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Sparse approximate inverses and target matrices. (English) Zbl 1077.65044
Summary: If $P$ has a prescribed sparsity and minimizes the Frobenius norm $|I-PA|_{F}$, it is called a sparse approximate inverse of $A$. It is well known that the computation of such a matrix $P$ is via the solution of independent linear least squares problems for the rows separately (and therefore in parallel). We consider the choice of other norms and introduce the idea of “target” matrices. A target matrix, $T$, is readily inverted and thus forms part of a preconditioner when $|T-PA|$ is minimized over some appropriate sparse matrices $P$. The use of alternatives to the Frobenius norm which maintain parallelizability, while discussed in early literature, does not appear to have been exploited.

65F50Sparse matrices (numerical linear algebra)
65F10Iterative methods for linear systems
65F35Matrix norms, conditioning, scaling (numerical linear algebra)
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