Sharp Convergence Estimates Research Articles

Regularized Solution of the Cauchy Problem for the Biharmonic Equation

In this paper, the Cauchy problem associated with the biharmonic equation is investigated. We prove that in principle, the problem is severely ill-posed in the sense of Hadamard. Therefore, we propose a quasi-boundary value-type regularization method for stabilizing the ill-posed problem. Very sharp convergence estimates are established based on some a priori information on the exact solution. Finally, several numerical examples with random data are provided to show the effectiveness of the proposed method.

Algebraic Two-Level Convergence Theory for Singular Systems

We consider the algebraic convergence theory that gives its theoretical foundations to classical algebraic multigrid methods. All the main results constitutive of the approach are properly extended to singular compatible systems, including the recent sharp convergence estimates for both symmetric and nonsymmetric systems. In fact, all results carry over smoothly to the singular case, which therefore does not require a specific treatment (except a proper handling of issues associated with singular coarse grid matrices). Regarding problems with a low-dimensional null space, the presented results thus mainly confirm what has been observed often at a more practical level in many previous works. As illustrated by the discussion of the application to the curl-curl equation, the potential impact is greater for problems with large-dimensional null space. Indeed, it turns out that the design of multilevel methods can then actually be easier for the singular case than it is for nearby regularized problems.

Iterative minimization of the Rayleigh quotient by block steepest descent iterations

The topic of this paper is the convergence analysis of subspace gradient iterations for the simultaneous computation of a few of the smallest eigenvalues plus eigenvectors of a symmetric and positive definite matrix pair (A,M). The methods are based on subspace iterations for A−1M and use the Rayleigh-Ritz procedure for convergence acceleration. New sharp convergence estimates are proved by generalizing estimates which have been presented for vectorial steepest descent iterations (see SIAM J. Matrix Anal. Appl., 32(2):443-456, 2011).

On optimal step‐length gradient eigensolvers

AbstractGradient iterations for the minimization of the Rayleigh quotient are robust and (with a proper preconditioning) fast iterations to compute approximations of the smallest eigenvalue of a self‐adjoint elliptic partial differential operator. Up to now sharp convergence estimates were only known for the basic fixed‐step size preconditioned gradient iteration (also called preconditioned inverse iteration). Recently sharp convergence estimates have been proved for optimal step size (preconditioned) gradient iterations. These new estimates are compared with previous results. (© 2011 Wiley‐VCH Verlag GmbH & Co. KGaA, Weinheim)

Convergence Analysis of Gradient Iterations for the Symmetric Eigenvalue Problem

Gradient iterations for the Rayleigh quotient are simple and robust solvers to determine a few of the smallest eigenvalues together with the associated eigenvectors of (generalized) matrix eigenvalue problems for symmetric matrices. Sharp convergence estimates for the Ritz values and Ritz vectors are derived for various steepest descent/ascent gradient iterations. The analysis shows that poorest convergence of the eigenvalue approximations is attained in a three-dimensional invariant subspace; explicit convergence estimates are then derived by means of a minidimensional analysis.

Uniform null controllability of the heat equation with rapidly oscillating periodic density

We consider the heat equation with fast oscillating periodic density, and an interior control in a bounded domain. First, we prove sharp convergence estimates depending explicitly on the initial data for the corresponding uncontrolled equation; these estimates are new, and their proof relies on a judicious smoothing of the initial data. Then we use those estimates to prove that the original equation is uniformly null controllable, provided a carefully chosen extra vanishing interior control is added to that equation. This uniform controllability result is the first in the multidimensional setting for the heat equation with oscillating density. Finally, we prove that the sequence of null controls converges to the optimal null control of the limit equation when the period tends to zero. To cite this article: L. Tebou, C. R. Acad. Sci. Paris, Ser. I 347 (2009).

A geometric theory for preconditioned inverse iteration applied to a subspace

The aim of this paper is to provide a convergence analysis for a preconditioned subspace iteration, which is designated to determine a modest number of the smallest eigenvalues and its corresponding invariant subspace of eigenvectors of a large, symmetric positive definite matrix. The algorithm is built upon a subspace implementation of preconditioned inverse iteration. i.e., the well-known inverse iteration procedure, where the associated system of linear equations is solved approximately by using a preconditioner. This step is followed by a Rayleigh-Ritz projection so that preconditioned inverse iteration is always applied to the Ritz vectors of the actual subspace of approximate eigenvectors. The given theory provides sharp convergence estimates for the Ritz values and is mainly built on arguments exploiting the geometry underlying preconditioned inverse iteration.

A geometric theory for preconditioned inverse iteration I: Extrema of the Rayleigh quotient

The discretization of eigenvalue problems for partial differential operators is a major source of matrix eigenvalue problems having very large dimensions, but only some of the smallest eigenvalues together with the eigenvectors are to be determined. Preconditioned inverse iteration (a “matrix-free” method) derives from the well-known inverse iteration procedure in such a way that the associated system of linear equations is solved approximately by using a (multigrid) preconditioner. A new convergence analysis for preconditioned inverse iteration is presented. The preconditioner is assumed to satisfy some bound for the spectral radius of the error propagation matrix resulting in a simple geometric setup. In this first part the case of poorest convergence depending on the choice of the preconditioner is analyzed. In the second part the dependence on all initial vectors having a fixed Rayleigh quotient is considered. The given theory provides sharp convergence estimates for the eigenvalue approximations showing that multigrid eigenvalue/vector computations can be done with comparable efficiency as known from multigrid methods for boundary value problems.

Construction d'un modèle d'évolution de plaques avec terme d'inerte de rotation

In a previous work, we have shown how the asymptotic expansion method (with the thickness as the « small » parameter) applied to the linear Hellinger-Reissner evolution model for an elastic plate ofRo3 yields a justification of the classical evolution equation for the transverse displacement of a plate with thickness 2e $$2\varrho e\frac{{\partial ^2 \zeta }}{{\partial t^2 }} + \frac{2}{3}\frac{E}{{1 - v^2 }}e^3 \Delta ^2 \zeta = \int\limits_{ - e}^e {f3.} $$ Pursuing farther the analysis of the asymptotic expansion and considering in particular the first corrector, we are led here to the plate model $$2\varrho e\frac{{\partial ^2 \zeta }}{{\partial t^2 }} + \frac{2}{3}\frac{E}{{1 - v^2 }}e^3 \Delta ^2 \zeta - \frac{{34 - 14v}}{{15(1 - v)}}\varrho e^2 \Delta \zeta '' = \int\limits_{ - e}^e {f3.} $$ Equations of the same form have been proposed notably by N. F. Morozov, G. Duvaut, J. L. Lions and are called equations with rotational inertia term. The applicability of such models is studied and sharp convergence estimates are given.