HSL_MI20: an efficient AMG preconditioner for finite element problems in 3D. Algebraic multigrid (AMG) is one of the most effective iterative methods for the solution of large, sparse linear systems obtained from the discretization of second-order scalar elliptic self-adjoint partial differential equations. It can also be used as a preconditioner for Krylov subspace methods. In this communication, we report on the design and development of a robust, effective and portable Fortran 95 implementation of the classical Ruge-St”uben AMG, which is available as package HSL_MI20 within the HSL mathematical software library. The routine can be used as a `black-box’ preconditioner, but it also offers the user a range of options and parameters. Proper tuning of these parameters for a particular application can significantly enhance the performance of an AMG-preconditioned Krylov solver. This is illustrated using a number of examples arising in the unstructured finite element discretization of the diffusion, the convection-diffusion, and the Stokes equations, as well as transient thermal convection problems associated with the Boussinesq approximation of the Navier-Stokes equations in 3D.

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  1. Benner, Peter; Onwunta, Akwum; Stoll, Martin: Block-diagonal preconditioning for optimal control problems constrained by PDEs with uncertain inputs (2016)
  2. Palitta, Davide; Simoncini, Valeria: Matrix-equation-based strategies for convection-diffusion equations (2016)
  3. Pearson, John W.: Fast iterative solvers for large matrix systems arising from time-dependent Stokes control problems (2016)
  4. Pearson, John W.: On the development of parameter-robust preconditioners and commutator arguments for solving Stokes control problems (2015)
  5. Porcelli, Margherita; Simoncini, Valeria; Tani, Mattia: Preconditioning of active-set Newton methods for PDE-constrained optimal control problems (2015)
  6. Wang, Zeng-Qi: Restrictively preconditioned Chebyshev method for solving systems of linear equations (2015)
  7. Ramage, Alison; Gartland, Eugene C.jun.: A preconditioned nullspace method for liquid crystal director modeling (2013)
  8. Sesana, Debora; Simoncini, Valeria: Spectral analysis of inexact constraint preconditioning for symmetric saddle point matrices (2013)
  9. Ahmed, Sarfraz; Goodyer, Christopher E.; Jimack, Peter K.: An efficient preconditioned iterative solution of fully-coupled elastohydrodynamic lubrication problems (2012)
  10. Gordon, Andrew D.; Powell, Catherine E.: On solving stochastic collocation systems with algebraic multigrid (2012)
  11. Powell, Catherine E.; Silvester, David J.: Preconditioning steady-state Navier-Stokes equations with random data (2012)
  12. Silvester, David J.; Bespalov, Alex; Powell, Catherine E.: A framework for the development of implicit solvers for incompressible flow problems (2012)
  13. Simoncini, V.: Reduced order solution of structured linear systems arising in certain PDE-constrained optimization problems (2012)
  14. Benzi, Michele; Ng, Michael; Niu, Qiang; Wang, Zhen: A relaxed dimensional factorization preconditioner for the incompressible Navier-Stokes equations (2011)
  15. Benzi, Michele; Wang, Zhen: Analysis of augmented Lagrangian-based preconditioners for the steady incompressible Navier-Stokes equations (2011)
  16. Druskin, V.; Simoncini, V.: Adaptive rational Krylov subspaces for large-scale dynamical systems (2011)
  17. Elman, Howard; Mihajlović, Milan; Silvester, David: Fast iterative solvers for buoyancy driven flow problems (2011)
  18. Pennacchio, Micol; Simoncini, Valeria: Fast structured AMG preconditioning for the bidomain model in electrocardiology (2011)
  19. Rees, Tyrone; Wathen, Andrew J.: Preconditioning iterative methods for the optimal control of the Stokes equations (2011)
  20. Benzi, Michele; Ferragut, Luis; Pennacchio, Micol; Simoncini, Valeria: Solution of linear systems from an optimal control problem arising in wind simulation (2010)

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