three-pt-stencil-solver.cpp

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/*****************************<DECSRIPTION>***********************************
This example solves a 1D Poisson equation:

    u : [0, 1] -> 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.
*****************************<DECSRIPTION>**********************************/

#include <ginkgo/ginkgo.hpp>
#include <iostream>
#include <map>
#include <string>
#include <vector>


// Creates a stencil matrix in CSR format for the given number of discretization
// points.
void generate_stencil_matrix(int discretization_points, int *row_ptrs,
                             int *col_idxs, double *values)
{
    int pos = 0;
    const double coefs[] = {-1, 2, -1};
    row_ptrs[0] = pos;
    for (int 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>
void generate_rhs(int discretization_points, Closure f, double u0, double u1,
                  double *rhs)
{
    const auto h = 1.0 / (discretization_points + 1);
    for (int i = 0; i < discretization_points; ++i) {
        const auto xi = (i + 1) * h;
        rhs[i] = -f(xi) * h * h;
    }
    rhs[0] += u0;
    rhs[discretization_points - 1] += u1;
}


// Prints the solution `u`.
void print_solution(int discretization_points, double u0, double u1,
                    const double *u)
{
    std::cout << u0 << '\n';
    for (int 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>
double calculate_error(int discretization_points, const double *u,
                       Closure correct_u)
{
    const auto h = 1.0 / (discretization_points + 1);
    auto error = 0.0;
    for (int i = 0; i < discretization_points; ++i) {
        using std::abs;
        const auto xi = (i + 1) * h;
        error += abs(u[i] - correct_u(xi)) / abs(correct_u(xi));
    }
    return error;
}


void solve_system(const std::string &executor_string,
                  unsigned int discretization_points, int *row_ptrs,
                  int *col_idxs, double *values, double *rhs, double *u,
                  double accuracy)
{
    // Some shortcuts
    using vec = gko::matrix::Dense<double>;
    using mtx = gko::matrix::Csr<double, int>;
    using cg = gko::solver::Cg<double>;
    using bj = gko::preconditioner::Jacobi<double, int>;
    using val_array = gko::Array<double>;
    using idx_array = gko::Array<int>;
    const auto &dp = discretization_points;

    // Figure out where to run the code
    const auto omp = gko::OmpExecutor::create();
    std::map<std::string, std::shared_ptr<gko::Executor>> exec_map{
        {"omp", omp},
        {"cuda", gko::CudaExecutor::create(0, omp)},
        {"reference", 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_map["omp"];

    // 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(dp).on(exec),
                gko::stop::ResidualNormReduction<>::build()
                    .with_reduction_factor(accuracy)
                    .on(exec))
            .with_preconditioner(bj::build().on(exec))
            .on(exec);
    auto solver = solver_gen->generate(gko::give(matrix));

    // Solve system
    solver->apply(gko::lend(b), gko::lend(x));
}


int main(int argc, char *argv[])
{
    if (argc < 2) {
        std::cerr << "Usage: " << argv[0] << " DISCRETIZATION_POINTS [executor]"
                  << std::endl;
        std::exit(-1);
    }

    const int discretization_points = argc >= 2 ? std::atoi(argv[1]) : 100;
    const auto executor_string = argc >= 3 ? argv[2] : "reference";

    // problem:
    auto correct_u = [](double x) { return x * x * x; };
    auto f = [](double x) { return 6 * x; };
    auto u0 = correct_u(0);
    auto u1 = correct_u(1);

    // matrix
    std::vector<int> row_ptrs(discretization_points + 1);
    std::vector<int> col_idxs(3 * discretization_points - 2);
    std::vector<double> values(3 * discretization_points - 2);
    // right hand side
    std::vector<double> rhs(discretization_points);
    // solution
    std::vector<double> u(discretization_points, 0.0);

    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(), 1e-12);

    print_solution(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;
}