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We discuss a construction of sparse approximate inverses (SPAI) of matrices
and matrix blocks in the context of linear systems of equations arising from finite element discretizations of partial differential equations.
The approximation is based on assembly of local, small-sized element matrix inverses, which are then assembled in the usual finite element manner and give raise to sparse matrix approximations. The method is particularly useful to obtain a good idea of a suitable sparsity pattern of the approximate inverse, an issue which is one of the bottlenecks for other known methods to construct approximate inverses, for instance based on minimization of some weighted Frobenius norm. Once the nonzero pattern of the approximate inverse matrix has been obtained, then the quality of the approximated inverse can be improved using some of the already available methods. An additional advantage of the method is that it processes a high degree of parallelism and is suitable for large scale numerical simulations. In the talk, we explain the proposed method, discuss the properties of the obtained matrix approximations and give illustrations of the numerical and the computational efficiency of the above SPAI technique. |