11,330 research outputs found
V-cycle optimal convergence for certain (multilevel) structured linear systems
In this paper we are interested in the solution by multigrid strategies of multilevel linear systems whose coefficient matrices belong to the circulant, Hartley, or \u3c4 algebras or to the Toeplitz class and are generated by (the Fourier expansion of) a nonnegative multivariate polynomial f. It is well known that these matrices are banded and have eigenvalues equally distributed as f, so they are ill-conditioned whenever f takes the zero value; they can even be singular and need a low-rank correction. We prove the V-cycle multigrid iteration to have a convergence rate independent of the dimension even in presence of ill-conditioning. If the (multilevel) coefficient matrix has partial dimension nr at level r, r = 1, . . . ,d, then the size of the algebraic system is N(n) = \u3a0r=1 d nr, O(N(n)) operations are required by our technique, and therefore the corresponding method is optimal. Some numerical experiments concerning linear systems arising in applications, such as elliptic PDEs with mixed boundary conditions and image restoration problems, are considered and discussed.cussed
V-cycle optimal convergence for DCT-III matrices
The paper analyzes a two-grid and a multigrid method for matrices belonging
to the DCT-III algebra and generated by a polynomial symbol. The aim is to
prove that the convergence rate of the considered multigrid method (V-cycle) is
constant independent of the size of the given matrix. Numerical examples from
differential and integral equations are considered to illustrate the claimed
convergence properties.Comment: 19 page
A Self-learning Algebraic Multigrid Method for Extremal Singular Triplets and Eigenpairs
A self-learning algebraic multigrid method for dominant and minimal singular
triplets and eigenpairs is described. The method consists of two multilevel
phases. In the first, multiplicative phase (setup phase), tentative singular
triplets are calculated along with a multigrid hierarchy of interpolation
operators that approximately fit the tentative singular vectors in a collective
and self-learning manner, using multiplicative update formulas. In the second,
additive phase (solve phase), the tentative singular triplets are improved up
to the desired accuracy by using an additive correction scheme with fixed
interpolation operators, combined with a Ritz update. A suitable generalization
of the singular value decomposition is formulated that applies to the coarse
levels of the multilevel cycles. The proposed algorithm combines and extends
two existing multigrid approaches for symmetric positive definite eigenvalue
problems to the case of dominant and minimal singular triplets. Numerical tests
on model problems from different areas show that the algorithm converges to
high accuracy in a modest number of iterations, and is flexible enough to deal
with a variety of problems due to its self-learning properties.Comment: 29 page
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