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Efficient Preconditioning of hp-FEM Matrices by Hierarchical Low-Rank Approximations

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Abstract

We introduce a preconditioner based on a hierarchical low-rank compression scheme of Schur complements. The construction is inspired by standard nested dissection, and relies on the assumption that the Schur complements can be approximated, to high precision, by Hierarchically-Semi-Separable matrices. We build the preconditioner as an approximate \(LDM^t\) factorization of a given matrix A, and no knowledge of A in assembled form is required by the construction. The \(LDM^t\) factorization is amenable to fast inversion, and the action of the inverse can be determined fast as well. We investigate the behavior of the preconditioner in the context of DG finite element approximations of elliptic and hyperbolic problems, with respect to both the mesh size and the order of approximation.

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Notes

  1. Strictly speaking, this is a tree with missing route, i.e., a forest. In fact, if we were to introduce a single top-level box holding the entirety of the dof’s, under a purely algebraic approach, no boundary dof’s could be identified.

  2. In fact, this is the compression threshold of each off-diagonal block of A.

References

  1. Ainsworth, M.: Dispersive and dissipative behaviour of high order discontinuous Galerkin finite element methods. J. Comput. Phys. 198(1), 106–130 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  2. Ainsworth, M., Monk, P., Muniz, W.: Dispersive and dissipative properties of discontinuous Galerkin finite element methods for the second-order wave equation. J. Sci. Comput. 27(1–3), 5–40 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  3. Amestoy, P., Ashcraft, C., Boiteau, O., Buttari, A., L’Excellent, J.-Y., Weisbecker, C.: Improving multifrontal methods by means of block low-rank representations. SIAM J. Sci. Comput. 37(3), A1451–A1474 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Aminfar, A., Darve, E.: A fast, memory efficient and robust sparse preconditioner based on a multifrontal approach with applications to finite-element matrices. Int. J. Numer. Methods Eng. 107(6), 520–540 (2016)

  5. Bebendorf, M.: Hierarchical Matrices. Springer, New York (2008)

    MATH  Google Scholar 

  6. Bebendorf, M., Hackbusch, W.: Existence of \({\cal{H}}\)-matrix approximants to the inverse fe-matrix of elliptic operators with \(l^\infty \)-coefficients. Numer. Math. 95(1), 1–28 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  7. Bendsøe, M.P., Sigmund, O.: Topology Optimization. World Scientific, Singapore (2009)

    Book  MATH  Google Scholar 

  8. Börm, S.: Approximation of solution operators of elliptic partial differential equations by \({\cal{H}}\)- and \({\cal{H}}^2\)-matrices. Numer. Math. 115(2), 165–193 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  9. Chan, T.F., Hansen, P.C.: Some applications of the rank revealing \(\mathit{QR}\) factorization. SIAM J. Sci. Stat. Comput. 13(3), 727–741 (1992)

    Article  MathSciNet  MATH  Google Scholar 

  10. Chandrasekaran, S., Dewilde, P., Gu, M., Somasunderam, N.: On the numerical rank of the off-diagonal blocks of schur complements of discretized elliptic pdes. SIAM J. Matrix Anal. Appl. 31(5), 2261–2290 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chandrasekaran, S., Gu, M., Pals, T.: A fast \(ulv\) decomposition solver for hierarchically semiseparable representations. SIAM J. Matrix Anal. Appl. 28(3), 603–622 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  12. Davis, T.A., Hu, Y.: The university of florida sparse matrix collection. ACM Trans. Math. Softw. (TOMS) 38(1), 1 (2011)

    MathSciNet  MATH  Google Scholar 

  13. Engquist, B., Ying, L.: Sweeping preconditioner for the Helmholtz equation: hierarchical matrix representation. Commun. Pure Appl. Math. 64(5), 697–735 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Engquist, B., Zhao, H.: Approximate separability of green’s function for high frequency helmholtz equations. Technical report, California Univ. Los Angeles Dept. of Mathematics (2014)

  15. Faustmann, T., Melenk, J.M., Praetorious, D.: \({\cal{H}}\)-matrix approximability of the inverses of fem matrices. Numer. Math. 131(4), 737–770 (2015)

