J. Brannick

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.

J. H. Adler, J. Brannick, C. Liu et al.

This paper develops a first-order system least-squares (FOSLS) formulation for equations of two-phase flow. The main goal is to show that this discretization, along with numerical techniques such as nested iteration, algebraic multigrid, and adaptive local refinement, can be used to solve these types of complex fluid flow problems. In addition, from an energetic variational approach, it can be shown that an important quantity to preserve in a given simulation is the energy law. We discuss the energy law and inherent structure for two-phase flow using the Allen-Cahn interface model and indicate how it is related to other complex fluid models, such as magnetohydrodynamics. Finally, we show that, using the FOSLS framework, one can still satisfy the appropriate energy law globally while using well-known numerical techniques.