledge, developed over the past two decades, can be found in the literature. Nonetheless many open questions remain concerning the meaning of long-time simulations performed by approximating dynamical systems. In recent years various attempts to analyse the statistical content of these long-time simulations have emerged, and the purpose of this article is to review some of that work. The subject area is far from complete; nonetheless a certain unity can be seen in what has been achieved to date and it is therefore of value to give an overview of the field. Some mathematical background concerning the propagation of probability measures by discrete and continuous time dynamical systems or Markov chains will be given. In particular the Frobenius-Perron and Fokker-Planck operators will be described. Using the notion of ergodicity two different approaches, direct and indirect, will be outlined. The majority of the review is concerned with indirect methods, where the initial value problem is simulated from a single initial condition and the statistical content of this trajectory studied. Three classes of problems will be studied: deterministic problems in fixed finite dimension, stochastic problems in fixed finite dimension, and deterministic problems with random data in dimension n → ∞; in the latter case ideas from statistical mechanics can be exploited to analyse or interpret numerical schemes. Throughout, the ideas are illustrated by simple numerical experiments. The emphasis is on understanding underlying concepts at a high level and mathematical detail will not be given a high priority in this review.


Item Type:Book Section
Related URLs:
URLURL TypeDescription
http://www.cambridge.org/catalogue/catalogue.asp?isbn=9781107107366&ss=tocPublisherArticle
http://www.cambridge.org/us/academic/subjects/mathematics/numerical-analysis/foundations-computational-mathematicsPublisherArticle
Additional Information:© 2001 Cambridge University Press. We are grateful to Michael Dellnitz, Gary Froyland, Oliver Jünge and Sebastian Reich for helpful readings of drafts of this manuscript.
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Other Numbering System NameOther Numbering System ID
Andrew StuartC8
Series Name:London Mathematical Society lecture note series
Issue or Number:284
Record Number:CaltechAUTHORS:20170614-073434784
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20170614-073434784
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:78187
Collection:CaltechAUTHORS
Deposited By: Ruth Sustaita
Deposited On:14 Jun 2017 16:14
Last Modified:03 Oct 2019 18:06

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