Parallel Preconditioning and Approximate Inverses on the Connection Machine
نویسندگان
چکیده
We present a new approach to preconditioning for very large, sparse, non-symmetric, linear systems. We explicitly compute an approximate inverse to our original matrix that can be applied most efficiently for iterative methods on massively parallel machines. The algorithm and its implementation on the Connection Machine CM-2 are discussed in detail and supported by timings obtained from real problem data.
منابع مشابه
Parallel Approximate Inverse Preconditioners
There has been much excitement recently over the use of approximate inverses for parallel preconditioning. The preconditioning operation is simply a matrix-vector product, and in the most popular formulations, the construction of the approximate inverse seems embarassingly parallel. However, diiculties arise in practical parallel implementations. This paper will survey approximate inverse preco...
متن کاملSparse Approximate Inverses for Preconditioning of Linear Equations
1. Sparse Approximate Inverses and Linear Equations We consider the problem of solving a system of linear equations Ax = b in a parallel environment. Here, the n n-matrix A is large, sparse, unstructured, nonsymmetric, and ill-conditioned. The solution method should be robust, easy to parallelize, and applicable as a black box solver. Direct solution methods like the Gaussian Elimination are no...
متن کاملPreconditioning Techniques for Large LinearSystems: A Survey
This article surveys preconditioning techniques for the iterative solution of large linear systems, with a focus on algebraic methods suitable for general sparse matrices. Covered topics include progress in incomplete factorization methods, sparse approximate inverses, reorderings, parallelization issues, and block and multilevel extensions. Some of the challenges ahead are also discussed. An e...
متن کاملStair Matrices and Their Generalizations with Applications to Iterative Methods Ii: Iteration Arithmetic and Preconditionings
Iteration arithmetic is formally introduced based on iteration multiplication and αaddition which is a special multisplitting. This part focuses on construction of convergent splittings and approximate inverses for Hermitian positive definite matrices by applying stair matrices, their generalizations and iteration arithmetic. Analysis of the splittings and the approximate inverses is also prese...
متن کاملSparse Approximate Inverse Smoother for Multigrid
Various forms of sparse approximate inverses (SAI) have been shown to be useful for preconditioning. Their potential usefulness in a parallel environment has motivated much interest in recent years. However, the capability of an approximate inverse in eliminating the local error has not yet been fully exploited in multigrid algorithms. A careful examination of the iteration matrices of these ap...
متن کامل