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The inverse fast multipole method: using a fast approximate direct dolver as a preconditioner for dense linear systems. (English) Zbl 1365.65068


MSC:

65F05 Direct numerical methods for linear systems and matrix inversion
65F08 Preconditioners for iterative methods
65F35 Numerical computation of matrix norms, conditioning, scaling

Software:

Eigen
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References:

[1] S. Ambikasaran and E. Darve, An \(\mathcal{O} (n \log n)\) fast direct solver for partial hierarchically semi-separable matrices, J. Sci. Comput., 57 (2013), pp. 477–501. · Zbl 1292.65030
[2] S. Ambikasaran and E. Darve, The Inverse Fast Multipole Method, preprint, , 2014. · Zbl 1292.65030
[3] A. Aminfar, S. Ambikasaran, and E. Darve, A fast block low-rank dense solver with applications to finite-element matrices, J. Comput. Phys., 304 (2016), pp. 170–188, . · Zbl 1349.65595
[4] M. Bebendorf, Hierarchical Matrices: A Means to Efficiently Solve Elliptic Boundary Value Problems, Springer, Berlin, 2008. · Zbl 1151.65090
[5] M. Bebendorf and S. Rjasanow, Adaptive low-rank approximation of collocation matrices, Computing, 70 (2003), pp. 1–24. · Zbl 1068.41052
[6] M. Bonnet, Boundary Integral Equation Methods for Solids and Fluids, John Wiley & Sons, Chichester, UK, 1995.
[7] S. Börm, Data-sparse approximation of non-local operators by \(\mathcal{H}^2\)-matrices, Linear Algebra Appl., 422 (2007), pp. 380–403. · Zbl 1115.65041
[8] S. Börm, L. Grasedyck, and W. Hackbusch, Introduction to hierarchical matrices with applications, Eng. Anal. Bound. Elem., 27 (2003), pp. 405–422. · Zbl 1035.65042
[9] M. D. Buhmann, Radial Basis Functions: Theory and Implementations, Cambridge University Press, Cambridge, UK, 2003. · Zbl 1038.41001
[10] B. Carpentieri, I. S. Duff, L. Giraud, and M. Magolu monga Made, Sparse symmetric preconditioners for dense linear systems in electromagnetism, Numer. Linear Algebra Appl., 11 (2004), pp. 753–771. · Zbl 1164.65340
[11] J. Carrier, L. Greengard, and V. Rokhlin, A fast adaptive multipole algorithm for particle simulations, SIAM J. Sci. Stat. Comput., 9 (1988), pp. 669–686. · Zbl 0656.65004
[12] M. Chandrasekaran, S. Gu and T. Pals, A fast ULV decomposition solver for hierarchically semiseparable representations, SIAM J. Matrix Anal. Appl., 28 (2006), pp. 603–622. · Zbl 1120.65031
[13] S. Chandrasekaran, P. Dewilde, M. Gu, W. Lyons, and T. Pals, A fast solver for HSS representations via sparse matrices, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 67–81. · Zbl 1135.65317
[14] S. H. Christiansen and J.-C. Nédélec, A preconditioner for the electric field integral equation based on Calderon formulas, SIAM J. Numer. Anal., 40 (2002), pp. 1100–1135. · Zbl 1021.78010
[15] E. Corona, P.-G. Martinsson, and D. Zorin, An \({O}({N})\) direct solver for integral equations on the plane, Appl. Comput. Harmon. Anal., 38 (2015), pp. 284–317. · Zbl 1307.65180
[16] M. Darbas, E. Darrigrand, and Y. Lafranche, Combining analytic preconditioner and fast multipole Method for the 3-D Helmholtz equation, J. Comput. Phys., 236 (2013), pp. 289–316. · Zbl 1286.78004
[17] E. Darve, The fast multipole method: Numerical implementation, J. Comput. Phys., 160 (2000), pp. 195–240. · Zbl 0974.78012
[18] S. Delong, F. B. Usabiaga, R. Delgado-Buscalioni, B. E. Griffith, and A. Donev, Brownian dynamics without Green’s functions, J. Chem. Phys., 140 (2014), 134110.
