The poisson solver example.

This example depends on simple-solver.

Table of contents
Introduction About the example The commented program	Results Comments about programming and debugging The plain program

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 used to build an application solving a real-world problem, which includes a solution of a large, sparse linear system as a component.

About the example

The commented program

#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(gko::matrix::Csr<> *matrix)
{
    const auto discretization_points = matrix->get_size()[0];
    auto row_ptrs = matrix->get_row_ptrs();
    auto col_idxs = matrix->get_col_idxs();
    auto values = matrix->get_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(Closure f, double u0, double u1, gko::matrix::Dense<> *rhs)
{
    const auto discretization_points = rhs->get_size()[0];
    auto values = rhs->get_values();
    const auto h = 1.0 / (discretization_points + 1);
    for (int i = 0; i < discretization_points; ++i) {
        const auto xi = (i + 1) * h;
        values[i] = -f(xi) * h * h;
    }
    values[0] += u0;
    values[discretization_points - 1] += u1;
}

Prints the solution u.

void print_solution(double u0, double u1, const gko::matrix::Dense<> *u)
{
    std::cout << u0 << '\n';
    for (int i = 0; i < u->get_size()[0]; ++i) {
        std::cout << u->get_const_values()[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 gko::matrix::Dense<> *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->get_const_values()[i] - correct_u(xi)) / abs(correct_u(xi));
    }
    return error;
}
int main(int argc, char *argv[])
{

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

Get number of discretization points

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

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 used by the application

const auto app_exec = exec_map["omp"];

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);

initialize matrix and vectors

auto matrix = mtx::create(app_exec, gko::dim<2>(discretization_points),
                          3 * discretization_points - 2);
generate_stencil_matrix(lend(matrix));
auto rhs = vec::create(app_exec, gko::dim<2>(discretization_points, 1));
generate_rhs(f, u0, u1, lend(rhs));
auto u = vec::create(app_exec, gko::dim<2>(discretization_points, 1));
for (int i = 0; i < u->get_size()[0]; ++i) {
    u->get_values()[i] = 0.0;
}

Generate solver and solve the system

    cg::build()
        .with_criteria(gko::stop::Iteration::build()
                           .with_max_iters(discretization_points)
                           .on(exec),
                       gko::stop::ResidualNormReduction<>::build()
                           .with_reduction_factor(1e-6)
                           .on(exec))
        .with_preconditioner(bj::build().on(exec))
        .on(exec)
        ->generate(clone(exec, matrix))  // copy the matrix to the executor
        ->apply(lend(rhs), lend(u));
    print_solution(u0, u1, lend(u));
    std::cout << "The average relative error is "
              << calculate_error(discretization_points, lend(u), correct_u) /
                     discretization_points
              << std::endl;
}

Results

This is the expected output:

0
00010798
000863838
00291545
0069107
0134975
0233236
037037
0552856
0787172
10798
143721
186589
237231
296296
364431
442285
530504
629738
740633
863838
1
The average relative error is 1.87318e-15

Comments about programming and debugging

The plain program

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#include <ginkgo/ginkgo.hpp>
#include <iostream>
#include <map>
#include <string>
#include <vector>
void generate_stencil_matrix(gko::matrix::Csr<> *matrix)
{
    const auto discretization_points = matrix->get_size()[0];
    auto row_ptrs = matrix->get_row_ptrs();
    auto col_idxs = matrix->get_col_idxs();
    auto values = matrix->get_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;
    }
}
template <typename Closure>
void generate_rhs(Closure f, double u0, double u1, gko::matrix::Dense<> *rhs)
{
    const auto discretization_points = rhs->get_size()[0];
    auto values = rhs->get_values();
    const auto h = 1.0 / (discretization_points + 1);
    for (int i = 0; i < discretization_points; ++i) {
        const auto xi = (i + 1) * h;
        values[i] = -f(xi) * h * h;
    }
    values[0] += u0;
    values[discretization_points - 1] += u1;
}
void print_solution(double u0, double u1, const gko::matrix::Dense<> *u)
{
    std::cout << u0 << '\n';
    for (int i = 0; i < u->get_size()[0]; ++i) {
        std::cout << u->get_const_values()[i] << '\n';
    }
    std::cout << u1 << std::endl;
}
template <typename Closure>
double calculate_error(int discretization_points, const gko::matrix::Dense<> *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->get_const_values()[i] - correct_u(xi)) / abs(correct_u(xi));
    }
    return error;
}
int main(int argc, char *argv[])
{
    using vec = gko::matrix::Dense<double>;
    using mtx = gko::matrix::Csr<double, int>;
    using cg = gko::solver::Cg<double>;
    using bj = gko::preconditioner::Jacobi<>;
    if (argc < 2) {
        std::cerr << "Usage: " << argv[0] << " DISCRETIZATION_POINTS [executor]"
                  << std::endl;
        std::exit(-1);
    }
    const unsigned int discretization_points =
        argc >= 2 ? std::atoi(argv[1]) : 100;
    const auto executor_string = argc >= 3 ? argv[2] : "reference";
    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()}};
    const auto exec = exec_map.at(executor_string);  // throws if not valid
    const auto app_exec = exec_map["omp"];
    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);
    auto matrix = mtx::create(app_exec, gko::dim<2>(discretization_points),
                              3 * discretization_points - 2);
    generate_stencil_matrix(lend(matrix));
    auto rhs = vec::create(app_exec, gko::dim<2>(discretization_points, 1));
    generate_rhs(f, u0, u1, lend(rhs));
    auto u = vec::create(app_exec, gko::dim<2>(discretization_points, 1));
    for (int i = 0; i < u->get_size()[0]; ++i) {
        u->get_values()[i] = 0.0;
    }
    cg::build()
        .with_criteria(gko::stop::Iteration::build()
                           .with_max_iters(discretization_points)
                           .on(exec),
                       gko::stop::ResidualNormReduction<>::build()
                           .with_reduction_factor(1e-6)
                           .on(exec))
        .with_preconditioner(bj::build().on(exec))
        .on(exec)
        ->generate(clone(exec, matrix))  // copy the matrix to the executor
        ->apply(lend(rhs), lend(u));
    print_solution(u0, u1, lend(u));
    std::cout << "The average relative error is "
              << calculate_error(discretization_points, lend(u), correct_u) /
                     discretization_points
              << std::endl;
}