MIT Libraries homeMIT Libraries logoDSpace@MIT

MIT
View Item 
  • DSpace@MIT Home
  • MIT Libraries
  • MIT Theses
  • Graduate Theses
  • View Item
  • DSpace@MIT Home
  • MIT Libraries
  • MIT Theses
  • Graduate Theses
  • View Item
JavaScript is disabled for your browser. Some features of this site may not work without it.

High-order retractions for reduced-order modeling and uncertainty quantification

Author(s)
Charous, Aaron (Aaron Solomon)
Thumbnail
Download1251767676-MIT.pdf (47.13Mb)
Other Contributors
Massachusetts Institute of Technology. Center for Computational Science & Engineering.
Advisor
Pierre F.J. Lermusiaux.
Terms of use
MIT theses may be protected by copyright. Please reuse MIT thesis content according to the MIT Libraries Permissions Policy, which is available through the URL provided. http://dspace.mit.edu/handle/1721.1/7582
Metadata
Show full item record
Abstract
Though computing power continues to grow quickly, our appetite to solve larger and larger problems grows just as fast. As a consequence, reduced-order modeling has become an essential technique in the computational scientist's toolbox. By reducing the dimensionality of a system, we are able to obtain approximate solutions to otherwise intractable problems. And because the methodology we develop is sufficiently general, we may agnostically apply it to a plethora of problems, whether the high dimensionality arises due to the sheer size of the computational domain, the fine resolution we require, or stochasticity of the dynamics. In this thesis, we develop time integration schemes, called retractions, to efficiently evolve the dynamics of a system's low-rank approximation. Through the study of differential geometry, we are able to analyze the error incurred at each time step. A novel, explicit, computationally inexpensive set of algorithms, which we call perturbative retractions, are proposed that converge to an ideal retraction that projects exactly to the manifold of fixed-rank matrices. Furthermore, each perturbative retraction itself exhibits high-order convergence to the best low-rank approximation of the full-rank solution. We show that these high-order retractions significantly reduce the numerical error incurred over time when compared to a naive Euler forward retraction. Through test cases, we demonstrate their efficacy in the cases of matrix addition, real-time data compression, and deterministic and stochastic differential equations.
Description
Thesis: S.M., Massachusetts Institute of Technology, Center for Computational Science & Engineering, February, 2021
 
Cataloged from the official PDF version of thesis.
 
Includes bibliographical references (pages 145-151).
 
Date issued
2021
URI
https://hdl.handle.net/1721.1/130904
Department
Massachusetts Institute of Technology. Center for Computational Science and Engineering
Publisher
Massachusetts Institute of Technology
Keywords
Computational Science, Engineering

Collections
  • Graduate Theses

Browse

All of DSpaceCommunities & CollectionsBy Issue DateAuthorsTitlesSubjectsThis CollectionBy Issue DateAuthorsTitlesSubjects

My Account

Login

Statistics

OA StatisticsStatistics by CountryStatistics by Department
MIT Libraries homeMIT Libraries logo

Find us on

Twitter Facebook Instagram YouTube RSS

MIT Libraries navigation

SearchHours & locationsBorrow & requestResearch supportAbout us
PrivacyPermissionsAccessibility
MIT
Massachusetts Institute of Technology
Content created by the MIT Libraries, CC BY-NC unless otherwise noted. Notify us about copyright concerns.