Bootstrap Algebraic Multigrid: status report, open problems, and outlook

This paper provides an overview of the main ideas driving the bootstrap algebraic multigrid methodology, including compatible relaxation and algebraic distances for defining effective coarsening strategies, the least squares method for computing accurate prolongation operators and the bootstrap cycles for computing the test vectors that are used in the least squares process. We review some recent research in the development, analysis and application of bootstrap algebraic multigrid and point to open problems in these areas. Results from our previous research as well as some new results for some model diffusion problems with highly oscillatory diffusion coefficient are presented to illustrate the basic components of the BAMG algorithm.

[1]  A. Brandt.  Algebraic multigrid theory: The symmetric case , 1986 .

[2]  J. W. Ruge,et al.  4. Algebraic Multigrid , 1987 .

[3]  Jacob B. Schroder,et al.  Smoothed aggregation solvers for anisotropic diffusion , 2012, Numer. Linear Algebra Appl..

[4]  Yousef Saad,et al.  A Greedy Strategy for Coarse-Grid Selection , 2007, SIAM J. Sci. Comput..

[5]  Ilya Safro,et al.  Multilevel algorithms for linear ordering problems , 2009, JEAL.

[6]  A. Brandt.  General highly accurate algebraic coarsening. , 2000 .

[7]  Panayot S. Vassilevski,et al.  A generalized eigensolver based on smoothed aggregation (GES-SA) for initializing smoothed aggregation (SA) multigrid , 2008, Numer. Linear Algebra Appl..

[8]  Achi Brandt,et al.  Bootstrap AMG , 2011, SIAM J. Sci. Comput..

[9]  Thomas A. Manteuffel,et al.  Relaxation‐corrected bootstrap algebraic multigrid (rBAMG) , 2012, Numer. Linear Algebra Appl..

[10]  Achi Brandt,et al.  Lean Algebraic Multigrid (LAMG): Fast Graph Laplacian Linear Solver , 2011, SIAM J. Sci. Comput..

[11]  A. Brandt.  Principles of Systematic Upscaling , 2008 .

[12]  J. H. Wilkinson.  The algebraic eigenvalue problem , 1966 .

[13]  D. Brandt,et al.  Multi-level adaptive solutions to boundary-value problems math comptr , 1977 .

[14]  William L. Briggs,et al.  8. Algebraic Multigrid (AMG) , 2000 .

[15]  Thomas A. Manteuffel,et al.  Adaptive reduction‐based AMG , 2006, Numer. Linear Algebra Appl..

[16]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[17]  I. LIVSHITS,et al.  One-Dimensional Algorithm for Finding Eigenbasis of the Schrödinger Operator , 2007, SIAM J. Sci. Comput..

[18]  Panayot S. Vassilevski,et al.  On Generalizing the Algebraic Multigrid Framework , 2004, SIAM J. Numer. Anal..

[19]  A. Brandt.  Multiscale Scientific Computation: Review 2001 , 2002 .

[20]  Panayot S. Vassilevski,et al.  Spectral AMGe (ρAMGe) , 2003, SIAM J. Sci. Comput..

[21]  O. E. Livne,et al.  Coarsening by compatible relaxation , 2004, Numer. Linear Algebra Appl..

[22]  D. Ron,et al.  Renormalization Multigrid (RMG): Statistically Optimal Renormalization Group Flow and Coarse-to-Fine Monte Carlo Acceleration , 2001 .

[23]  L. Zikatanov,et al.  Algebraic Multigrid Methods Based on Compatible Relaxation and Energy Minimization , 2007 .

[24]  Robert D. Falgout,et al.  Compatible Relaxation and Coarsening in Algebraic Multigrid , 2009, SIAM J. Sci. Comput..

[25]  Thomas A. Manteuffel,et al.  Operator‐based interpolation for bootstrap algebraic multigrid , 2010, Numer. Linear Algebra Appl..

[26]  Bobby Philip,et al.  Adaptive algebraic smoothers , 2012, J. Comput. Appl. Math..

[27]  Nathan Halko,et al.  Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions , 2009, SIAM Rev..

[28]  James J. Brannick,et al.  Bootstrap Algebraic Multigrid for the 2D Wilson Dirac system , 2013, SIAM J. Sci. Comput..

[29]  Irene Livshits,et al.  The least squares AMG Solver for the one-dimensional Helmholtz operator , 2011, Comput. Vis. Sci..

[30]  Ludmil T. Zikatanov,et al.  An algebraic multilevel method for anisotropic elliptic equations based on subgraph matching , 2012, Numer. Linear Algebra Appl..

[31]  P. Vassilevski,et al.  ON GENERALIZING THE AMG FRAMEWORK , 2003 .

[32]  Achi Brandt,et al.  A Bootstrap Algebraic Multilevel Method for Markov Chains , 2010, SIAM J. Sci. Comput..

[33]  Achi Brandt,et al.  An algebraic distances measure of AMG strength of connection , 2011, 1106.5990.

[34]  Ilya Safro,et al.  Relaxation-based coarsening and multiscale graph organization , 2010, Multiscale Model. Simul..

[35]  Thomas A. Manteuffel,et al.  Adaptive Smoothed Aggregation (αSA) , 2004, SIAM J. Sci. Comput..