Communications in Mathematical Sciences

Volume 15 (2017)

Number 7

Filter based methods for statistical linear inverse problems

Pages: 1867 – 1896



Marco A. Iglesias (School of Mathematical Sciences, University of Nottingham, United Kingdom)

Kui Lin (School of Mathematical Sciences, Fudan University, Fudan, China)

Shuai Lu (School of Mathematical Sciences, Fudan University, Fudan, China)

Andrew M. Stuart (Computing and Mathematical Sciences, California Institute of Technology, Pasadena, Calif., U.S.A.)


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.


Kalman filter, 3DVAR, statistical inverse problems, artificial dynamics

2010 Mathematics Subject Classification

47A52, 65J22, 93E11

Received 12 June 2016

Accepted 6 May 2017

Published 16 October 2017