Approximate Inverse Pre-conditions Son

Solving sparse linear systems is an important issue in scientific computing. With the advent and gradual popularity of parallel and distributed architectures, the search for parallelizable preconditioners becomes an important issue. Owing to the good parallelizability, sparse approximate inverse techniques are becoming popular.At first, the paper introduces the main theory of SAI and some specific algorithms that proved successful. According to the numerical tests, we compare and analyze these algorithms from aspects (robustness, efficiency, parallelism). And we draw the conclusion that, in general, the efficiency and robustness of SAI is not as good as ILU class of preconditioners, but it has better parallelizability. The numerical tests showed that each SAI algorithm has its own advantages and disadvantages. In other words, none can be the best algorithm for all the aspects should be considered.Besides, the paper investigates the use of sparse approximate inverse techniques in a multilevel block ILU preconditioner to design a robust and efficient parallelizable preconditioner for solving general sparse matrices. The preconditioner retains robustness of the BILUM and has better ability of control sparsity and increased parallelism. Numerical experiments show the effectiveness and efficiency of the proposed variant of BILUM.The paper also establishes a parallelizable approximate inverse preconditioner using the block constant pattern given by PH. Guillaume et al. The main idea of this precondition technique consisted in approximating the inverse of a matrix by a block constant matrix instead of a sparse matrix like in SAI methods. They do not require more storage, or even less, and are well adapted to paralleled computing, both for the construction of the preconditioner and for matrix-vector products. The parallelizable implements of BCAI are given in the paper. According to the numerical tests, a conclusion could be reached that BCAI is suitable for SPDs, and behaves badly for non-SPD matrices.