Algebraic Multigrid (AMG)

This package lets you solve sparse linear systems using Algebraic Multigrid (AMG). This works especially well for symmetric positive definite matrices.

Usage

using AMG

A = poisson(1000) # Creates a sample symmetric positive definite sparse matrix
ml = ruge_stuben(A) # Construct a Ruge-Stuben solver
# Multilevel Solver
# -----------------
# Operator Complexity: 1.9859906604402935
# Grid Complexity: 1.99
# No. of Levels: 8
# Coarse Solver: AMG.Pinv()
# Level     Unknowns     NonZeros
# -----     --------     --------
#     1         1000         2998 [50.35%]
#     2          500         1498 [25.16%]
#     3          250          748 [12.56%]
#     4          125          373 [ 6.26%]
#     5           62          184 [ 3.09%]
#     6           31           91 [ 1.53%]
#     7           15           43 [ 0.72%]
#     8            7           19 [ 0.32%]


solve(ml, A * ones(1000)) # should return ones(1000)

As a Preconditioner

You can use AMG as a preconditioner for Krylov methods such as Conjugate Gradients.

import IterativeSolvers: cg
p = aspreconditioner(ml)
c = cg(A, A*ones(1000), Pl = p) 

Features and Roadmap

This package currently supports:

AMG Styles:

• Ruge-Stuben Solver
• Smoothed Aggregation (SA)

Strength of Connection:

• Classical Strength of Connection
• Symmetric Strength of Connection

Smoothers:

• Gauss Seidel (Symmetric, Forward, Backward)

Cycling:

• V cycle

In the future, this package will support:

1. Other splitting methods (like CLJP)
2. SOR, Jacobi smoothers
3. W, F, AMLI cycles

Acknowledgements

This package has been heavily inspired by the PyAMG project.