Keywords: incomplete factorization, shifted linear systems, preconditioning, conjugate gradient, wind modelling, mass consistent models, genetic algorithms.

The efficiency of a mass-consistent model for wind field adjustment [1,2] depends on the stability parameter alpha which permits from a strictly horizontal wind adjustment to a pure vertical one. Each simulation with the wind model leads to the resolution of a linear system of equations, the matrix of which depends on a function epsilon(alpha), i.e., (M + epsilon N) x_epsilon = b_epsilon, where M and N are constant, symmetric and positive definite for a given level of discretization. The estimation of this parameter can be carried out using genetic algorithms, such that some of the wind velocities observed at the measurement station are regenerated as accurately as possible by the model. This requires the evaluation of a fitness function for each individual of the poputation of alpha, that is, the resolution of the above linear systems of equations for each value of alpha. It is well known that the preconditioned conjugate gradient algorithm (PCG) provides the best convergence results in the resolution of these types of linear systems. Since each value of epsilon(alpha) yields a different linear system, we have to solve a set of them. So we could either construct a different preconditioner for each of them and improve the convergence of PCG at the expense of a high computational cost related to the construction of each preconditioner. On the other hand, a single preconditioner constructed with the first value of epsilon may be used for all the systems. In this latter case the convergence will get worse as the value of the parameter moves away from the initial value. Here, an intermediate solution is proposed by using a preconditioner, which is constructed once at first and updated for each epsilon at a low computational cost. The rate of convergence of this strategy is between the above extreme options. An incomplete Cholesky factorization of A_epsilon that can be updated in terms epsilon was proposed by Meurant [3] for N being a diagonal matrix. In this paper, we generalise that algorithm to the case of a non diagonal N. Numerical experiments related to realistic wind field are presented in order to show the efficiency of the proposed preconditioner.

1

G. Montero, R. Montenegro, J.M. Escobar, "A 3-D Diagnostic Model for Wind Field Adjustment", J. Wind Eng. Ind. Aer., 74-76, 249-261, 1998. doi:10.1016/S0167-6105(98)00022-1

2

G. Montero, E. Rodríguez, R. Montenegro, J.M. Escobar, J.M. González- Yuste, "Genetic Algorithms for an Improved Parameter Estimation with Local Refinenent of Tetrahedral Meshes in a Wind Model", Adv. Engng. Soft., 36, 3-10, 2005. doi:10.1016/j.advengsoft.2004.03.011

3

G. Meurant, "On the incomplete Cholesky Decomposition of a class of perturbed matrices", SIAM J. Sci. Comput., 23, 2, 419-429, 2001. doi:10.1137/S1064827500371244

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