Robust Algebraic Multigrid Methods and Their Parallelization

Funding agency	Austrian Science Fund (FWF)
Project number	P 14953
Principal investigator	Urlich Langer
Co-Investigator	Gundolf Haase
Members	Florian Bachinger Clemens Pechstein David Pusch Stefan Reitzinger Markus Wabro
Duration	2001-10-01 – 2005-09-30

Objectives

• efficient solution of positive definite and indefinite equations with AMG
• parallelization of AMG
• software development
• real life applications

Abstract

The aim of this project is the development and construction of Algebraic MultiGrid (AMG) methods. AMG methods try to mimic geometric multigrid methods just by knowing single grid information but retaining the optimality of the method. Since the construction of an universal AMG method for all arising situations in practice is an unrealistic dream, we will assemble efficient and robust solvers for special classes of matrices such as large scale systems arising from the (FEM, FDM, FIT, BEM) discretization of:

• Anisotropic Potential Equation
• Maxwell's Equations
• Stokes / Navier-Stokes Equations

Moreover, parallelization of AMG is an important task because it is the main source for a further enhancement of efficiency in real life applications.

Publications

• U. Langer, D. Pusch
• Data-sparse Algebraic Multigrid Methods for Large Scale Boundary Element Equations
• Applied Numerical Mathematics, Vol. 54, Issues 3-4, 2005, p. 406-424
• doi
• U. Langer, D. Pusch
• Comparison of Geometrical and Algebraic Multigrid Preconditioners for Data-sparse Boundary Element Matrices
• 5th International Conference on Large-Scale Scientific Computations, LSSC 2005, Sozopol, Bulgaria, submitted
•  
• C. Pechstein
• Multigrid-Newton-Methods For Nonlinear Magnetostatic Problems
• Diploma Thesis, Johannes Kepler University Linz, 2004
• thesis
• U. Langer, A. Pohoata, O. Steinbach
• Application of Preconditioned Coupled FETI/BETI Solvers to 2D Magnetic Field Problems
• SFB-Report 2004-23, SFB F013, Johannes Kepler University Linz, September, 2004
• report
• U. Langer, D. Pusch
• Comparison of Geometrical and Algebraic Multigrid Preconditioners for Data-Sparse Boundary Element Matrices
• SFB-Report 2004-24, SFB F013, Johannes Kepler University Linz, September, 2004
• Report
• U. Langer, D. Pusch
• Data-sparse Algebraic Multigrid Methods for Large Scale Boundary Element Equations
• Applied Numerical Mathematics, 2004, in press
• doi
• C. Pechstein, B. Jüttler
• Monotonicity-preserving Interproximation of B-H-Curves
• SFB-Report 2004-15, SFB F013, Johannes Kepler University Linz, August, 2004
• report
• M. Wabro
• Coupled algebraic Multigrid methods for the Oseen problem
• Computing and Visualization in Science 7, No. 3-4, 141-151, 2004
• doi
• M. Wabro
• AMGe - Coarsening Strategies and Application to the Oseen-Equations
• SFB-Report 2004-12, SFB F013, Johannes Kepler University Linz, June, 2004
• report
• M. Wabro
• AMGe - Coarsening Strategies and Application to the Oseen-Equations
• accepted for publication in SIAM Journal for Scientific Computing
•  
• D. Pusch, Efficient Algebraic Multigrid Preconditioners for Boundary Element Matrices
• Diploma Thesis, Johannes Kepler University Linz, 2003
•  
• M. Wabro
• Algebraic Multigrid Methods for the Numerical Solution of the Incompressible Navier-Stokes Equations
• PhD Thesis, Johannes Kepler University Linz, Trauner Verlag, 2003
•  
• F. Bachinger
• Multigrid Solvers for 3D Multiharmonic Nonlinear Magnetic Field Computations
• Diploma Thesis at the University of Linz, 2003
•  
• U. Langer, D. Pusch
• Algebraic Multigrid Preconditioners for Adaptive-Cross-Approximated Boundary Element Matrices
• Industrial Mathematics and Mathematical Modelling IMAMM 2003, submitted
•  
• U. Langer, D. Pusch
• Sparse Algebraic Multigrid Methods for Large Scale Boundary Element Equations
• SFB-Report 2003-47, SFB F013, Johannes Kepler University Linz, December, 2003
• report
