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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.
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)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)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)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:
- Other splitting methods (like CLJP)
- SOR smoother
- AMLI cycles
This package has been heavily inspired by the PyAMG project.