Geoffrey Sanders

Smoothed Aggregation Multigrid for Markov Chains

A smoothed aggregation multigrid method is presented for the numerical calculation of the stationary probability vector of an irreducible sparse Markov chain. It is shown how smoothing the interpolation and restriction operators can dramatically increase the efficiency of aggregation multigrid methods for Markov chains that have been proposed in the literature. The proposed smoothing approach is inspired by smoothed aggregation multigrid for linear systems, supplemented with a new lumping technique that assures well-posedness of the coarse-level problems: the coarse-level operators are singular M-matrices on all levels, resulting in strictly positive coarse-level corrections on all levels. Numerical results show how these methods lead to nearly optimal multigrid efficiency for an extensive set of test problems, both when geometric and algebraic aggregation strategies are used.

Towards Adaptive Smoothed Aggregation (

Applying smoothed aggregation (SA) multigrid to solve a nonsymmetric linear system,

\alpha

A\mathbf{x} =\mathbf{b}

SA) for Nonsymmetric Problems

, is often impeded by the lack of a minimization principle that can be used as a basis for the coarse-grid correction process. This paper proposes a Petrov-Galerkin (PG) approach based on applying SA to either of two symmetric positive definite (SPD) matrices,

Recursively Accelerated Multilevel Aggregation for Markov Chains

\sqrt{A^{t}A}

Algebraic Multigrid for Markov Chains

or

Top-level acceleration of adaptive algebraic multilevel methods for steady-state solution to Markov chains

\sqrt{AA^{t}}

Nested Iteration and First-Order System Least Squares for Incompressible, Resistive Magnetohydrodynamics

. These matrices, however, are typically full and difficult to compute, so it is not computationally efficient to use them directly to form a coarse-grid correction. The proposed approach approximates these coarse-grid corrections by using SA to accurately approximate the right and left singular vectors of

A generalized eigensolver based on smoothed aggregation (GES‐SA) for initializing smoothed aggregation (SA) multigrid

A

Multilevel space-time aggregation for bright field cell microscopy segmentation and tracking

that correspond to the lowest singular value. These left and right singular vectors are used to construct the restriction and interpolation operators, respectively. A preliminary two-level convergence theory is presented, suggesting that more relaxation should be applied than for an SPD problem. Additionally, a nonsymmetric version of adaptive SA (