    Article  MathSciNet  Google Scholar 

  16. George, A.: Nested dissection of a regular finite element mesh. SIAM J. Numer. Anal. 10(2), 345–363 (1973)

    Article  MathSciNet  MATH  Google Scholar 

  17. Gillman, A., Martinsson, P.G.: An \(\text{ O }(n)\) algorithm for constructing the solution operator to 2D elliptic boundary value problems in the absence of body loads. Adv. Comput. Math. 40(4), 773–796 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  18. Gillman, A., Young, P.M., Martinsson, P.G.: A direct solver with \(\text{ O } (n)\) complexity for integral equations on one-dimensional domains. Front. Math. China 7(2), 217–247 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  19. Grasedyck, L., Kriemann, R., LeBorne, S.: Domain decomposition based \({\cal{H}}\)-LU preconditioning. Numer. Math. 112(4), 565–600 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  20. Hackbusch, W.: A sparse matrix arithmetic based on \({\cal{H}}\)-matrices. Part I: introduction to \({\cal{H}}\)-matrices. Computing 62(2), 89–108 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  21. Hackbusch, W., Khoromskij, B., Sauter, S.A.: On \({\cal{H}}^2\)-matrices. In: Bungartz, H.-J., Hoppe, R.H.W., Zenger, C. (eds.) Lectures on Applied Mathematics: Proceedings of the Symposium Organized by the Sonderforschungsbereich 438 on the Occasion of Karl-Heinz Hoffmann’s 60th Birthday, Munich, June 30—July 1, 1999, pp. 9–29. Springer Berlin Heidelberg (2000)

  22. Hackbusch, W., Khoromskij, B.N., Kriemann, R.: Hierarchical matrices based on a weak admissibility criterion. Computing 73(3), 207–243 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  23. Hesthaven, J.S., Warburton, T.: Nodal Discontinuous Galerkin Methods: Algorithms, Analysis, and Applications. Springer, New York (2007)

    MATH  Google Scholar 

  24. Karypis, G., Kumar, V.: A fast and high quality multilevel scheme for partitioning irregular graphs. SIAM J. Sci. Comput. 20(1), 359–392 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  25. Martinsson, P.G.: A fast randomized algorithm for computing a hierarchically semiseparable representation of a matrix. SIAM J. Matrix Anal. Appl. 32(4), 1251–1274 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  26. Martinsson, P.G., Rokhlin, V.: A fast direct solver for scattering problems involving elongated structures. J. Comput. Phys. 221(1), 288–302 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  27. Oden, J.T., Demkowicz, L.F.: Applied Functional Analysis. CRC Press, Boca Raton (2010)

    MATH  Google Scholar 

  28. Sheng, Z., Dewilde, P., Chandrasekaran, S.: Algorithms to solve hierarchically semi-separable systems. System Theory The Schur Algorithm and Multidimensional Analysis, pp. 255–294. Springer, New York (2007)

    Chapter  Google Scholar 

  29. Woolfe, F., Liberty, E., Rokhlin, V., Tygert, M.: A fast randomized algorithm for the approximation of matrices. Appl. Comput. Harmon. Anal. 25(3), 335–366 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  30. Xia, J.: Efficient structured multifrontal factorization for general large sparse matrices. SIAM J. Sci. Comput. 35(2), A832–A860 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  31. Xia, J.: Randomized sparse direct solvers. SIAM J. Matrix Anal. Appl. 34(1), 197–227 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  32. Xia, J., Chandrasekaran, S., Gu, M., Li, X.S.: Superfast multifrontal method for large structured linear systems of equations. SIAM J. Matrix Anal. Appl. 31(3), 1382–1411 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  33. Xia, J., Chandrasekaran, S., Gu, M., Li, X.S.: Fast algorithms for hierarchically semiseparable matrices. Numer. Linear Algebra Appl. 17(6), 953–976 (2010)

    Article  MathSciNet  MATH  Google Scholar 

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Authors and Affiliations

  1. MATHCCES, Department of Mathematics, RWTH Aachen University, Schinkelstr. 2, 52062, Aachen, Germany

    P. Gatto

  2. MATHICSE, École Polytechnique Fédérale de Lausanne (EPFL), MA C2 652 (Batiment MA), Station 8, 1015, Lausanne, Switzerland

    J. S. Hesthaven

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Gatto, P., Hesthaven, J.S. Efficient Preconditioning of hp-FEM Matrices by Hierarchical Low-Rank Approximations. J Sci Comput 72, 49–80 (2017). https://doi.org/10.1007/s10915-016-0347-x

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