[19] W. Fong and E. Darve, The black-box fast multipole method, J. Comput. Phys., 228 (2009), pp. 8712–8725. · Zbl 1177.65009
[20] A. Gillman and P.-G. Martinsson, An O(N) algorithm for constructing the solution operator to 2D elliptic boundary value problems in the absence of body loads, Adv. Comput. Math., 40 (2014), pp. 773–796. · Zbl 1295.65107
[21] G. H. Golub and C. F. Van Loan, Matrix Computations, 3rd ed., John Hopkins University Press, Baltimore, MD, 1996. · Zbl 0865.65009
[22] L. Greengard and V. Rokhlin, A fast algorithm for particle simulations, J. Comput. Phys., 73 (1987), pp. 325–348. · Zbl 0629.65005
[23] G. Guennebaud et al., Eigen v3, , 2010.
[24] N. A. Gumerov and R. Duraiswami, Fast radial basis function interpolation via preconditioned Krylov iteration, SIAM J. Sci. Comput., 29 (2007), pp. 1876–1899. · Zbl 1154.65303
[25] W. Hackbusch, A sparse matrix arithmetic based on \(\mathscr{H}\)-matrices. Part I: Introduction to \(\mathscr{H}\)-matrices, Computing, 62 (1999), pp. 89–108.
[26] W. Hackbusch and S. Börm, Data-sparse approximation by adaptive \(\mathcal{H}^2\)-matrices, Computing, 69 (2002), pp. 1–35.
[27] W. Hackbusch and S. Börm, \(\mathscr{H}^2\)-matrix approximation of integral operators by interpolation, Appl. Numer. Math., 43 (2002), pp. 129–143. · Zbl 1019.65103
[28] W. Hackbusch and B. Khoromskij, A sparse \(\mathscr{H}\)-matrix arithmetic: General complexity estimates, J. Comput. Appl. Math., 125 (2000), pp. 479–501. · Zbl 0977.65036
[29] W. Hackbusch, B. N. Khoromskij, and R. Kriemann, Hierarchical matrices based on a weak admissibility criterion, Computing, 73 (2004), pp. 207–243. · Zbl 1063.65035
[30] W. Hackbusch and Z. Nowak, On the fast matrix multiplication in the boundary element method by panel clustering, Numer. Math., 54 (1989), pp. 463–491. · Zbl 0641.65038
[31] N. Halko, P. G. Martinsson, and J. A. Tropp, Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53 (2011), pp. 217–288. · Zbl 1269.65043
[32] K. L. Ho and L. Greengard, A fast direct solver for structured linear systems by recursive skeletonization, SIAM J. Sci. Comput., 34 (2012), pp. 2507–2532. · Zbl 1259.65062
[33] K. L. Ho and L. Ying, Hierarchical interpolative factorization for elliptic operators: Differential equations, Comm. Pure Appl. Math., 69 (2016), pp. 1415–1451. · Zbl 1353.35142
[34] K. L. Ho and L. Ying, Hierarchical interpolative factorization for elliptic operators: Integral equations, Comm. Pure Appl. Math., 69 (2016), pp. 1314–1353. · Zbl 1344.65123
[35] G. C. Hsiao, Boundary element methods—an overview, Appl. Numer. Math., 56 (2006), pp. 1356–1369. · Zbl 1237.65133
[36] B. Kallemov, A. Pal Singh Bhalla, B. E. Griffith, and A. Donev, An Immersed Boundary Method for Rigid Bodies, Commun. Appl. Math. Comput. Sci., 11 (2016), pp. 79–141, . · Zbl 1382.76191
[37] R. E. Kalman, Mathematical description of linear dynamical systems, SIAM J. Control Optim., 1 (1963), pp. 152–192. · Zbl 0145.34301
[38] W. Y. Kong, J. Bremer, and V. Rokhlin, An adaptive fast direct solver for boundary integral equations in two dimensions, Appl. Comput. Harmon. Anal., 31 (2011), pp. 346–369. · Zbl 1227.65118
[39] U. Langer and D. Pusch, Data-sparse algebraic multigrid methods for large scale boundary element equations, Appl. Numer. Math., 54 (2005), pp. 406–424. · Zbl 1073.65135
[40] J. Lee, J. Zhang, and C. Lu, Incomplete LU preconditioning for large scale dense complex linear systems from electromagnetic wave scattering problems, J. Comput. Phys., 185 (2003), pp. 158–175. · Zbl 1017.65027
