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There are variety of computational algorithms need sequential sweeping;sweeping based on specific order; on a structured grid, e.g., preconditioning(smoothing) by SOR or ILU methods and solution of eikonal equation by fastsweeping algorithm. Due to sequential nature, parallel implementation of thesealgorithms usually leads to miss of efficiency; e.g. a significant convergencerate decay. Therefore, there is an interest to parallelize sequential sweepingprocedures, keeping the efficiency of the original method simultaneously. Thispaper goals to parallelize sequential sweeping algorithms on structured grids,with emphasis on SOR and ILU preconditioners. The presented method can beaccounted as an overlapping domain decomposition method combined to amulti-frontal sweeping procedure. The implementation of method in one and twodimensions are discussed in details. The extension to higher dimensions andgeneral structured n-diagonal matrices is outlined. Introducing notion ofalternatively block upper-lower triangular matrices, the convergence theory isestablished in general cases. Numerical results on model problems show that,unlike related alternative parallel methods, the convergence rate andefficiency of the presented method is close to the original sequential method.Numerical results also support successful use of the presented method as acache efficient solver in sequential computations as well.

Parallelizing Sequential Sweeping on Structured Grids -- Fully Parallel SOR/ILU preconditioners for Structured n-Diagonal Matrices