A Robust Abstract Multigrid Solver Goes Algebraic
Craig C. Douglas (Univ. of Kentucky)
(joint work with Ryan McKenzie and Adam Zornes, Univ. Kentucky, Lexington)
Abstract multigrid is a term coined by the author some years ago to describe
multigrid methods that are defined strictly in terms of matrix operations.
For example, on each level Ax = b is the system of linear equations that
must be solved. Level transfers are described in terms of matrixvector
operations, e.g., fine to coarse is just c = Rf and coarse to fine is just
f = Pc. While the matrices may or may not have a matrix multiply relation
in abstract multigrid (though usually they do), in algebraic multigrid there
is a relation, namely the Galerkin relation B = RAP to define a coarse
level's
matrix from a finer level's matrix.
Abstract multigrid leads to quite general solvers for problems in which
there
is traditional theory leading to convergence results. The Madpack
solvers are
really general sparse matrix solvers in an abstract multigrid setting.
Madpack has recently been extended to algebraic multigrid. Memory cache
usage
is also an issue that is being addressed.
