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Keywords:
nonlinear multigrid; exact interpolation scheme
nonlinear multigrid; exact interpolation scheme
Summary:
We extend the analysis of the recently proposed nonlinear EIS scheme applied to the partial eigenvalue problem. We address the case where the Rayleigh quotient iteration is used as the smoother on the fine-level. Unlike in our previous theoretical results, where the smoother given by the linear inverse power method is assumed, we prove nonlinear speed-up when the approximation becomes close to the exact solution. The speed-up is cubic. Unlike existent convergence estimates for the Rayleigh quotient iteration, our estimates take advantage of the powerful effect of the coarse-space.
References:
[1] Brandt, A., Ron, D.: Multigrid solvers and multilevel Optimization Strategies. Multilevel Optimization in VLSICAD (J. Cong et al., eds) Comb. Optim. 14, Kluwer Academic Publishers, Dordrecht 1-69 (2003). DOI 10.1007/978-1-4757-3748-6_1 | MR 2021995 | Zbl 1046.65043
[2] Ciarlet, P. G.: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications 4, North-Holland Publishing, Amsterdam (1978). DOI 10.1016/s0168-2024(08)x7014-6 | MR 0520174 | Zbl 0383.65058
[3] Crouzeix, M., Philippe, B., Sadkane, M.: The Davidson method. SIAM J. Sci. Comput. 15 (1994), 62-76. DOI 10.1137/0915004 | MR 1257154 | Zbl 0803.65042
[4] Fraňková, P., Hanuš, M., Kopincová, H., Kužel, R., Marek, I., Pultarová, I., Vaněk, P., Vastl, Z.: Convergence theory for the exact interpolation scheme with approximation vector as the first column of the prolongator: the partial eigenvalue problem. Submitted to Numer. Math.
[5] Knyazev, A. V.: Convergence rate estimates for iterative methods for a mesh symmetric eigenvalue problem. Sov. J. Numer. Anal. Math. Model. 2 371-396 (1987). DOI 10.1515/rnam.1987.2.5.371 | MR 0915330 | Zbl 0825.65034
[6] Kushnir, D., Galun, M., Brandt, A.: Efficient multilevel eigensolvers with applications to data analysis tasks. IEEE Trans. Pattern Anal. Mach. Intell. 32 (2010), 1377-1391. DOI 10.1109/TPAMI.2009.147
[7] Kužel, R., Vaněk, P.: Exact interpolation scheme with approximation vector used as a column of the prolongator. Numer. Linear Algebra Appl. (electronic only) 22 (2015), 950-964. DOI 10.1002/nla.1975 | MR 3426323 | Zbl 06604517
[8] Mandel, J., Sekerka, B.: A local convergence proof for the iterative aggregation method. Linear Algebra Appl. 51 (1983), 163-172. DOI 10.1016/0024-3795(83)90157-X | MR 0699731 | Zbl 0494.65014
[9] Notay, Y.: Combination of Jacobi-Davidson and conjugate gradients for the partial symmetric eigenproblem. Numer. Linear Algebra Appl. 9 (2002), 21-44. DOI 10.1002/nla.246 | MR 1874781 | Zbl 1071.65516
[10] Oliveira, S.: On the convergence rate of a preconditioned subspace eigensolver. Computing 63 (1999), 219-231. DOI 10.1007/s006070050032 | MR 1738755 | Zbl 0944.65039
[11] Ovtchinnikov, E.: Convergence estimates for the generalized Davidson method for symmetric eigenvalue problems. II: The subspace acceleration. SIAM J. Numer. Anal. 41 (2003), 272-286. DOI 10.1137/S0036142902411768 | MR 1974502 | Zbl 1078.65538
[12] Parlett, B. N.: The Symmetric Eigenvalue Problem. Classics in Applied Mathematics 20, Society for Industrial and Applied Mathematics, Philadelphia (1987). MR 1490034
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