A line search algorithm for wind field adjustment with incomplete data and RBF approximation
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A line search algorithm for wind field adjustment with incomplete data and RBF approximation Daniel A. Cervantes1 · Pedro González Casanova1 · Christian Gout2 · Miguel Ángel Moreles3
Received: 20 December 2016 / Revised: 2 May 2017 / Accepted: 28 May 2017 © SBMAC - Sociedade Brasileira de Matemática Aplicada e Computacional 2017
Abstract The problem of concern in this work is the construction of free divergence fields given scattered horizontal components. As customary, the problem is formulated as a PDE constrained least squares problem. The novelty of our approach is to construct the so-called adjusted field, as the unique solution along an appropriately chosen descent direction. The latter is obtained by the adjoint equation technique. It is shown that the classical adjusted field of Sasaki’s is a particular case. On choosing descent directions, the underlying mass consistent model leads to the solution of an elliptic problem which is solved by means of a radial basis functions method. Finally, some numerical results for wind field adjustment are presented. Keywords Wind adjustment · RBF methods · Line search Mathematics Subject Classification 65K10 · 65N35 · 35Q86
Communicated by Frederic Valentin.
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Miguel Ángel Moreles [email protected] Daniel A. Cervantes [email protected] Pedro González Casanova [email protected] Christian Gout [email protected]
1
Instituto de Matemáticas, UNAM, Ciudad Universitaria, 04510 Mexico, CDMX, Mexico
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INSA Rouen, LMI, Av. de lUniversité, BP 08, 76801 St Etienne du Rouvray Cedex, France
3
CIMAT, Jalisco S/N, 36240 Valenciana, Guanajuato, GTO, Mexico
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1 Introduction The problem of recovering atmospheric wind fields from prescribed horizontal data is of great interest in meteorological applications. In practice, the vertical component is unavailable. Consequently, the measured data are complemented with mass consistency to pose a variational problem for approximation. This approach dates back to Sasaki (1958). Literature on the subject is vast, and a timely review is presented in Ratto et al. (1994). The numerical approximation of the variational problem requires the solution of Poisson boundary value problems. Numerical approximations of the solutions can be obtained conventionally using the finite element method, finite volume method, finite differences, etc. In these methods, mesh managing is computationally expensive. For the wind field adjustment problem, a case can be made for a mesh-free approach in terms of radial basis functions (RBF). See Pepper et al. (2014) and Cervantes et al. (2013). In the context of RBF, the problem of wind field recovery has been also considered as a smoothing problem. Mass consistency is introduced by penalizing with the norm of the divergence of the vector field. For the case of polyharmonic splines, Benbourhim and Bouhamidi (2008) developed a smoothing algorithm for a given set of prescribed data in two and three dimensions. A full convergence analysis is provided. These RBF techniq
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