• U. Langer, D. Pusch, S.Reitzinger
• Efficient Preconditioners for Boundary Element Matrices based on Grey-Box Algebraic Multigrid Methods
• Int. J. Num. Meth. Eng., Vol. 58, Issue 13: 1937-1953, 2003
•  
• U. Langer, D. Pusch, S. Reitzinger
• Algebraic Multigrid Preconditioners for Sparse Approximations of Boundary Element Matrices
• PAMM, Vol. 3, Issue 1, 2003, p. 579-580
•  
• U. Langer, O. Steinbach
• Boundary Element Tearing and Interconnecting Methods
• Computing 71, 205-228, 2003
•  
• U. Langer, O.Steinbach
• Coupled Boundary and Finite Element Tearing and Interconnecting Methods
• SFB-Report 2003-45, SFB F013, Johannes Kepler University Linz, November 2003
• report
• U. Langer, S. Reitzinger, J. Schicho
• Symbolic methods for the element preconditioning technique, Symbolic and numerical scientific computation
• Second international conference, SNSC 2001, Hagenberg, Austria, September 12-14, 2001. Revised papers. Berlin: Springer. Lect. Notes Comput. Sci. 2630, 293-308, 2003.
•  
• S. Reitzinger, U. Schreiber, U. van Rienen
• Algebraic multigrid for complex symmetric matrices and applications
• J. Comput. Appl. Math. 155, No.2, 405-421, 2003
•  
• M. Wabro
• Coupled Algebraic Multigrid Methods for the Oseen Problem
• SFB-Report No. 03-2, SFB F013, J.K. Univ. Linz, February, 2003
•  
• M. Wabro
• Coupled Algebraic Multigrid Methods for the Oseen Problem
• accepted for publication in Computing and Visualization in Science
•  
• C. Wolters, A. Anwander, M. Koch, S. Reitzinger, M. Kuhn, M. Svensen
• Influence of head tissue conductivity anisotropy on human EEG and MEG using fast high resolution finite element modeling, based on a parallel algebraic multigrid solver
• T. Plesser and P. Wittenburg (eds.): "Forschung und wissenschaftliches Rechnen" Heinz-Billing Award for the Advancement of Scientific Computing, Gesellschaft für wissenschaftliche Datenverarbeitung mbH Göttingen, 54 pages, 2002
•  
• S. Reitzinger, J. Schöberl
• An algebraic multigrid method for finite element discretizations with edge elements
• Numerical Linear Algebra with Applications, Volume 9, No 3, pp 223-238, 2002
•  
• U. Langer, S. Reitzinger, J. Schicho
• Symbolic Methods for the Element Preconditioning Technique
• Technical SFB-Report 02-03
•  
• S. Reitzinger, U. Schreiber, U. van Rienen
• Algebraic Multigrid for Complex Symmetric Matrices: Numerical Studies
• SFB-Report 02-01, University of Linz, 2002
•  
• M. Kaltenbacher, S. Reitzinger
• Nonlinear 3D Magnetic Field Computations using Lagrange FE-functions and Algebraic Multigrid
• IEEE Transactions on Magnetics, Volume 38, No 3, pp 1489-1496, May 2002
•  
• M. Kaltenbacher, S. Reitzinger
• Algebraic Multigrid for Magnetostatic Field Problems
• IEEE Transactions on Magnetics, accepted for publication, COMPUMAG, Volume 38, No 2, 2002
•  
• G. Haase, M. Kuhn and S. Reitzinger
• Parallel AMG on Distributed Memory Computers
• SIAM Journal of Scientific Computing, Volume 24, No 2, pp 410-427, 2002
•  
• B. Kaltenbacher, M. Kaltenbacher, S. Reitzinger
• Identification of Nonlinear B-H Curves based on Magnetic Field Computations and Multigrid Methods for Ill-posed Problems
• European Journal of Applied Mathematics 2002, in press
•  
• A. Anwander, M. Kuhn, S. Reitzinger, C. Wolters
• A Parallel Algebraic Multigrid Solver for the Finite Element Method based Source Localization in the Human Brain
• Computing and Visualization in Science, in press
•  
• S. Reitzinger
• Algebraic Multigrid Methods for Large Scale Finite Element Equations
• Reihe C - Technik und Naturwissenschaften 36, Universitätsverlag Rudolf Trauner, 2001
•  
• U. Langer, D. Pusch, S. Reitzinger
• An Efficient Preconditioner for Boundary Element Matrices based on Algebraic Multigrid Methods
• SFB-Report 01-30, University of Linz, 2001
•  
• G. Haase, U. Langer, S. Reitzinger, J. Schöberl
• Algebraic Multigrid Methods based on Element Preconditioning
