CaltechAUTHORS
  A Caltech Library Service

Filter Based Methods For Statistical Linear Inverse Problems

Iglesias, Marco A. and Lin, Kui and Lu, Shuai and Stuart, Andrew M. (2017) Filter Based Methods For Statistical Linear Inverse Problems. Communications in Mathematical Sciences , 15 (7). pp. 1867-1896. ISSN 1539-6746. https://resolver.caltech.edu/CaltechAUTHORS:20161221-113147238

[img] PDF - Submitted Version
See Usage Policy.

2074Kb

Use this Persistent URL to link to this item: https://resolver.caltech.edu/CaltechAUTHORS:20161221-113147238

Abstract

Ill-posed inverse problems are ubiquitous in applications. Understanding of algorithms for their solution has been greatly enhanced by a deep understanding of the linear inverse problem. In the applied communities ensemble-based filtering methods have recently been used to solve inverse problems by introducing an artificial dynamical system. This opens up the possibility of using a range of other filtering methods, such as 3DVAR and Kalman based methods, to solve inverse problems, again by introducing an artificial dynamical system. The aim of this paper is to analyze such methods in the context of the linear inverse problem. Statistical linear inverse problems are studied in the sense that the observational noise is assumed to be derived via realization of a Gaussian random variable. We investigate the asymptotic behavior of filter based methods for these inverse problems. Rigorous convergence rates are established for 3DVAR and for the Kalman filters, including minimax rates in some instances. Blowup of 3DVAR and a variant of its basic form is also presented, and optimality of the Kalman filter is discussed. These analyses reveal a close connection between (iterated) regularization schemes in deterministic inverse problems and filter based methods in data assimilation. Numerical experiments are presented to illustrate the theory.


Item Type:Article
Related URLs:
URLURL TypeDescription
http://dx.doi.org/10.4310/CMS.2017.v15.n7.a4DOIArticle
http://www.intlpress.com/site/pub/pages/journals/items/cms/content/vols/0015/0007/a004PublisherArticle
https://arxiv.org/abs/1512.01955arXivDiscussion Paper
Additional Information:© 2017 International Press. Paper received on 12 June 2016; Paper accepted on 6 May 2017. Andrew M. Stuart is funded by EPSRC (under the Programme Grant EQUIP), DARPA (under EQUiPS) and ONR. Shuai Lu is supported by Special Funds for Major State Basic Research Projects of China (2015CB856003), NSFC (91130004, 11522108), Shanghai Science and Technology Commission Grant (14QA1400400) and the Programme of Introducing Talents of Discipline to Universities (number B08018), China.
Funders:
Funding AgencyGrant Number
Engineering and Physical Sciences Research Council (EPSRC)UNSPECIFIED
Defense Advanced Research Projects Agency (DARPA)UNSPECIFIED
Office of Naval Research (ONR)UNSPECIFIED
Special Funds for Major State Basic Research Projects of China2015CB856003
National Natural Science Foundation of China91130004
National Natural Science Foundation of China11522108
Shanghai Science and Technology Commission14QA1400400
Programme of Introducing Talents of Discipline to Universities (China)B08018
Subject Keywords:Kalman filter, 3DVAR, statistical inverse problems, artificial dynamics
Issue or Number:7
Classification Code:2010 Mathematics Subject Classification: 47A52, 65J22, 93E11
Record Number:CaltechAUTHORS:20161221-113147238
Persistent URL:https://resolver.caltech.edu/CaltechAUTHORS:20161221-113147238
Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:73078
Collection:CaltechAUTHORS
Deposited By: Linda Taddeo
Deposited On:21 Dec 2016 19:49
Last Modified:03 Oct 2019 16:24

Repository Staff Only: item control page