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Parallel inexact constraint preconditioning for ill-conditioned consolidation problems

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Abstract

Constraint preconditioners have proved very efficient for the solution of ill-conditioned finite element (FE) coupled consolidation problems in a sequential computing environment. Their implementation on parallel computers, however, is not straightforward because of their inherent sequentiality. The present paper describes a novel parallel inexact constraint preconditioner (ParICP) for the efficient solution of linear algebraic systems arising from the FE discretization of the coupled poro-elasticity equations. The ParICP implementation is based on the use of the block factorized sparse approximate inverse incomplete Cholesky preconditioner, which is a very recent and effective development for the parallel preconditioning of symmetric positive definite matrices. The ParICP performance is experimented with in real 3D coupled consolidation problems, proving a scalable and efficient implementation of the constraint preconditioning for high-performance computing. ParICP appears to be a very robust algorithm for solving ill-conditioned large-size coupled models in a parallel computing environment.

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References

  1. Lukšan, L., Vlček, J.: Indefinitely preconditioned inexact Newton method for large sparse equality constrained nonlinear programming problems. Numer. Linear Algebr. Appl. 5, 219–247 (1998)

    Article  MATH  Google Scholar 

  2. Keller, C., Gould, N.I.M., Wathen, A.J.: Constraint preconditioning for indefinite linear systems. SIAM J. Math. Anal. Appl. 21, 1300–1317 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  3. Perugia, I., Simoncini, V.: Block-diagonal and indefinite symmetric preconditioners for mixed finite elements formulations. Numer. Linear. Algebr. Appl. 7, 585–616 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  4. Silvester, D.J., Elman, H.C., Kay, D., Wathen, A.J.: Efficient preconditioning of the linearized Navier–Stokes equations for incompressible flow. J. Comput. Appl. Math. 128, 261–279 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  5. Elman, H.C., Silvester, D.J., Wathen, A.J.: Performance and analysis of saddle point preconditioners for the discrete steady-state Navier–Stokes equations. Numer. Math. 90, 665–688 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Bergamaschi, L., Gondzio, J., Zilli, G.: Preconditioning indefinite systems in interior point methods for optimization. Comput. Optim. Appl. 28, 149–171 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bergamaschi, L., Gondzio, J., Venturin, M., Zilli, G.: Inexact constraint preconditioners for linear systems arising in interior point methods. Comput. Optim. Appl. 36, 136–147 (2007)

    Article  MathSciNet  Google Scholar 

  8. Toh, K.C., Phoon, K.K., Chan, S.H.: Block preconditioners for symmetric indefinite linear systems. Int. J. Numer. Methods Eng. 60, 1361–1381 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  9. Bergamaschi, L., Ferronato, M., Gambolati, G.: Novel preconditioners for the iterative solution to FE-discretized coupled consolidation equations. Comput. Meth. Appl. Mech. Eng. 196, 2647–2656 (2007)

    Article  MATH  Google Scholar 

  10. Bergamaschi, L., Ferronato, M., Gambolati, G.: Mixed constraint preconditioners for the iterative solution of FE coupled consolidation equations. J. Comput. Phys. 227, 9885–9897 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  11. Ferronato, M., Janna, C., Gambolati, G.: Mixed constraint preconditioning in computational contact mechanics. Comput. Methods Appl. Mech. Eng. 197, 3922–3931 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. Benzi, M., Golub, G., Liesen, J.: Numerical solution of saddle point problems. Acta Numer. 14, 1–137 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  13. Mardal, K.A., Winther, R.: Preconditioning discretizations of systems of partial differential equations. Numer. Linear Algebr. Appl. 18, 1–40 (2011)

    Article  MathSciNet  Google Scholar 

  14. Rozložník, M., Simoncini, V.: Krylov subspace methods for saddle point problems with indefinite preconditioning. SIAM J. Math Anal. Appl. 24, 368–391 (2002)

    Article  MATH  Google Scholar 

  15. Durazzi, C., Ruggiero, V.: Indefinitely preconditioned conjugate gradient method for large sparse equality and inequality constrained quadratic problems. Numer. Linear Algebr. Appl. 10, 673–688 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. Janna, C., Ferronato, M., Gambolati, G.: Multilevel incomplete factorizations for the iterative solution of non-linear FE problems. Int. J. Numer. Methods Eng. 80, 651–670 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  17. Benzi, M., Cullum, J.K., Tůma, M.: Robust approximate inverse preconditioning for the conjugate gradient method. SIAM J. Sci. Comput. 22, 1318–1332 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  18. Saad, Y.: ILUM: a parallel multielimination ILU preconditioner for general sparse matrices. Technical Report CS-92-241, Department of Computer Science, University of Minnesota (1992)

