Three Pt Stencil Solver#
The 3-point stencil example.
Kind: basic
Builds on: simple-solver, poisson-solver
Upstream source: examples/three-pt-stencil-solver/three-pt-stencil-solver.cpp in the Ginkgo repository.
Introduction#
This example solves a 1D Poisson equation:
\( u : [0, 1] \rightarrow R\\ u'' = f\\ u(0) = u0\\ u(1) = u1 \)
using a finite difference method on an equidistant grid with K discretization
points (K can be controlled with a command line parameter). The discretization
is done via the second order Taylor polynomial:
$ u(x + h) = u(x) - u’(x)h + 1/2 u’’(x)h^2 + O(h^3)\ u(x - h) = u(x) + u’(x)h + 1/2 u’’(x)h^2 + O(h^3) / +\ ———————- \ -u(x - h) + 2u(x) + -u(x + h) = -f(x)h^2 + O(h^3)
$
For an equidistant grid with K “inner” discretization points \(x1, ..., xk, \)and step size\( h = 1 / (K + 1)\), the formula produces a system of linear equations
$
2u_1 - u_2 = -f_1 h^2 + u0\\
-u_(k-1) + 2u_k - u_(k+1) = -f_k h^2, k = 2, …, K - 1\ -u_(K-1) + 2u_K = -f_K h^2 + u1\
$
which is then solved using Ginkgo’s implementation of the CG method
preconditioned with block-Jacobi. It is also possible to specify on which
executor Ginkgo will solve the system via the command line.
The function \(`f` \)is set to \(`f(x) = 6x`\) (making the solution \(`u(x) = x^3`\)), but
that can be changed in the main function.
The intention of the example is to show how Ginkgo can be integrated into
existing software - the generate_stencil_matrix, generate_rhs,
print_solution, compute_error and main function do not reference Ginkgo at
all (i.e. they could have been there before the application developer decided to
use Ginkgo, and the only part where Ginkgo is introduced is inside the
solve_system function.
The commented program#
#include <iostream>
#include <map>
#include <string>
#include <vector>
#include <ginkgo/ginkgo.hpp>
Creates a stencil matrix in CSR format for the given number of discretization points.
template <typename ValueType, typename IndexType>
void generate_stencil_matrix(IndexType discretization_points,
IndexType* row_ptrs, IndexType* col_idxs,
ValueType* values)
{
IndexType pos = 0;
const ValueType coefs[] = {-1, 2, -1};
row_ptrs[0] = pos;
for (IndexType i = 0; i < discretization_points; ++i) {
for (auto ofs : {-1, 0, 1}) {
if (0 <= i + ofs && i + ofs < discretization_points) {
values[pos] = coefs[ofs + 1];
col_idxs[pos] = i + ofs;
++pos;
}
}
row_ptrs[i + 1] = pos;
}
}
Generates the RHS vector given f and the boundary conditions.
template <typename Closure, typename ValueType, typename IndexType>
void generate_rhs(IndexType discretization_points, Closure f, ValueType u0,
ValueType u1, ValueType* rhs)
{
const ValueType h = 1.0 / (discretization_points + 1);
for (IndexType i = 0; i < discretization_points; ++i) {
const ValueType xi = ValueType(i + 1) * h;
rhs[i] = -f(xi) * h * h;
}
rhs[0] += u0;
rhs[discretization_points - 1] += u1;
}
Prints the solution u.
template <typename ValueType, typename IndexType>
void print_solution(IndexType discretization_points, ValueType u0, ValueType u1,
const ValueType* u)
{
std::cout << u0 << '\n';
for (IndexType i = 0; i < discretization_points; ++i) {
std::cout << u[i] << '\n';
}
std::cout << u1 << std::endl;
}
Computes the 1-norm of the error given the computed u and the correct
solution function correct_u.
