On the Relations between ILUs and Factored Approximate Inverses

This paper discusses some relationships between ILU factorization techniques and factored sparse approximate inverse techniques. While ILU factorizations compute approximate LU factors of the coefficient matrix A, approximate inverse techniques aim at building triangular matrices Z and W such that $W^\top AZ$ is approximately diagonal. The paper shows that certain forms of approximate inverse techniques amount to approximately inverting the triangular factors obtained from some variants of ILU factorization of the original matrix. A few useful applications of these relationships will be discussed.

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