[41] B. Lizé, Fast Direct Solver for the Boundary Element Method in Electromagnetism and Acoustics: \(\mathcal{H}\)-Matrices. Parallelism and Industrial Applications, Ph.D. thesis, Université Paris 13, Paris, 2014.
[42] P. G. Martinsson and V. Rokhlin, A fast direct solver for boundary integral equations in two dimensions, J. Comput. Phys., 205 (2005), pp. 1–23. · Zbl 1078.65112
[43] N. Nishimura, Fast multipole accelerated boundary integral equation methods, Appl. Mech. Rev., 55 (2002), pp. 299–324.
[44] T. Pals, Multipole for Scattering Computations: Spectral Discretization, Stabilization, Fast Solvers, Ph.D. thesis, University of California, Santa Barbara, CA, 2004.
[45] A. Piche, G.-P. Piau, O. Urrea, G. Sabanowski, B. Lize, J. Robert, G. Sylvand, P. Benjamin, A. Thain, R. Perraud, and G. Peres, BEM/MoM fast direct computation for antenna sitting and antenna coupling on large aeronautic plateforms, in Proceedings of the 2014 IEEE Conference on Antenna Measurements & Applications (CAMA), 2014, , 2014, 7003376.
[46] H. Pouransari, P. Coulier, and E. Darve, Fast Hierarchical Solvers for Sparse Matrices, SIAM J. Sci. Comput., accepted. · Zbl 1365.65072
[47] H. Pouransari and E. Darve, Optimizing the adaptive fast multipole method for fractal sets, SIAM J. Sci. Comput., 37 (2015), pp. A1040–A1066. · Zbl 1316.28010
[48] S. Rjasanow and O. Steinbach, The Fast Solution of Boundary Integral Equations, Mathematical and Analytical Techniques with Applications to Engineering, Springer, New York, 2007. · Zbl 1119.65119
[49] V. Rokhlin, Rapid solution of integral equations of classical potential theory, J. Comput. Phys., 60 (1985), pp. 187–207. · Zbl 0629.65122
[50] J. Rotne and S. Prager, Variational treatment of hydrodynamic interaction in polymers, J. Chem. Phys., 50 (1969), pp. 4831–4837.
[51] Y. Saad, Krylov subspace methods for solving large unsymmetric linear systems, Math. Comp., 37 (1981), pp. 105–126. · Zbl 0474.65019
[52] Y. Saad, A flexible inner-outer preconditioned GMRES algorithm, SIAM J. Sci. Comput., 14 (1993), pp. 461–469. · Zbl 0780.65022
[53] Y. Saad and M. H. Schultz, GMRES: A generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7 (1986), pp. 856–869. · Zbl 0599.65018
[54] S. Sauter, The panel clustering method in 3-d BEM, in Wave Propagation in Complex Media, G. Papanicolaou, ed., The IMA Volumes in Mathematics and its Applications 96, Springer, New York, 1998, pp. 199–224. · Zbl 0901.65070
[55] Z. Sheng, P. Dewilde, and S. Chandrasekaran, Algorithms to solve hierarchically semi-separable systems, in System Theory, the Schur Algorithm and Multidimensional Analysis, D. Alpay and V. Vinnikov, eds., Operator Theory: Advances and Applications 176, Birkhäuser, Basel, 2007, pp. 255–294. · Zbl 1123.65020
[56] U. Trottenberg, C. W. Oosterlee, and A. Schuller, Multigrid, Academic Press, London, 2000.
[57] J. Xia, S. Chandrasekaran, M. Gu, and X. S. Li, Fast algorithms for hierarchically semiseparable matrices, Numer. Linear Algebra Appl., 17 (2010), pp. 953–976. · Zbl 1240.65087
[58] J. Xia, S. Chandrasekaran, M. Gu, and X. S. Li, Superfast multifrontal method for large structured linear systems of equations, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 1382–1411. · Zbl 1195.65031
[59] L. Ying, G. Biros, and D. Zorin, A kernel-independent adaptive fast multipole algorithm in two and three dimensions, J. Comput. Phys., 196 (2004), pp. 591–626. · Zbl 1053.65095
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