template <typename Closure, typename ValueType, typename IndexType>
gko::remove_complex<ValueType> calculate_error(IndexType discretization_points,
const ValueType* u,
Closure correct_u)
{
const ValueType h = 1.0 / (discretization_points + 1);
gko::remove_complex<ValueType> error = 0.0;
for (IndexType i = 0; i < discretization_points; ++i) {
using std::abs;
const ValueType xi = ValueType(i + 1) * h;
error += abs(u[i] - correct_u(xi)) / abs(correct_u(xi));
}
return error;
}
template <typename ValueType, typename IndexType>
void solve_system(const std::string& executor_string,
IndexType discretization_points, IndexType* row_ptrs,
IndexType* col_idxs, ValueType* values, ValueType* rhs,
ValueType* u, gko::remove_complex<ValueType> reduction_factor)
{
Some shortcuts
using vec = gko::matrix::Dense<ValueType>;
using mtx = gko::matrix::Csr<ValueType, IndexType>;
using cg = gko::solver::Cg<ValueType>;
using bj = gko::preconditioner::Jacobi<ValueType, IndexType>;
using val_array = gko::array<ValueType>;
using idx_array = gko::array<IndexType>;
const auto& dp = discretization_points;
Figure out where to run the code
std::map<std::string, std::function<std::shared_ptr<gko::Executor>()>>
exec_map{
{"omp", [] { return gko::OmpExecutor::create(); }},
{"cuda",
[] {
return gko::CudaExecutor::create(0,
gko::OmpExecutor::create());
}},
{"hip",
[] {
return gko::HipExecutor::create(0, gko::OmpExecutor::create());
}},
{"dpcpp",
[] {
return gko::DpcppExecutor::create(0,
gko::OmpExecutor::create());
}},
{"reference", [] { return gko::ReferenceExecutor::create(); }}};
executor where Ginkgo will perform the computation
const auto exec = exec_map.at(executor_string)(); // throws if not valid
executor where the application initialized the data
const auto app_exec = exec->get_master();
Tell Ginkgo to use the data in our application
Matrix: we have to set the executor of the matrix to the one where we
want SpMVs to run (in this case exec). When creating array views, we
have to specify the executor where the data is (in this case app_exec).
If the two do not match, Ginkgo will automatically create a copy of the
data on exec (however, it will not copy the data back once it is done
here this is not important since we are not modifying the matrix).
auto matrix = mtx::create(exec, gko::dim<2>(dp),
val_array::view(app_exec, 3 * dp - 2, values),
idx_array::view(app_exec, 3 * dp - 2, col_idxs),
idx_array::view(app_exec, dp + 1, row_ptrs));
RHS: similar to matrix
auto b = vec::create(exec, gko::dim<2>(dp, 1),
val_array::view(app_exec, dp, rhs), 1);
Solution: we have to be careful here - if the executors are different,
once we compute the solution the array will not be automatically copied
back to the original memory locations. Fortunately, whenever apply is
called on a linear operator (e.g. matrix, solver) the arguments
automatically get copied to the executor where the operator is, and
copied back once the operation is completed. Thus, in this case, we can
just define the solution on app_exec, and it will be automatically
transferred to/from exec if needed.
auto x = vec::create(app_exec, gko::dim<2>(dp, 1),
val_array::view(app_exec, dp, u), 1);
Generate solver
auto solver_gen =
cg::build()
.with_criteria(gko::stop::Iteration::build().with_max_iters(
gko::size_type(dp)),
gko::stop::ResidualNorm<ValueType>::build()
.with_reduction_factor(reduction_factor))
.with_preconditioner(bj::build())
.on(exec);
auto solver = solver_gen->generate(gko::give(matrix));
Solve system
solver->apply(b, x);
}
int main(int argc, char* argv[])
{
using ValueType = double;
using IndexType = int;
Print version information
std::cout << gko::version_info::get() << std::endl;
if (argc == 2 && std::string(argv[1]) == "--help") {
std::cerr << "Usage: " << argv[0]
<< " [executor] [DISCRETIZATION_POINTS]" << std::endl;
std::exit(-1);
}
const auto executor_string = argc >= 2 ? argv[1] : "reference";
const IndexType discretization_points =
argc >= 3 ? std::atoi(argv[2]) : 100;
problem:
auto correct_u = [](ValueType x) { return x * x * x; };
auto f = [](ValueType x) { return ValueType(6) * x; };
auto u0 = correct_u(0);
auto u1 = correct_u(1);
matrix
std::vector<IndexType> row_ptrs(discretization_points + 1);
std::vector<IndexType> col_idxs(3 * discretization_points - 2);
std::vector<ValueType> values(3 * discretization_points - 2);
right hand side
std::vector<ValueType> rhs(discretization_points);
solution
std::vector<ValueType> u(discretization_points, 0.0);
const gko::remove_complex<ValueType> reduction_factor = 1e-7;
generate_stencil_matrix(discretization_points, row_ptrs.data(),
col_idxs.data(), values.data());
looking for solution u = x^3: f = 6x, u(0) = 0, u(1) = 1
generate_rhs(discretization_points, f, u0, u1, rhs.data());
solve_system(executor_string, discretization_points, row_ptrs.data(),
col_idxs.data(), values.data(), rhs.data(), u.data(),
reduction_factor);
Uncomment to print the solution print_solution<ValueType, IndexType>(discretization_points, 0, 1, u.data());
std::cout << "The average relative error is "
<< calculate_error(discretization_points, u.data(), correct_u) /
discretization_points
<< std::endl;
}
Results#
This is the expected output:
The average relative error is 2.52236e-11