Updating incomplete factorization preconditioners for shifted linear systems arising in a wind model

The efficiency of a finite element mass-consistent model for wind field adjustment depends on the stability parameter α which allows 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 ε(α), i.e., (M + εN)xε = bε, where M and N are constant, symmetric and positive definite matrices with the same sparsity pattern for a given level of discretization. The estimation of this parameter may be carried out by using genetic algorithms. This procedure requires the evaluation of a fitness function for each individual of the population defined in the searching space of α, that is, the resolution of one linear system of equations for each value of α. Preconditioned Conjugate Gradient algorithm (PCG) is usually applied for the resolution of this type of linear systems due to its good convergence results. In order to solve this set of linear systems, we could either construct a different preconditioner for each of them or use a single preconditioner constructed from the first value of ε to solve all the systems. In this paper, an intermediate approach is proposed. An incomplete Cholesky factorization of matrix Aε is constructed for the first linear system and it is updated for each ε at a low computational cost. Numerical experiments related to realistic wind field are presented in order to show the performance of the proposed preconditioning strategy.