• Int. Journal of Computer Mathematics Vol 78, no 4, pp.575 - 598, 2001
•  
• M. Kaltenbacher, S. Reitzinger, M. Schinnerl, J. Schöberl, H. Landes
• Multigrid Methods for the Computation of 3D Electromagnetic Field Problems
• Proceedings of the IGTE 2000, COMPEL 20, 2001
•  
• M. Kaltenbacher, S. Reitzinger
• Algebraic Multigrid for Static Nonlinear 3D Electromagnetic Field Computations
• SFB-Report 00-07, University of Linz, 2000
•  
• S. Reitzinger, J. Schöberl
• Algebraic Multigrid for Edge Elements
• SFB-Report 00-15, University of Linz, 2000. accepted for publication in Numerical Linear Algebra and Applications
•  
• G. Haase, M. Kuhn, S. Reitzinger
• Parallel AMG on Distributed Memory Computers
• SFB-Report 00-16, University of Linz, 2000. accepted for publication in SIAM Journal of Scientific Computing
•  
• G. Haase, U. Langer, S. Reitzinger, J. Schöberl
• A General Approach to Algebraic Multigrid
• SFB-Report 00-33, University of Linz, 2000
•  
• G. Haase
• A parallel AMG for overlapping and non-overlapping domain decomposition
• Electronic Transactions on Numerical Analysis Vol. 10, pp. 41-55, 2000
•  
• G. Haase, M. Kuhn, U. Langer, S. Reitzinger, J. Schöberl
• Parallel Maxwell Solvers
• Proceedings of the 3rd International Workshop on "Scientific Computing in Electrical Engineering" (ed. by U. van Rienen), Warnemünde, Germany, August 20-22, 2000, Springer-Verlag, Heidelberg. pp. 71 - 78
•  
• M. Kaltenbacher, S. Reitzinger
• Algebraic Multigrid for Solving Electromechanical Problems
• Multigrid Methods VI (E. Dick, K. Rienslagh, and J. Vierendeels eds.), Lecture Notes in Compuational Science and Engineering 2000, pp. 129-135
•  
• M. Kaltenbacher, S. Reitzinger, J. Schöberl
• Algebraic Multigrid for Solving Nonlinear Electrostatic and Magnetostatic Field Problems
• IEEE Trans. on Magnetics, Vol 36, no 4, pp. 1561 - 1564, 2000
•  
• C. Wolters, S. Reitzinger, A. Basermann, S. Burkhardt, U. Hartmann, F. Kruggel, A. Anwander
• Improved tissue modelling and fast solver methods for high resolution FE-modelling in EEG/MEG-source localization
• Biomag 2000, Proc. of the 12 th Internat. Conf. on Biomagnetism. J. Neonen, R.J. Ilmoniemi, and T. Katila, eds., 2000
•  
• F. Kickinger
• Algebraic Multigrid for Discrete Elliptic second-order Problems
• Multigrid Methods V, Proceedings of the 5th European MG Conference, (W. Hackbusch ed.), Springer Lecture Notes in Computational Science and Engineering, Vol 3, 1998, pp. 157 - 172
•  
• F. Kickinger, U. Langer
• A note on the global extraction element-by-element method
• ZAMM (1998), no 78, pp. 965 - 966
•  
• S. Reitzinger
• Robust Algebraic Multigrid Methods in Magnetic Shielding Problems
• Diploma Theses, Johannes Kepler University Linz, 1998
•  
• S. Reitzinger
• Algebraic Multigrid and Element Preconditioning I
• SFB-Report 98-15, Johannes Kepler University Linz, 1998
•  
• S. Reitzinger
• Algebraic Multigrid and Element Preconditioning II
• SFB-Report 99-18, Johannes Kepler University Linz, 1999
•  
• G. Haase
• Algebraic Multigrid with Local Support
• SFB-Report 99-29, Johannes Kepler University Linz, 1999
•  

A special issue (Vol. 8 (3-4), Dec. 2005) of the journal Computing and Visualization in Science was dedicated to Fast Boundary Element Methods in Industrial Applications.

Cooperations

• Institute of Sensor Technology, Erlangen, Germany (Prof. R. Lerch, Dr. M. Kaltenbacher)
• Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany (C. Wolters)
• Institute of General Electrical Engineering, Rostock, Germany (Prof. U. van Rienen, Dr. U. Schreiber, Dr. G. Pöplau)
• Working Group MAGNET, Vienna University of Technology, Vienna, Austria (Dr. T. Schrefl)
• Institute of Applied Mathematics and Numerical Simulation, Stuttgart, Germany (PD Dr. Olaf Steinbach)
• START-Project "3D hp-Finite Elements: Fast Solvers and Adaptivity", Linz (Dr. J. Schöberl, ...)
• Radon Institute for Computational and Applied Mathematics (RICAM), Linz (Dr. J. Kraus)