amgcl is a simple and generic algebraic multigrid (AMG) hierarchy builder (and a work in progress). The constructed hierarchy may be used as a standalone solver or as a preconditioner with some iterative solver. Conjugate Gradient and Stabilized BiConjugate Gradient iterative solvers are provided. It is also possible to use generic solvers from other libraries, e.g. ViennaCL.
The setup phase is completely CPU-based. The constructed levels of AMG hierarchy may be stored and used through several backends. This allows for transparent acceleration of the solution phase with help of OpenCL, CUDA, or OpenMP technologies. See examples/vexcl.cpp, examples/viennacl.cpp and examples/eigen.cpp for examples of using amgcl with VexCL, ViennaCL, or Eigen backends.
Doxygen-generated documentation is available at http://ddemidov.github.com/amgcl.
You can use amgcl to solve large sparse system of linear equations in three simple steps: first, you have to select method components (this is a compile time decision); second, the AMG hierarchy has to be constructed from a system matrix; and third, the hierarchy is used to solve the equation system for a given right-hand side.
The list of interpolation schemes and available backends may be found in Interpolation and Level Storage Backends documentation modules. The aggregation and smoothed-aggregation interpolation schemes use less memory and are set up faster than classic interpolation, but their convergence rate is slower. They are well suited for GPU-accelerated backends, where the cost of the setup phase is much more important.
Here is the complete code example showing each step in action:
// First, we need to include relevant headers. Each header basically
// corresponds to an AMG component. Let's say we want to use conjugate gradient
// method preconditioned with smoothed aggregation AMG with VexCL backend:
// This is generic hierarchy builder.
#include <amgcl/amgcl.hpp>
// It will use the following components:
// Interpolation scheme based on smoothed aggregation.
#include <amgcl/interp_smoothed_aggr.hpp>
// Aggregates will be constructed with plain aggregation:
#include <amgcl/aggr_plain.hpp>
// VexCL will be used as a backend:
#include <amgcl/level_vexcl.hpp>
// The definition of conjugate gradient method:
#include <amgcl/cg.hpp>
int main() {
// VexCL context initialization (let's use all GPUs that support double precision):
vex::Context ctx( vex::Filter::Type(CL_DEVICE_TYPE_GPU) && vex::Filter::DoublePrecision );
// Here, the system matrix and right-hand side are somehow constructed. The
// system matrix data is stored in compressed row storage format in vectors
// row, col, and val.
int size;
std::vector<int> row, col;
std::vector<double> val, rhs;
// We wrap the matrix data into amgcl-compatible type.
// No data is copied here:
auto A = amgcl::sparse::map(size, size, row.data(), col.data(), val.data());
// The AMG builder type:
typedef amgcl::solver<
double /* matrix value type */, int /* matrix index type */,
amgcl::interp::smoothed_aggregation<amgcl::aggr::plain>,
amgcl::level::vexcl
> AMG;
// The parameters. Most of the parameters have some reasonable defaults.
// VexCL backend needs to know what context to use:
typename AMG::params prm;
prm.level.ctx = &ctx;
// Here we construct the hierarchy:
AMG amg(A, prm);
// Now let's solve the system of equations. We need to transfer matrix,
// right-hand side, and initial approximation to GPUs. The matrix part may
// be omitted though, since AMG already has it as part of the hierarchy:
std::vector<double> x(size, 0.0);
vex::vector<double> f(ctx.queue(), rhs);
vex::vector<double> u(ctx.queue(), x);
// Call AMG-preconditioned CG method:
auto cnv = amgcl::solve(amg.top_matrix(), f, amg, u, amgcl::cg_tag());
std::cout << "Iterations: " << std::get<0>(cnv) << std::endl
<< "Error: " << std::get<1>(cnv) << std::endl;
// Copy the solution back to host:
vex::copy(u, x);
}The following command line would compile the example:
g++ -o example -std=c++0x -O3 -fopenmp example.cpp -I<path/to/vexcl> -I<path/to/amgcl> -lOpenCL
The C++11 support is enabled here (by -std=c++0x flag) because it is required
by VexCL library. amgcl relies on Boost instead. Also note the use of
-fopenmp switch. It enables an OpenMP-based parallelization of the setup
stage.
Here is output of utest benchmark (see examples folder), solving 2D 2048x2048
Poisson problem generated with ./genproblem 2048.
The first run is CPU-only (--level=2, see ./utest -help for the options
list). The CPU is Intel Core i7 920:
$ ./utest --level 2
Reading "problem.dat"...
Done
Number of levels: 6
Operator complexity: 1.34
Grid complexity: 1.19
level unknowns nonzeros
---------------------------------
0 4194304 20938768 (74.75%)
1 698198 6278320 (22.41%)
2 77749 701425 ( 2.50%)
3 8814 82110 ( 0.29%)
4 988 9362 ( 0.03%)
5 115 1149 ( 0.00%)
Iterations: 25
Error: 6.679105e-09
[Profile: 7.562 sec.] (100.00%)
[ self: 0.011 sec.] ( 0.15%)
[ Read problem: 0.130 sec.] ( 1.72%)
[ setup: 1.020 sec.] ( 13.49%)
[ solve: 6.401 sec.] ( 84.65%)
The second run is VexCL-based, the GPU is NVIDIA Tesla C2075:
$ ./utest --level=1
Reading "problem.dat"...
Done
1. Tesla C2075
Number of levels: 6
Operator complexity: 1.34
Grid complexity: 1.19
level unknowns nonzeros
---------------------------------
0 4194304 20938768 (74.75%)
1 698198 6278320 (22.41%)
2 77749 701425 ( 2.50%)
3 8814 82110 ( 0.29%)
4 988 9362 ( 0.03%)
5 115 1149 ( 0.00%)
Iterations: 25
Error: 6.679105e-09
[Profile: 3.592 sec.] (100.00%)
[ self: 0.437 sec.] ( 12.16%)
[ OpenCL initialization: 0.051 sec.] ( 1.41%)
[ Read problem: 0.130 sec.] ( 3.61%)
[ setup: 2.179 sec.] ( 60.65%)
[ solve: 0.796 sec.] ( 22.16%)
Setup time has increased, because data structures have to be transfered to GPU memory. But due to the accelerated solution the total time is reduced. Further time savings may be expected if the preconditioner is reused for solution with different right-hand sides.
The library is header-only, so there is nothing to compile or link to. You just need to copy amgcl folder somewhere and tell your compiler to scan it for include files.