A Bit-Compatible Parallelization for ILU(k) Preconditioning
ILU(k) is a commonly used preconditioner for iterative linear solvers for sparse, non-symmetric systems. It is often preferred for the sake of its stability. We present TPILU(k), the first efficiently parallelized ILU(k) preconditioner that maintains this important stability property. Even better, TPILU(k) preconditioning produces an answer that is bit-compatible with the sequential ILU(k) preconditioning. In terms of performance, the TPILU(k) preconditioning is shown to run faster whenever more cores are made available to it — while continuing to be as stable as sequential ILU(k). This is in contrast to some competing methods that may become unstable if the degree of thread parallelism is raised too far. Where Block Jacobi ILU(k) fails in an application, it can be replaced by TPILU(k) in order to maintain good performance, while also achieving full stability. As a further optimization, TPILU(k) offers an optional level-based incomplete inverse method as a fast approximation for the original ILU(k) preconditioned matrix. Although this enhancement is not bit-compatible with classical ILU(k), it is bit-compatible with the output from the single-threaded version of the same algorithm. In experiments on a 16-core computer, the enhanced TPILU(k)-based iterative linear solver performed up to 9 times faster. As we approach an era of many-core computing, the ability to efficiently take advantage of many cores will become ever more important.
KeywordsDomain Decomposition Gaussian Elimination Iterative Solver Sparse Linear System Preconditioned Matrix
Unable to display preview. Download preview PDF.
- 4.Hysom, D., Pothen, A.: Efficient Parallel Computation of ILU(k) Preconditioners. In: Supercomputing 1999 (1999)Google Scholar
- 7.Dong, X., Cooperman, G., Apostolakis, J.: Multithreaded Geant4: Semi-Automatic Transformation into Scalable Thread-Parallel Software. In: Euro-Par 2010 (2010)Google Scholar
- 10.hypre: High Performance Preconditioners. User’s Manual, version 2.6.0b, https://computation.llnl.gov/casc/hypre/download/hypre-2.6.0b_usr_manual.pdf
- 11.Matrix Market.: Driven Cavity from the SPARSKIT Collection, http://math.nist.gov/MatrixMarket/data/SPARSKIT/drivcav/drivcav.html
- 12.UF Sparse Matrix Collection, http://www.cise.ufl.edu/research/sparse/matrices/
- 13.Anderson, E.: Parallel Implementation of Preconditioned Conjugate Gradient Methods for Solving Sparse Systems of Linear Equations. Master’s Thesis, Center for Supercomputing Research and Development, University of Illinois (1988)Google Scholar