@Article{NMTMA-10-520,
author = {Salvatore Cuomo, Ardelio Galletti, Giulio Giunta and Livia Marcellino},
title = {Numerical Effects of the Gaussian Recursive Filters in Solving Linear Systems in the 3Dvar Case Study},
journal = {Numerical Mathematics: Theory, Methods and Applications},
year = {2017},
volume = {10},
number = {3},
pages = {520--540},
abstract = {<p style="text-align: justify;">In many applications, the Gaussian convolution is approximately computed
by means of recursive filters, with a significant improvement of computational efficiency.
We are interested in theoretical and numerical issues related to such an use of recursive
filters in a three-dimensional variational data assimilation (3Dvar) scheme as it appears
in the software OceanVar. In that context, the main numerical problem consists in
solving large linear systems with high efficiency, so that an iterative solver, namely
the conjugate gradient method, is equipped with a recursive filter in order to compute
matrix-vector multiplications that in fact are Gaussian convolutions. Here we present
an error analysis that gives effective bounds for the perturbation on the solution of such
linear systems, when is computed by means of recursive filters. We first prove that such
a solution can be seen as the exact solution of a perturbed linear system. Then we study
the related perturbation on the solution and we demonstrate that it can be bounded
in terms of the difference between the two linear operators associated to the Gaussian
convolution and the recursive filter, respectively. Moreover, we show through numerical
experiments that the error on the solution, which exhibits a kind of edge effect, i.e.
most of the error is localized in the first and last few entries of the computed solution,
is due to the structure of the difference of the two linear operators.</p>},
issn = {2079-7338},
doi = {https://doi.org/10.4208/nmtma.2017.m1528},
url = {http://global-sci.org/intro/article_detail/nmtma/12357.html}
}