K. Kahl

M. Bolten, A. Brandt, J. Brannick et al.

This work concerns the development of an Algebraic Multilevel method for computing stationary vectors of Markov chains. We present an efficient Bootstrap Algebraic Multilevel method for this task. In our proposed approach, we employ a multilevel eigensolver, with interpolation built using ideas based on compatible relaxation, algebraic distances, and least squares fitting of test vectors. Our adaptive variational strategy for computation of the state vector of a given Markov chain is then a combination of this multilevel eigensolver and associated multilevel preconditioned GMRES iterations. We show that the Bootstrap AMG eigensolver by itself can efficiently compute accurate approximations to the state vector. An additional benefit of the Bootstrap approach is that it yields an accurate interpolation operator for many other eigenmodes. This in turn allows for the use of the resulting AMG hierarchy to accelerate the MLE steps using standard multigrid correction steps. The proposed approach is applied to a range of test problems, involving non-symmetric stochastic M-matrices, showing promising results for all problems considered.

A. Brandt, J. Brannick, K. Kahl et al.

In this paper we motivate, discuss the implementation and present the resulting numerics for a new definition of strength of connection which is based on the notion of algebraic distance. This algebraic distance measure, combined with compatible relaxation, is used to choose suitable coarse grids and accurate interpolation operators for algebraic multigrid algorithms. The main tool of the proposed measure is the least squares functional defined using a set of relaxed test vectors. The motivating application is the anisotropic diffusion problem, in particular problems with non-grid aligned anisotropy. We demonstrate numerically that the measure yields a robust technique for determining strength of connectivity among variables, for both two-grid and multigrid solvers. %We illustrate the use of the measure to construct, in addition, an adaptive aggregation form of interpolation for the targeted anisotropic problems. %Our approach is not a two-level approach -- we provide preliminary results that show its extendability to multigrid. The proposed algebraic distance measure can also be used in any other coarsening procedure, assuming a rich enough set of the near-kernel components of the matrix for the targeted system is known or computed.

A. Frommer, K. Kahl, Th. Lippert et al.

The Lanczos process constructs a sequence of orthonormal vectors v_m spanning a nested sequence of Krylov subspaces generated by a hermitian matrix A and some starting vector b. In this paper we show how to cheaply recover a secondary Lanczos process starting at an arbitrary Lanczos vector v_m. This secondary process is then used to efficiently obtain computable error estimates and error bounds for the Lanczos approximations to the action of a rational matrix function on a vector. This includes, as a special case, the Lanczos approximation to the solution of a linear system Ax = b. Our approach uses the relation between the Lanczos process and quadrature as developed by Golub and Meurant. It is different from methods known so far because of its use of the secondary Lanczos process. With our approach, it is now in particular possible to efficiently obtain {\em upper bounds} for the error in the {\em 2-norm}, provided a lower bound on the smallest eigenvalue of $A$ is known. This holds in particular for a large class of rational matrix functions including best rational approximations to the inverse square root and the sign function. We compare our approach to other existing error estimates and bounds known from the literature and include results of several numerical experiments.

K. Kahl, H. Rittich

Deflation techniques for Krylov subspace methods have seen a lot of attention in recent years. They provide means to improve the convergence speed of these methods by enriching the Krylov subspace with a deflation subspace. The most common approach for the construction of deflation subspaces is to use (approximate) eigenvectors, but also more general subspaces are applicable. In this paper we discuss two results concerning the accuracy requirements within the deflated CG method. First we show that the effective condition number which bounds the convergence rate of the deflated conjugate gradient method depends asymptotically linearly on the size of the perturbations in the deflation subspace. Second, we discuss the accuracy required in calculating the deflating projection. This is crucial concerning the overall convergence of the method, and also allows to save some computational work. To show these results, we use the fact that as a projection approach deflation has many similarities to multigrid methods. In particular, recent results relate the spectra of the deflated matrix to the spectra of the error propagator of twogrid methods. In the spirit of these results we show that the effective condition number can be bounded by the constant of a weak approximation property.