A Self-Learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs

A self-learning algebraic multigrid method for dominant and minimal singular triplets and eigenpairs is described. The method consists of two multilevel phases. In the first, multiplicative phase (setup phase), tentative singular triplets are calculated along with a multigrid hierarchy of interpolation operators that approximately fit the tentative singular vectors in a collective and self-learning manner, using multiplicative update formulas. In the second, additive phase (solve phase), the tentative singular triplets are improved up to the desired accuracy by using an additive correction scheme with fixed interpolation operators, combined with a Ritz update. A suitable generalization of the singular value decomposition is formulated that applies to the coarse levels of the multilevel cycles. The proposed algorithm combines and extends two existing multigrid approaches for symmetric positive definite eigenvalue problems to the case of dominant and minimal singular triplets. Numerical tests on model probl...

[1]  Thomas A. Manteuffel,et al.  Towards Adaptive Smoothed Aggregation (AlphaSA) for Nonsymmetric Problems , 2010, SIAM J. Sci. Comput..

[2]  Lothar Reichel,et al.  Augmented Implicitly Restarted Lanczos Bidiagonalization Methods , 2005, SIAM J. Sci. Comput..

[3]  S. McCormick,et al.  Towards Adaptive Smoothed Aggregation (αsa) for Nonsymmetric Problems * , 2022 .

[4]  Robert H. Halstead,et al.  Matrix Computations , 2011, Encyclopedia of Parallel Computing.

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

[6]  William J. Stewart,et al.  Introduction to the numerical solution of Markov Chains , 1994 .

[7]  R. Larsen.  Lanczos Bidiagonalization With Partial Reorthogonalization , 1998 .

[8]  Thomas A. Manteuffel,et al.  Smoothed Aggregation Multigrid for Markov Chains , 2010, SIAM J. Sci. Comput..

[9]  Hans De Sterck,et al.  Augmented high order finite volume element method for elliptic PDEs in non-smooth domains: Convergence study , 2011, J. Comput. Appl. Math..

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

[11]  D. Ron,et al.  Multigrid Solvers and Multilevel Optimization Strategies , 2003 .

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

[13]  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..

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

[15]  Richard B. Lehoucq,et al.  Anasazi software for the numerical solution of large-scale eigenvalue problems , 2009, TOMS.

[16]  T. Manteuffel,et al.  Adaptive Smoothed Aggregation ( α SA ) Multigrid ∗ , 2005 .

[17]  Richard A. Harshman,et al.  Indexing by Latent Semantic Analysis , 1990, J. Am. Soc. Inf. Sci..

[18]  Jinchao Xu,et al.  A generalization of the vertex-centered finite volume scheme to arbitrary high order , 2010, Comput. Vis. Sci..

[19]  Thomas A. Manteuffel,et al.  Adaptive Algebraic Multigrid , 2005, SIAM J. Sci. Comput..

[20]  StübenKlaus.  Algebraic multigrid (AMG) , 1983 .

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

[22]  Van Emden Henson,et al.  Multilevel Aggregation Methods for Small-World Graphs with Application to Random-Walk Ranking , 2011, Comput. Informatics.

[23]  Andrew V. Knyazev,et al.  Toward the Optimal Preconditioned Eigensolver: Locally Optimal Block Preconditioned Conjugate Gradient Method , 2001, SIAM J. Sci. Comput..

[24]  Achi Brandt,et al.  Efficient Multilevel Eigensolvers with Applications to Data Analysis Tasks , 2010, IEEE Transactions on Pattern Analysis and Machine Intelligence.

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

[26]  Andreas Stathopoulos,et al.  PRIMME: preconditioned iterative multimethod eigensolver—methods and software description , 2010, TOMS.

[27]  Ulrich Hetmaniuk.  A Rayleigh quotient minimization algorithm based on algebraic multigrid , 2007, Numer. Linear Algebra Appl..

[28]  Thomas A. Manteuffel,et al.  Algebraic Multigrid for Markov Chains , 2010, SIAM J. Sci. Comput..

[29]  Heng Tao Shen,et al.  Principal Component Analysis , 2009, Encyclopedia of Biometrics.

[30]  K. Tanabe.  Projection method for solving a singular system of linear equations and its applications , 1971 .

[31]  Irad Yavneh,et al.  Square and stretch multigrid for stochastic matrix eigenproblems , 2010, Numer. Linear Algebra Appl..

[32]  U. Hetmaniuk,et al.  A comparison of eigensolvers for large‐scale 3D modal analysis using AMG‐preconditioned iterative methods , 2005 .

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

[34]  Graham Horton,et al.  A multi-level solution algorithm for steady-state Markov chains , 1994, SIGMETRICS.

[35]  A. Borzì,et al.  Algebraic multigrid methods for solving generalized eigenvalue problems , 2006 .

[36]  Irad Yavneh,et al.  Fast multilevel methods for Markov chains , 2011, Numer. Linear Algebra Appl..

[37]  S. McCormick,et al.  Multigrid Methods for Differential Eigenproblems , 1983 .

[38]  William L. Briggs,et al.  A multigrid tutorial , 1987 .

[39]  Efstratios Gallopoulos,et al.  Computing smallest singular triplets with implicitly restarted Lanczos bidiagonalization , 2004, Applied Numerical Mathematics.

[40]  K. Stuben,et al.  Algebraic Multigrid (AMG) : An Introduction With Applications , 2000 .