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.
nonlinear multigrid exact interpolation scheme
This is a preview of subscription content, log in to check access.
A. Brandt, D. Ron: Multigrid solvers and multilevel Optimization Strategies. Multilevel Optimization in VLSICAD (J. Cong et al., eds). Comb. Optim. 14, Kluwer Academic Publishers, Dordrecht, 2003, pp. 1–69.CrossRefMATHGoogle Scholar
P. G. Ciarlet: The Finite Element Method for Elliptic Problems. Studies in Mathematics and Its Applications 4, North-Holland Publishing, Amsterdam, 1978.Google Scholar
P. Fraňková, M. Hanuš, H. Kopincová, R. Kužel, I. Marek, I. Pultarová, P. Vaněk, Z. Vastl: 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.Google Scholar
A. V. Knyazev: Convergence rate estimates for iterative methods for a mesh symmetric eigenvalue problem. Sov. J. Numer. Anal. Math. Model. 2 (1987), 371–396.MathSciNetMATHGoogle Scholar
D. Kushnir, M. Galun, A. Brandt: Efficient multilevel eigensolvers with applications to data analysis tasks. IEEE Trans. Pattern Anal. Mach. Intell. 32 (2010), 1377–1391.CrossRefGoogle Scholar
R. Kužel, P. Vaněk: Exact interpolation scheme with approximation vector used as a column of the prolongator. Numer. Linear Algebra Appl. (electronic only) 22 (2015), 950–964.MathSciNetCrossRefMATHGoogle Scholar
E. Ovtchinnikov: Convergence estimates for the generalized Davidson method for symmetric eigenvalue problems. II: The subspace acceleration. SIAM J. Numer. Anal. 41 (2003), 272–286.MathSciNetCrossRefMATHGoogle Scholar
B. N. Parlett: The Symmetric Eigenvalue Problem. Classics in Applied Mathematics 20, Society for Industrial and Applied Mathematics, Philadelphia, 1987.Google Scholar