Incomplete LU Preconditioner Based on Max-Plus Approximation of LU Factorization

We present a new method for the a priori approximation of the orders of magnitude of the entries in the LU factors of a complex or real matrix $A$. This approximation can be used to quickly determine the positions of the largest entries in the LU factors of $A$ and these positions can then be used as the sparsity pattern for an incomplete LU factorization preconditioner. Our method uses max-plus algebra and is based solely on the moduli of the entries of $A$. We also present techniques for predicting which permutation matrices will be chosen by Gaussian elimination with partial pivoting. We exploit the strong connection between the field of Puiseux series and the max-plus semiring to prove properties of the max-plus LU factors.\udExperiments with a set of test matrices from the University of Florida sparse matrix collection show that our max-plus LU preconditioners outperform traditional level of fill methods and have similar performance to those preconditioners computed with more expensive threshold-based methods