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Numerical Effects of the Gaussian Recursive Filters in Solving Linear Systems in the 3Dvar Case Study

Part of: Error analysis and interval analysis Computer aspects of numerical algorithms Multivariate analysis

Published online by Cambridge University Press:  20 June 2017

Salvatore Cuomo ,
Ardelio Galletti ,
Giulio Giunta  and
Livia Marcellino
Salvatore Cuomo*
Affiliation:
Department of Mathematics and Applications, University of Naples Federico II, Naples 80126, Italy
Ardelio Galletti*
Affiliation:
Department of Science and Technology, Parthenope University of Naples, Naples 80143, Italy
Giulio Giunta*
Affiliation:
Department of Science and Technology, Parthenope University of Naples, Naples 80143, Italy
Livia Marcellino*
Affiliation:
Department of Science and Technology, Parthenope University of Naples, Naples 80143, Italy
*
*Corresponding author. Email addresses:salvatore.cuomo@unina.it (S. Cuomo), ardelio.galletti@uniparthenope.it (A. Galletti), giulio.giunta@uniparthenope.it (G. Giunta), livia.marcellino@uniparthenope.it (L. Marcellino)
*Corresponding author. Email addresses:salvatore.cuomo@unina.it (S. Cuomo), ardelio.galletti@uniparthenope.it (A. Galletti), giulio.giunta@uniparthenope.it (G. Giunta), livia.marcellino@uniparthenope.it (L. Marcellino)
*Corresponding author. Email addresses:salvatore.cuomo@unina.it (S. Cuomo), ardelio.galletti@uniparthenope.it (A. Galletti), giulio.giunta@uniparthenope.it (G. Giunta), livia.marcellino@uniparthenope.it (L. Marcellino)
*Corresponding author. Email addresses:salvatore.cuomo@unina.it (S. Cuomo), ardelio.galletti@uniparthenope.it (A. Galletti), giulio.giunta@uniparthenope.it (G. Giunta), livia.marcellino@uniparthenope.it (L. Marcellino)
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Abstract

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.

Keywords

Data assimilationrecursive filtersGaussian convolutionperturbation analysis

MSC classification

Secondary: 65G20: Algorithms with automatic result verification 65Y04: Algorithms for computer arithmetic, etc. 62H99: None of the above, but in this section
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 3 , August 2017 , pp. 520 - 540
Copyright
Copyright © Global-Science Press 2017 

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