×

zbMATH — the first resource for mathematics

A fast hierarchically preconditioned eigensolver based on multiresolution matrix decomposition. (English) Zbl 1455.65053

MSC:
65F15 Numerical computation of eigenvalues and eigenvectors of matrices
15A18 Eigenvalues, singular values, and eigenvectors
15A12 Conditioning of matrices
65F08 Preconditioners for iterative methods
Software:
ARPACK; eigs; IRAM
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] L. Bergamaschi and E. Bozzo, Computing the smallest eigenpairs of the graph Laplacian, SeMA J., 75 (2018), pp. 1–16. · Zbl 1392.65087
[2] L. Bergamaschi, G. Gambolati, and G. Pini, Asymptotic convergence of conjugate gradient methods for the partial symmetric eigenproblem, Numer. Linear Algebra Appl., 4 (1997), pp. 69–84. · Zbl 0889.65032
[3] E. Bozzo and M. Franceschet, Effective and Efficient Approximations of the Generalized Inverse of the Graph Laplacian Matrix with an Application to Current-Flow Betweenness Centrality, preprint, arXiv:1205.4894, 2012. · Zbl 1258.05068
[4] E. Bozzo and M. Franceschet, Resistance distance, closeness, and betweenness, Soc. Netw., 35 (2013), pp. 460–469.
[5] D. Calvetti, L. Reichel, and D. C. Sorensen, An implicitly restarted Lanczos method for large symmetric eigenvalue problems, Electron. Trans. Numer. Anal., 2 (1994), pp. 1–21. · Zbl 0809.65030
[6] F. R. Chung, Spectral Graph Theory, CBMS Reg. Conf. Ser. Math. 92, American Mathematical Society, Providence, RI, 1997.
[7] S. Cocco, R. Monasson, and M. Weigt, From principal component to direct coupling analysis of coevolution in proteins: Low-eigenvalue modes are needed for structure prediction, PLoS Comput. Biol., 9 (2013), e1003176.
[8] J. Francis, The transformation: A unitary analogue to the transformation. I, Comput. J., 4 (1961), pp. 265–271. · Zbl 0104.34304
[9] S. Goedecker, Low complexity algorithms for electronic structure calculations, J. Comput. Phys., 118 (1995), pp. 261–268. · Zbl 0827.65082
[10] T. Hou, D. Huang, K. Lam, and P. Zhang, An adaptive fast solver for a general class of positive definite matrices via energy decomposition, Multiscale Model. Simul., 16 (2018), pp. 615–678. · Zbl 1391.65066
[11] Y. T. Hou and P. Zhang, Sparse operator compression of higher-order elliptic operators with rough coefficients, Res. Math. Sci., 4, (2017), 24. · Zbl 1412.65215
[12] R. B. Lehoucq and D. C. Sorensen, Deflation techniques for an implicitly restarted Arnoldi iteration, SIAM J. Matrix Anal. Appl., 17 (1996), pp. 789–821. · Zbl 0863.65016
[13] R. B. Lehoucq, D. C. Sorensen, and C. Yang, ARPACK Users’ Guide: Solution of Large-Scale Eigenvalue Problems with Implicitly Restarted Arnoldi Methods, SIAM, Philadelphia, 1998. · Zbl 0901.65021
[14] Q. Lin and H. Xie, A multi-level correction scheme for eigenvalue problems, Math. Comp., 84 (2015), pp. 71–88. · Zbl 1307.65159
[15] A. M\aalqvist and D. Peterseim, Localization of elliptic multiscale problems, Math. Comp., 83 (2014), pp. 2583–2603. · Zbl 1301.65123
[16] Á. Martínez, Tuned preconditioners for the eigensolution of large SPD matrices arising in engineering problems, Numer. Linear Algebra Appl., 23 (2016), pp. 427–443. · Zbl 1413.65101
[17] L. Meirovitch, Elements of Vibration Analysis, McGraw-Hill, New York, 1975. · Zbl 0359.70039
[18] M. Newman, Networks: An Introduction, Oxford University Press, Oxford, 2010. · Zbl 1195.94003
[19] A. Y. Ng, M. I. Jordan, and Y. Weiss, On spectral clustering: Analysis and an algorithm, in 14th NIPS’01, MIT Press, Cambridge, MA, 2001, pp. 849–856.
[20] H. Owhadi, Multigrid with rough coefficients and multiresolution operator decomposition from hierarchical information games, SIAM Rev., 59 (2017), pp. 99–149. · Zbl 1358.65071
[21] H. Owhadi and C. Scovel, Universal Scalable Robust Solvers from Computational Information Games and Fast Eigenspace Adapted Multiresolution Analysis, preprint, arXiv:1703.10761, 2017.
[22] V. Ozoliņš, R. Lai, R. Caflisch, and S. Osher, Compressed modes for variational problems in mathematics and physics, Proc. Natl. Acad. Sci. USA, 110 (2013), pp. 18368–18373. · Zbl 1292.81024
[23] E. Romero, M. B. Cruz, J. E. Roman, and P. B. Vasconcelos, A parallel implementation of the Jacobi-Davidson eigensolver for unsymmetric matrices, in VECPAR 2010, Springer, Berlin, 2010, pp. 380–393. · Zbl 1323.65132
[24] F. Schäfer, T. Sullivan, and H. Owhadi, Compression, Inversion, and Approximate PCA of Dense Kernel Matrices at Near-Linear Computational Complexity, preprint, arXiv:1706.02205, 2017.
[25] J. Shi and J. Malik, Normalized cuts and image segmentation, IEEE Trans. Pattern Anal. Mach. Intell., 22 (2000), pp. 888–905.
[26] G. L. Sleijpen and H. A. Van der Vorst, A Jacobi–Davidson iteration method for linear eigenvalue problems, SIAM Rev., 42 (2000), pp. 267–293. · Zbl 0949.65028
[27] D. C. Sorensen, Implicit application of polynomial filters in a k-step Arnoldi method, SIAM J. Matrix Anal. Appl., 13 (1992), pp. 357–385. · Zbl 0763.65025
[28] D. C. Sorensen, Implicitly restarted Arnoldi/Lanczos methods for large scale eigenvalue calculations, in Parallel Numerical Algorithms, Springer, Dordrecht, The Netherlands, 1997, pp. 119–165. · Zbl 0865.65019
[29] G. Stewart, Accelerating the orthogonal iteration for the eigenvectors of a Hermitian matrix, Numer. Math., 13 (1969), pp. 362–376. · Zbl 0185.40203
[30] H. Xie, A multigrid method for eigenvalue problem, J. Comput. Phys., 274 (2014), pp. 550–561. · Zbl 1352.65631
[31] H. Xie, L. Zhang, and H. Owhadi, Fast Eigenpairs Computation with Operator Adapted Wavelets and Hierarchical Subspace Correction, preprint, arXiv:1806.00565, 2018.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.