The easiest way to solve a problem with AMGCL is to use the
amgcl::make_solver
class. It has two
template parameters: the first one specifies a preconditioner to use, and the second chooses an iterative solver. The class constructor takes the system matrix in one of supported
formats and parameters for the chosen algorithms and for the
backend.
Let us consider a simple example of Poisson’s equation in a unit square. Here is how the problem may be solved with AMGCL. We will use BiCGStab solver preconditioned with smoothed aggregation multigrid with SPAI(0) for relaxation (smoothing). First, we include the necessary headers. Each of those brings in the corresponding component of the method:
#include <amgcl/make_solver.hpp>
#include <amgcl/solver/bicgstab.hpp>
#include <amgcl/amg.hpp>
#include <amgcl/coarsening/smoothed_aggregation.hpp>
#include <amgcl/relaxation/spai0.hpp>
#include <amgcl/adapter/crs_tuple.hpp>
Next, we assemble sparse matrix for the Poisson’s equation on a uniform
1000x1000 grid. See Assembling matrix for Poisson’s equation for the source code of the
poisson()
function:
std::vector<int> ptr, col;
std::vector<double> val, rhs;
int n = poisson(1000, ptr, col, val, rhs);
For this example, we select the builtin
backend with double precision numbers as value type:
typedef amgcl::backend::builtin<double> Backend;
Now we can construct the solver for our system matrix. We use the convenient adapter for boost tuples here and just tie together the matrix size and its CRS components:
typedef amgcl::make_solver<
// Use AMG as preconditioner:
amgcl::amg<
Backend,
amgcl::coarsening::smoothed_aggregation,
amgcl::relaxation::spai0
>,
// And BiCGStab as iterative solver:
amgcl::solver::bicgstab<Backend>
> Solver;
Solver solve( boost::tie(n, ptr, col, val) );
Once the solver is constructed, we can apply it to the right-hand side to obtain the solution. This may be repeated multiple times for different right-hand sides. Here we start with a zero initial approximation. The solver returns a boost tuple with number of iterations and norm of the achieved residual:
std::vector<double> x(n, 0.0);
int iters;
double error;
boost::tie(iters, error) = solve(rhs, x);
That’s it! Vector x
contains the solution of our problem now.
We used STL vectors to store the matrix components in the above axample. This may seem too restrictive if you want to use AMGCL with your own types. But the crs_tuple adapter will take anything that the Boost.Range library recognizes as a random access range. For example, you can wrap raw pointers to your data into a boost::iterator_range:
Solver solve( boost::make_tuple(
n,
boost::make_iterator_range(ptr.data(), ptr.data() + ptr.size()),
boost::make_iterator_range(col.data(), col.data() + col.size()),
boost::make_iterator_range(val.data(), val.data() + val.size())
) );
Same applies to the right-hand side and the solution vectors. And if that is still not general enough, you can provide your own adapter for your matrix type. See Matrix adapters for further information on this.
Any component in AMGCL defines its own parameters by declaring a param
subtype. When a class wraps several subclasses, it includes parameters of its
children into its own param
. For example, parameters for the
amgcl::make_solver<Precond, Solver>
are declared as
struct params {
typename Precond::params precond;
typename Solver::params solver;
};
Knowing that, we can easily set the parameters for individual components. For example, we can set the desired tolerance for the iterative solver in the above example like this:
Solver::params prm;
prm.solver.tol = 1e-3;
Solver solve( boost::tie(n, ptr, col, val), prm );
Parameters may also be initialized with a boost::property_tree::ptree. This is especially convenient when Runtime interface is used, and the exact structure of the parameters is not known at compile time:
boost::property_tree::ptree prm;
prm.put("solver.tol", 1e-3);
Solver solve( boost::tie(n, ptr, col, val), prm );
make_solver
class¶amgcl::
make_solver
¶Convenience class that bundles together a preconditioner and an iterative solver.
Public Functions
make_solver
(const Matrix &A, const params &prm = params (), const backend_params &bprm = backend_params())¶Sets up the preconditioner and creates the iterative solver.
operator()
(Matrix const &A, Vec1 const &rhs, Vec2 &&x) const¶Computes the solution for the given system matrix A
and the right-hand side rhs
. Returns the number of iterations made and the achieved residual as a boost::tuple
. The solution vector x
provides initial approximation in input and holds the computed solution on output.
The system matrix may differ from the matrix used during initialization. This may be used for the solution of non-stationary problems with slowly changing coefficients. There is a strong chance that a preconditioner built for a time step will act as a reasonably good preconditioner for several subsequent time steps [DeSh12].
operator()
(Vec1 const &rhs, Vec2 &&x) const¶Computes the solution for the given right-hand side rhs
. Returns the number of iterations made and the achieved residual as a boost::tuple
. The solution vector x
provides initial approximation in input and holds the computed solution on output.
apply
(const Vec1 &rhs, Vec2 &&x) const¶Acts as a preconditioner. That is, applies the solver to the right-hand side rhs
to get the solution x
with zero initial approximation. Iterative methods usually use estimated residual for exit condition. For some problems the value of the estimated residual can get too far from the true residual due to round-off errors. Nesting iterative solvers in this way may allow to shave the last bits off the error. The method should not be used directly but rather allows nesting make_solver
classes as in the following example:
typedef amgcl::make_solver<
amgcl::make_solver<
amgcl::amg<
Backend, amgcl::coarsening::smoothed_aggregation, amgcl::relaxation::spai0
>,
amgcl::solver::cg<Backend>
>,
amgcl::solver::cg<Backend>
> NestedSolver;
precond
() const¶Returns reference to the constructed preconditioner.
solver
() const¶Returns reference to the constructed iterative solver.
system_matrix
() const¶Returns the system matrix in the backend format.
get_params
(boost::property_tree::ptree &p) const¶Stores the parameters used during construction into the property tree p
.
size
() const¶Returns the size of the system matrix.
params
¶Combined parameters of the bundled preconditioner and the iterative solver.