  19. Karypis, G., Kumar, V.: Parallel threshold-based ILU factorization. In: Proceedings of the 1997 ACM/IEEE Conference and Supercomputing, pp. 1–24. San Jose, CA (1997)

  20. Benzi, M., Tůma, M.: A parallel solver for large-scale Markov chains. Appl. Numer. Math. 41, 135–153 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hackbush, W.: Multigrid methods and applications. Springer, Berlin (1985)

    Google Scholar 

  22. Trottember, U., Oosterlee, C., Schüller, A.: Multigrid. Academic, London (2001)

    Google Scholar 

  23. Vaněk, P.: Acceleration of convergence of a two-level algorithm by smoothing transfer operator. Appl. Math. 37, 265–274 (1992)

    MathSciNet  MATH  Google Scholar 

  24. Vaněk, P., Mandel, J., Brezina, M.: Algebraic multigrid by smoothed aggregation for second and fourth order elliptic problems. Computing 56, 179–196 (1996)

    Article  MathSciNet  MATH  Google Scholar 

  25. Sala, M., Tuminaro, R.S.: A new Petrov–Galerkin smoothed aggregation preconditioner for nonsymmetric linear systems. SIAM J. Sci. Comput. 31, 143–166 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  26. Chan, T.F., Mathew, T.P.: Domain decomposition algorithms. Acta Numer. 3, 61–143 (1994)

    Article  MathSciNet  Google Scholar 

  27. Cai, X.C., Sarkis, M.: A restrictive additive Schwarz preconditioner for general sparse linear systems. SIAM J. Sci. Comput. 21, 792–797 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  28. Farhat, C., Lesoinne, M., Le Tallec, P., Pierson, K., Rixen, D.: FETI-DP: A dual-primal unified FETI method—part I: a faster alternative to the two-level FETI method. Int. J. Numer. Methods Eng. 50, 1523–1544 (2001)

    Article  MATH  Google Scholar 

  29. Elman, H., Silvester, D., Wathen, A.: Finite Elements and Fast Iterative Solvers. Oxford University Press, Oxford (2005)

    MATH  Google Scholar 

  30. Kolotilina, L.Y., Yeremin, A.Y.: Factorized sparse approximate inverse preconditioning. I. Theory. SIAM J. Mat. Anal. Appl. 14, 45–58 (1993)

    Article  MathSciNet  MATH  Google Scholar 

  31. Ferronato, M., Janna, C., Pini, G.: Parallel solution to ill-conditioned FE geomechanical problems. Int. J. Numer. Anal. Methods Geomech. 36, 422–437 (2012)

    Article  Google Scholar 

  32. Janna, C., Ferronato, M., Gambolati, G.: A Block FSAI-ILU parallel preconditioner for symmetric positive definite linear systems. SIAM J. Sci. Comput. 32, 2468–2484 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  33. Janna, C., Ferronato, M.: Adaptive pattern research for block FSAI preconditioning. SIAM J. Sci. Comput. 33, 3357–3380 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  34. Freund, R.W., Nachtingal, N.M.: A new Krylov-subspace method for symmetric indefinite linear systems. In: Proceedings of the 14th IMACS World Congress on Computational and Applied Mathematics, pp. 1253–1256. (1994)

  35. Biot, M.A.: General theory of three-dimensional consolidation. J. Appl. Phys. 12, 155–164 (1941)

    Article  MATH  Google Scholar 

  36. Ferronato, M., Gambolati, G., Teatini, P.: Ill-conditioning of finite element poroelasticity equations. Int. J. Solids Struct. 38, 5995–6014 (2001)

    Article  MATH  Google Scholar 

  37. Ferronato, M., Pini, G., Gambolati, G.: The role of preconditioning in the solution to FE coupled consolidation equations by Krylov subspace methods. Int. J. Numer. Anal. Methods Geomech. 33, 405–423 (2009)

    Article  Google Scholar 

  38. Phoon, K.K., Toh, K.C., Chan, S.H., Lee, F.H.: An efficient diagonal preconditioner for finite element solution of Biot’s consolidation equations. Int. J. Numer. Methods Eng. 55, 377–400 (2002)

    Article  MATH  Google Scholar 

  39. Chen, X., Toh, K.C., Phoon, K.K.: A modified SSOR preconditioner for sparse symmetric indefinite linear systems of equations. Int. J. Numer. Methods Eng. 65, 785–807 (2006)

