Algebraic Multigrid Methods for Constrained Systems of Linear Equations

Mark Adams

Sandia National Laboratories
PO Box 969.
Livermore CA 94551-9217


Constrained systems of linear equations (KKT systems) arise in many applications from contact in solid mechanics and incompressible flow, to problems in mathematical optimization for PDE systems.   Several approaches have been developed for solving these systems, given a solver for the primal part of the system: 1) constraint reduction, 2) projection methods, and 3) Uzawa/augmented Lagrange.   All of these methods have advantages and disadvantages.   Uzawa methods are robust and scalable though they require several approximate solves, as they iterate on the Shur complement.   Constraint reduction is effective for certain types of constraints, but is limited in its applicability.   Projection methods are attractive as they handle general constraints and are simple to use with an existing solver or preconditioner, but they are generally not scalable nor are they robust when used with powerful preconditioners, such as multigrid.

Algebraic multigrid is a popular method for solving the equations that arise from discretized PDEs, the primal part of KKT systems, but the application of multigrid to the entire KKT system has not been investigated to a great degree.   V.H. Schulz has recently applied structured geometric multigrid to optimization problems, S.P. Vanka has developed structured geometric multigrid techniques for incompressible flow problems, and a commercial product (TaskFlow) has developed an algebraic (aggregation) multigrid method for unstructured incompressible flow problems.   Multigrid methods are attractive for constrained systems as they have the potential to be both highly optimal like constraint reduction and general like projection methods by maintaining tight coupling of the constraints in the preconditioning process while not requiring that the primal part of the system be augmented by regularization as in Uzawa methods or a Shur complement as in constraint reduction.

This talk discusses the application of algebraic multigrid techniques to discretized PDEs with constraints.   We describe a framework for developing algebraic multigrid methods for KKT systems and discuss coarsening techniques for the constraint equations.   We cover several approaches to constructing multigrid smoothers and present examples of application of some of these ideas to contact problems in solid mechanics.