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 | PyAMG is developed by Nathan Bell, Luke Olson, and Jacob Schroder, in the Deparment of Computer Science at the University of Illinois at Urbana-Champaign. Portions of the project were partially supported by the NSF under award DMS-0612448. |
Introduction
PyAMG is a library of Algebraic Multigrid (AMG) solvers with a convenient Python interface.
What is AMG?
AMG is a multilevel technique for solving large-scale linear systems with optimal or near-optimal efficiency. Unlike geometric multigrid, AMG requires little or no geometric information about the underlying problem and develops a sequence of coarser grids directly from the input matrix. This feature is especially important for problems discretized on unstructured meshes and irregular grids.
Features
PyAMG features implementations of:
- Ruge-Stuben (RS) or Classical AMG
- AMG based on Smoothed Aggregation (SA)
- Adaptive Smoothed Aggregation (αSA)
- Compatible Relaxation (CR)
The predominant portion of PyAMG is written in Python with a smaller amount of supporting C++ code for performance critical operations.
Objectives
- ease of use
- interface is accessible to non-experts
- extensive documentation and references
- speed
- solves problems with millions of unknowns efficiently
- core multigrid algorithms are implemented in C++ and translated through SWIG
- sparse matrix support provided by scipy.sparse
- readability
- source code is organized into intuitive components
- extensibility
- core components can be reused to implement additional techniques
- new features are easy integrated
- experimentation
- facilitates rapid prototyping and analysis of multigrid methods
- portability
- tested on several platforms
- relies only on Python, NumPy, and SciPy
Example Usage
PyAMG is easy to use! The following code constructs a two-dimensional Poisson problem and solves the resulting linear system with Classical AMG.
from scipy import * from scipy.linalg import * from pyamg import * from pyamg.gallery import * A = poisson((500,500), format='csr') # 2D Poisson problem on 500x500 grid ml = ruge_stuben_solver(A) # construct the multigrid hierarchy print ml # print hierarchy information b = rand(A.shape[0]) # pick a random right hand side x = ml.solve(b, tol=1e-10) # solve Ax=b to a tolerance of 1e-8 print "residual norm is", norm(b - A*x) # compute norm of residual vector
Program output:
multilevel_solver
Number of Levels: 6
Operator Complexity: 2.198
Grid Complexity: 1.666
Coarse Solver: 'pinv2'
level unknowns nonzeros
0 250000 1248000 [45.50%]
1 125000 1121002 [40.87%]
2 31252 280662 [10.23%]
3 7825 70657 [ 2.58%]
4 1937 17973 [ 0.66%]
5 484 4728 [ 0.17%]
residual norm is 1.86112114946e-06Refer to Tutorial or Examples for more applications of PyAMG. Complete code documentation can be found here.