    Article  MATH  Google Scholar 

  40. Gambolati, G., Pini, G., Ferronato, M.: Scaling improves stability of preconditioned CG-like solvers for FE consolidation equations. Int. J. Numer. Anal. Methods Geomech. 27, 1043–1056 (2003)

    Article  MATH  Google Scholar 

  41. Gambolati, G., Ferronato, M., Janna, C.: Preconditioners in computational geomechanics: a survey. Int. J. Numer. Anal. Methods Geomech. 35, 980–996 (2011)

    Article  Google Scholar 

  42. Ferronato, M., Bergamaschi, L., Gambolati, G.: Performance and robustness of block constraint preconditioners in FE coupled consolidation problems. Int. J. Numer. Methods Eng. 81, 381–402 (2010)

    MathSciNet  MATH  Google Scholar 

  43. Benzi, M., Simoncini, V.: On the eigenvalues of a class of saddle point matrices. Numer. Math. 103, 173–196 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  44. Saad, Y.: ILUT: a dual threshold incomplete ILU factorization. Numer. Linear Algebr. Appl. 1, 387–402 (1994)

    Article  MathSciNet  MATH  Google Scholar 

  45. Li, N., Saad, Y., Chow, E.: Crout version of ILU for general sparse matrices. SIAM J. Sci. Comput. 25, 716–728 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  46. Janna, C., Comerlati, A., Gambolati, G.: A comparison of projective and direct solvers for finite elements in elastostatics. Adv. Eng. Softw. 40, 675–685 (2009)

    Article  MATH  Google Scholar 

  47. Huckle, T.: Approximate sparsity patterns for the inverse of a matrix and preconditioning. Appl. Numer. Math. 30, 291–303 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  48. Chow, E.: A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J. Sci. Comput. 21, 1804–1822 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  49. Janna, C., Ferronato, M., Castelletto, N.: BFSAI-IC OpenMP implementation. http://www.dmsa.unipd.it/~ferronat/software.html (2011)

  50. Hendrickson, B., Kolda, T.G.: Graph partitioning models for parallel computing. Paral. Comput. 26, 1519–1534 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  51. Uçar, B., Aykanat, C.: Partitioning sparse matrices for parallel preconditioned iterative methods. SIAM J. Sci. Comput. 29, 1683–1709 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  52. Cuthill, E., McKee, J.: Reducing the bandwidth of sparse symmetric matrices. In: Proceedings of the 1969 24th National Conference, pp. 157–172. (1969)

  53. Vermeer, P.A., Verruijt, A.: An accuracy condition for consolidation by finite elements. Int. J. Numer. Anal. Methods Geomech. 5, 1–14 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  54. Booker, J.R., Small, J.C.: An investigation of the stability of numerical solutions of Biot’s equations of consolidation. Int. J. Solids Struct. 11, 907–917 (1975)

    Article  MATH  Google Scholar 

  55. Bergamaschi, L., Martinez, A.: FSAI-based parallel mixed constraint preconditioners for saddle point problems arising in geomechanics. J. Comput. Appl. Math. 236, 308–318 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  56. Chan, S.H., Phoon, K.K., Lee, F.H.: A modified Jacobi preconditioner for solving ill-conditioned Biot’s consolidation equations using symmetric quasi-minimal residual method. Int. J. Numer. Anal. Methods Geomech. 25, 1001–1025 (2001)

    Article  MATH  Google Scholar 

  57. Baù, D., Ferronato, M., Gambolati, G., Teatini, P.: Basin-scale compressibility of the Northern Adriatic by the radioactive marker technique. Géotechnique 52, 605–616 (2002)

    Article  Google Scholar 

  58. Teatini, P., Castelletto, N., Ferronato, M., Gambolati, G., Janna, C., Cairo, E., Marzorati, D., Colombo, D., Ferretti, A., Bagliani, A., Bottazzi, F.: Geomechanical response to seasonal gas storage in depleted reservoirs: a case study in the Po River basin, Italy. J. Geophys. Res. 116, F02002 (2011). doi:10.1029/2010JF001793

    Article  Google Scholar 

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  1. Department Mathematical Models and Methods for Scientific Applications, University of Padova, via Trieste 63, 35121, Padua, Italy

    Carlo Janna, Massimiliano Ferronato & Giuseppe Gambolati

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  1. Carlo Janna
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  2. Massimiliano Ferronato
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  3. Giuseppe Gambolati
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Correspondence to Massimiliano Ferronato.

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Janna, C., Ferronato, M. & Gambolati, G. Parallel inexact constraint preconditioning for ill-conditioned consolidation problems. Comput Geosci 16, 661–675 (2012). https://doi.org/10.1007/s10596-012-9276-4

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