BILUM is a set of programs designed for solving general sparse linear systems by using Krylov subspace methods preconditioned by some multi-level block ILU (BILUM) preconditioning techniques. BILUM combines the benefits of generality and robustness of ILU preconditioning techniques with those of grid-independent convergence of multigrid methods. The multi-level algorithms implemented by BILUM are based on the block independent set ordering and multi-elimination techniques. At each level, a block independent set is found by some greedy algorithms such that each block is decoupled with other blocks in the independent set. There is an inherited parallelism associated with this technique. The coefficient matrix is then re-ordered according to the independent set ordering and an approximate block ILU factorization is performed with a reduced system of smaller size. The multi-level structure is constructed by recursively applying the above idea to the approximate Schur complement (the reduced system) until the last reduced system is small enough to be solved by a direct method or a preconditioned iterative method.

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  1. de Almeida, Moisés Ceni; da Cruz, Julia Sekiguchi; Goldfeld, Paulo; Carvalho, Luiz Mariano; Souza, Michael: Supporting theory for a block approximate inverse preconditioner (2021)
  2. Cambier, Léopold; Chen, Chao; Boman, Erik G.; Rajamanickam, Sivasankaran; Tuminaro, Raymond S.; Darve, Eric: An algebraic sparsified nested dissection algorithm using low-rank approximations (2020)
  3. Buranay, Suzan C.; Iyikal, Ovgu C.: Approximate Schur-block ILU preconditioners for regularized solution of discrete ill-posed problems (2019)
  4. Xi, Yuanzhe; Li, Ruipeng; Saad, Yousef: An algebraic multilevel preconditioner with low-rank corrections for sparse symmetric matrices (2016)
  5. Carpentieri, Bruno; Liao, Jia; Sosonkina, Masha: VBARMS: a variable block algebraic recursive multilevel solver for sparse linear systems (2014)
  6. Vannieuwenhoven, Nick; Meerbergen, Karl: IMF: an incomplete multifrontal (LU)-factorization for element-structured sparse linear systems (2013)
  7. Ferronato, Massimiliano: Preconditioning for sparse linear systems at the dawn of the 21st century: history, current developments, and future perspectives (2012)
  8. Wu, Jian Ping; Zhao, Jun; Song, Jun Qiang; Li, Xiao Mei: A parallelization technique based on factor combination and graph partitioning for general incomplete Lu factorization (2012)
  9. Rivera, Christian A.; Heniche, Mourad; Glowinski, Roland; Tanguy, Philippe A.: Parallel finite element simulations of incompressible viscous fluid flow by domain decomposition with Lagrange multipliers (2010)
  10. Maclachlan, Scott; Saad, Yousef: Greedy coarsening strategies for nonsymmetric problems (2007)
  11. Jiang, Peng; Yang, Geng: Performance analysis of preconditioners based on Broyden method (2006)
  12. Notay, Yvan: Aggregation-based algebraic multilevel preconditioning (2006)
  13. Shen, Chi; Zhang, Jun: Performance study and analysis of parallel multilevel preconditioners (2006)
  14. Notay, Y.: Algebraic multigrid and algebraic multilevel methods: a theoretical comparison. (2005)
  15. Shen, Chi; Zhang, Jun; Wang, Kai: Distributed block independent set algorithms and parallel multilevel ILU preconditioners (2005)
  16. Shi, Yuying; Chang, Qianshun: Remark on convergence of algebraic multigrid in the form of matrix decomposition (2005)
  17. Gu, Tongxiang; Chi, Xuebin; Liu, Xingping: AINV and BILUM preconditioning techniques (2004)
  18. Saad, Yousef; Soulaimani, Azzeddine; Touihri, Ridha: Variations on algebraic recursive multilevel solvers (ARMS) for the solution of CFD problems (2004)
  19. Chow, Edmond; Vassilevski, Panayot S.: Multilevel block factorizations in generalized hierarchical bases. (2003)
  20. Li, Zhongze; Saad, Yousef; Sosonkina, Masha: pARMS: a parallel version of the algebraic recursive multilevel solver. (2003)

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