Mathematical Background

Abstract

Conditional and Marginal DistributionsLet (a, b) ∈ R ℓ × R m denote a jointly varying random variable.Definition 1.7 The marginal pdf of a, P(a), is given in terms of the pdf of (a, b), P(a, b), byThus the marginal pdf P(a) is indeed the pdf for a in situations where we have no information about the random variable b, other than that it is in R m . ♠ Thus ρ ∞ is the density of a Gaussian N (0, σ 2 ∞ ). We then have,where µ J and µ ′ J denote the filtering distributions at time J corresponding to data Y J , Y ′ J respectively (i.e. the marginals of µ and µ ′ on the coordinate at time J).

Filter based methods for statistical linear inverse problems

“…All of these facts concerning the Kalman filter may be found in Chapter 4 of [13]. Expression (5) requires careful justification in infinite dimensions, and this is provided in [1] in certain settings.…”

“…However we will only use (5) as a quick method for deriving useful formulae, not expressed in terms of precision operators, which can be justified directly under the assumptions we make. A simplification of the Kalman filter method is the 3DVAR algorithm [13] which is not, strictly speaking, a probabilistic filter because it does not attempt to accurately track covariance information. Instead the covariance is fixed in time at…”

“…The paper [10] gives further insight into the development of regularization for these ensemble Kalman inversion methods, drawing on links with the Levenberg-Marquardt scheme [7]. In this paper our aim is to further the study of filters for the solution of inverse problems, going beyond the ensemble Kalman filter to encompass the study of other filters such as 3DVAR and the Kalman filter itself -see [13] for an overview of these filtering methods. A key issue will be the implementation of regularization with the aim of deriving optimal error estimates.…”

“…from N (0, γ 2 I). Note, however, that whilst the data sequence {y n } n≥0 we use in Data Model 1 is of this form, the assumption is not compatible with Data Model 2; thus for Data Model 2 we have a form of model error or model mis-specification [13].…”

“…In this paper, we establish the connection between iterative regularization methods (c.f. [6,8]) and filter based methods [13] with respect to an artificial dynamic system. More precisely, for a linear inverse problem, we verify that the iterative Tikhonov regularization method…”