An error estimate of Gaussian Recursive Filter in 3Dvar problem

Abstract

Abstract-Computational kernel of the three-dimensional variational data assimilation (3D-Var) problem is a linear system, generally solved by means of an iterative method. The most costly part of each iterative step is a matrix-vector product with a very large covariance matrix having Gaussian correlation structure. This operation may be interpreted as a Gaussian convolution, that is a very expensive numerical kernel. Recursive Filters (RFs) are a well known way to approximate the Gaussian convolution and are intensively applied in the meteorology, in the oceanography and in forecast models. In this paper, we deal with an oceanographic 3D-Var data assimilation scheme, named OceanVar, where the linear system is solved by using the Conjugate Gradient (GC) method by replacing, at each step, the Gaussian convolution with RFs. Here we give theoretical issues on the discrete convolution approximation with a first order (1st-RF) and a third order (3rd-RF) recursive filters. Numerical experiments confirm given error bounds and show the benefits, in terms of accuracy and performance, of the 3-rd RF.

A K-iterated scheme for the first-order Gaussian recursive filter with boundary conditions

“…So far, a comprehensive study of the conditions for the K-iterated RFs is still lacking. Moreover, as noticed in [3], and recalled in this work (see Section 3), edge effects reappear when the number K of filter iterations increases, even though the classic RF end conditions are used. In this context, the purpose of this work is to provide an algorithm which combines classic end conditions with a suitable oversizing and reduction of both the input and the output signals, so that the edge effects are strongly mitigated.…”

“…Among RFs, the Gaussian RFs are particularly suitable for digital image processing [13] and applications of the scalespace theory [8], [15]. Gaussian RFs are an efficient computational tool for approximating Gaussian-based convolutions [3], [14], [10], [11], [12]. Gaussian RFs are mainly derived in three different ways: the Deriche strategy uses an approximation of the Gaussian function in the space domain [5]; the approximation procedure of Jin et al is carried out in the z-domain, i.e.…”

“…In particular, these solutions present an error that affects the entries near to the boundaries. This phenomenon has been recognized as an edge effect in [3] and some authors, such as Purser et al [10] and Triggs et al [14], suggest how to avoid this by simulating the effect of the continuation of the neglected equations. This results in inserting the so-called boundary conditions, or end conditions, i.e.…”

“…In some real applications [3], [6], [7], Gaussian RFs are used iteratively. This relies on a more general definition of RFs, named K-iterated RFs, which have been formally introduced in [3].…”

“…This relies on a more general definition of RFs, named K-iterated RFs, which have been formally introduced in [3]. So far, a comprehensive study of the conditions for the K-iterated RFs is still lacking.…”