https://github.com/julialinearalgebra/algebraicmultigrid.jl
Algebraic Multigrid in Julia
https://github.com/julialinearalgebra/algebraicmultigrid.jl
iterative-methods julia-language sparse-linear-systems
Last synced: about 1 month ago
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Algebraic Multigrid in Julia
- Host: GitHub
- URL: https://github.com/julialinearalgebra/algebraicmultigrid.jl
- Owner: JuliaLinearAlgebra
- License: other
- Created: 2017-09-15T04:30:18.000Z (almost 8 years ago)
- Default Branch: master
- Last Pushed: 2025-03-23T14:31:19.000Z (3 months ago)
- Last Synced: 2025-05-16T08:23:48.504Z (about 1 month ago)
- Topics: iterative-methods, julia-language, sparse-linear-systems
- Language: Julia
- Homepage:
- Size: 403 KB
- Stars: 124
- Watchers: 5
- Forks: 24
- Open Issues: 13
-
Metadata Files:
- Readme: README.md
- License: LICENSE.md
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README
# Algebraic Multigrid (AMG)
| **Build Status**|
|:----------------------------------------------------------------:|
| [](https://github.com/JuliaLinearAlgebra/AlgebraicMultigrid.jl/actions?query=workflow%3ACI) |
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`.
```julia
A = poisson(100);
b = rand(100);
solve(A, b, RugeStubenAMG(), maxiter = 1, abstol = 1e-6)
```
### Multigrid cycling
```julia
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.
```julia
import IterativeSolvers: cg
p = aspreconditioner(ml)
c = cg(A, A*ones(1000), Pl = p)
```
### As a preconditioner with LinearSolve.jl
`RugeStubenPreconBuilder` and `SmoothedAggregationPreconBuilder` work with the
[`precs` API](https://docs.sciml.ai/LinearSolve/stable/basics/Preconditioners/#Specifying-Preconditioners)
of LinearSolve.jl
```julia
A = poisson( (100,100) )
u0= rand(size(A,1))
b=A*u0
prob = LinearProblem(A, b)
strategy = KrylovJL_CG(precs = RugeStubenPreconBuilder())
sol = solve(prob, strategy, atol=1.0e-14)
strategy = KrylovJL_CG(precs = SmoothedAggregationPreconBuilder())
sol = solve(prob, strategy, atol=1.0e-14)
```
## 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`](http://github.com/pyamg/pyamg) project.