Parallel Implementation of a Two-level Algebraic ILU(k)-based Domain Decomposition Preconditioner

We discuss the parallel implementation of a two-level algebraic ILU(k)-based domain decomposition preconditioner using the PETSc library. We present strategies to improve performance and minimize communication among processes during setup and application phases. We compare our implementation with an off-the-shelf preconditioner in PETSc for solving linear systems arising in reservoir simulation problems, and show that for some cases our implementation performs better.

[1]  Luc Giraud,et al.  Algebraic Two-Level Preconditioners for the Schur Complement Method , 2000, SIAM J. Sci. Comput..

[2]  Cornelis Vuik,et al.  Coarse grid acceleration of a parallel block preconditioner , 2001, Future Gener. Comput. Syst..

[3]  Barry F. Smith,et al.  Domain Decomposition: Parallel Multilevel Methods for Elliptic Partial Differential Equations , 1996 .

[4]  Hamdi A. Tchelepi,et al.  Parallel Scalable Unstructured CPR-Type Linear Solver for Reservoir Simulation , 2005 .

[5]  Hua Xiang,et al.  A two level domain decomposition preconditioner based on local Dirichlet-to-Neumann maps , 2010 .

[6]  François Pellegrini,et al.  PT-Scotch: A tool for efficient parallel graph ordering , 2008, Parallel Comput..

[7]  Cornelis Vuik,et al.  A Comparison of Deflation and Coarse Grid Correction Applied to Porous Media Flow , 2004, SIAM J. Numer. Anal..

[8]  H. Tchelepi,et al.  Multi-scale finite-volume method for elliptic problems in subsurface flow simulation , 2003 .

[9]  Klaus Stüben,et al.  Preconditioning for Efficiently Applying Algebraic Multigrid in Fully Implicit Reservoir Simulations , 2013, ANSS 2013.

[10]  Jinchao Xu,et al.  An agglomeration multigrid method for unstructured grids , 1998 .

[11]  Renato Spigler,et al.  Applied and Industrial Mathematics. Venice-1, 1989 , 1991 .

[12]  Patrick Jenny,et al.  Adaptive fully implicit multi-scale finite-volume method for multi-phase flow and transport in heterogeneous porous media , 2006, J. Comput. Phys..

[13]  Yvan Notay,et al.  Algebraic analysis of two‐grid methods: The nonsymmetric case , 2010, Numer. Linear Algebra Appl..

[14]  Mary F. Wheeler,et al.  Studies of Robust Two Stage Preconditioners for the Solution of Fully Implicit Multiphase Flow Problems , 2009 .

[15]  Michael Andrew Christie,et al.  Tenth SPE Comparative Solution Project: a comparison of upscaling techniques , 2001 .

[16]  Yvan Notay,et al.  Algebraic multigrid and algebraic multilevel methods: a theoretical comparison , 2005, Numer. Linear Algebra Appl..

[17]  Jun Zhang,et al.  BILUM: Block Versions of Multielimination and Multilevel ILU Preconditioner for General Sparse Linear Systems , 1999, SIAM J. Sci. Comput..

[18]  R. P. Kendall,et al.  Constrained Residual Acceleration of Conjugate Residual Methods , 1985 .

[19]  Cornelis Vuik,et al.  Theoretical and numerical comparison of various projection methods derived from deflation, domain decomposition and multigrid methods , 2007 .

[20]  Yvan Notay,et al.  Analysis of Aggregation-Based Multigrid , 2008, SIAM J. Sci. Comput..

[21]  Y. Saad,et al.  GMRES: a generalized minimal residual algorithm for solving nonsymmetric linear systems , 1986 .

[22]  Luiz Mariano Carvalho,et al.  An Algebraic ILU(k) Based Two-Level Domain Decomposition Preconditioner , 2015 .

[23]  Xiao-Chuan Cai,et al.  A Restricted Additive Schwarz Preconditioner for General Sparse Linear Systems , 1999, SIAM J. Sci. Comput..

[24]  Volker Mehrmann,et al.  Algebraic Multilevel Methods and Sparse Approximate Inverses , 2002, SIAM J. Matrix Anal. Appl..

[25]  David A. Collins,et al.  A Shared-Memory Parallel Black-Oil Simulator with a Parallel ILU Linear Solver , 2003 .

[26]  L. Formaggia,et al.  Algebraic coarse grid operators for domain decomposition based preconditioners , 2002 .

[27]  Yousef Saad,et al.  Iterative methods for sparse linear systems , 2003 .

[28]  Patrick Jenny,et al.  Adaptive Multiscale Finite-Volume Method for Multiphase Flow and Transport in Porous Media , 2005, Multiscale Model. Simul..

[29]  Haran Jackson,et al.  A Two-Level Variant of Additive Schwarz Preconditioning for Use in Reservoir Simulation , 2014, ArXiv.