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Algebraic Multigrid (AMG)

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

Usage

Using the CommonSolve interface

This is highest level API. It internally creates the multilevel object and calls the multigrid cycling _solve.

A = poisson(100); 
b = rand(100);
solve(A, b, RugeStubenAMG(), maxiter = 1, abstol = 1e-6)

Multigrid cycling

using AlgebraicMultigrid

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%]


AlgebraicMultigrid._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)
  • Damped Jacobi

Cycling:

  • V, W and F cycles

In the future, this package will support:

  1. Other splitting methods (like CLJP)
  2. SOR smoother
  3. AMLI cycles

Acknowledgements

This package has been heavily inspired by the PyAMG project.