BILUM: Block Versions of Multielimination and Multilevel ILU Preconditioner for General Sparse Linear Systems

Abstract

We introduce block versions of the multielimination incomplete LU (ILUM) factorization preconditioning technique for solving general sparse unstructured linear systems. These preconditioners have a multilevel structure and, for certain types of problems, may exhibit properties that are typically enjoyed by multigrid methods. Several heuristic strategies for forming blocks of independent sets are introduced and their relative merits are discussed. The advantages of block ILUM over point ILUM include increased robustness and efficiency. We compare several versions of the block ILUM, point ILUM, and the dual-threshold-based ILUT preconditioners. In particular, tests with some convection-diffusion problems show that it may be possible to obtain convergence that is nearly independent of the Reynolds number as well as of the grid size.

Self-adaptive Multiprecision Preconditioners on Multicore and Manycore Architectures

“…Additional fill-in usually reduces the amount of parallelism in ILU(m) compared to ILU(0), but there are a number of techniques designed to retain it, such as the level-scheduling techniques [15,11] or the multi-coloring algorithms for the ILU factorization with levels based on the power(q)-pattern method [9]. Another workaround is given by the idea of multi-elimination [14,16], which is based on successive independent set coloring [6]. The motivation is that in a step of the Gaussian elimination, there usually exists a large set of rows that can be processed in parallel.…”

An Element-by-Element Multilevel Block ILU Preconditioner Using GLAS

“…In [3], a multilevel block ILU-factorization was proposed, inspired by BILUM [4]. At level k, A k is approximately factored as…”

“…The elements in this set should form a block independent set [4]. That is, there should be no edge between any two elements in the set of diagonal elements.…”

On residual smoothing in ILUM-type preconditioning

“…For instance the classical ilu preconditioner is not suitable for parallel architectures. Block versions of ilu, like ilum [9], bilum [12] and amli [1], have a much better data flow for parallel architectures.…”