Filter based methods for statistical linear inverse problems

Marco A. Iglesias,Shuai Lu,Andrew M. Stuart,Kui Lin

doi:10.4310/cms.2017.v15.n7.a4

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.

Highlights

In many geophysical applications, in particular in the petroleum industry and in hydrology, distributed parameter estimation problems are often solved by means of iterative ensemble Kalman filters [15]
We investigate the asymptotic behavior of filter based methods for these inverse problems
In particular in the petroleum industry and in hydrology, distributed parameter estimation problems are often solved by means of iterative ensemble Kalman filters [15]

Summary

Introduction

In particular in the petroleum industry and in hydrology, distributed parameter estimation problems are often solved by means of iterative ensemble Kalman filters [15]. The methodology is described in a basic, abstract form, applicable to a general, possibly nonlinear, inverse problem in [9] In this basic form of the algorithm regularization is present due to dynamical preservation of a subspace spanned by the ensemble during the iteration. By studying the application of filtering methods to the solution of the linear inverse problem our aim is to open up the possibility of employing the filtering methodology to (static) inverse problems of the form 1, and nonlinear generalizations. Is closely related to filtering methods such as 3DVAR and the Kalman filter when applied to the partially observed linear dynamical system 2 The similarity between both schemes provides a probabilistic interpretation of iterative regularization methods, and allows the possibility of quantifying uncertainty via the variance.

Filter Definitions

Asymptotic Behaviour of Filters

Assumptions

Filter Properties

Asymptotic Analysis of the Kalman Filter

Kalman Filter and Data Model 1

Kalman Filter and Data Model 2

Classical 3DVAR

Variant of 3DVAR for Data Model 1

Numerical Illustrations

Set-Up

Using Kalman Filter and 3DVAR for solving linear inverse problems