Abstract
We present a new method for constructing incomplete Cholesky factorization preconditioners for use in solving large sparse symmetric positive-definite linear systems. This method uses max-plus algebra to predict the positions of the largest entries in the Cholesky factor and then uses these positions as the sparsity pattern for the preconditioner. Our method builds on the max-plus incomplete LU factorization preconditioner recently proposed in [J. Hook and F. Tisseur, Incomplete LU preconditioner based on max-plus approximation of LU factorization, MIMS Eprint 2016.47, Manchester, 2016] but applied to symmetric positive-definite matrices, which comprise an important special case for the method and its application. A attractive feature of our approach is that the sparsity pattern of each column of the preconditioner can be computed in parallel. Numerical comparisons are made with other incomplete Cholesky factorization preconditioners using problems from a range of practical applications. We demonstrate that the new preconditioner can outperform traditional level-based preconditioners and offer a parallel alternative to a serial limited-memory based approach.
Highlights
Incomplete Cholesky (IC) factorizations are an important tool in the solution of large sparse symmetric positive-definite linear systems of equations Ax = b
We present a new method for constructing incomplete Cholesky factorization preconditioners for use in solving large sparse symmetric positive-definite linear systems
The aim of this paper is to present an algorithm for constructing IC preconditioners for large sparse positive-definite problems using max-plus algebra to predict the positions of the largest entries in the Cholesky factor
Summary
Incomplete Cholesky (IC) factorizations are an important tool in the solution of large sparse symmetric positive-definite linear systems of equations Ax = b. The basic idea is to compute a factorization A ≈ LU (or A ≈ LLT in the positive-definite case) with L and U sparse triangular matrices with the fill-in (that is, the entries in L and U that lie outside the sparsity pattern of A) restricted to some sparsity pattern S. Hook and Tisseur [20] have shown how max-plus algebra can be used to approximate the order of magnitude of the moduli of the entries in the LU factors of A and have used this to construct the sparsity pattern of ILU preconditioners. Max-plus algebra is the analogue of linear algebra developed
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