The 9-point stencil example..

This example depends on simple-solver, three-pt-stencil-solver, poisson-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 2D Poisson equation:

[ \Omega = (0,1)^2 \ \Omega_b = [0,1]^2 \text{ (with boundary)} \ \partial\Omega = \Omega_b \backslash \Omega \ u : \Omega_b -> R \ u'' = f \in \Omega \ u = u_D \in \partial\Omega \ ]

using a finite difference method on an equidistant grid with K discretization points (K can be controlled with a command line parameter). The discretization may be done by any order Taylor polynomial. For an equidistant grid with K "inner" discretization points ((x1,y1), \ldots, (xk,y1),(x1,y2), \ldots, (xk,yk,z1)) step size (h = 1 / (K + 1)) and a stencil (\in \mathbb{R}^{3 \times 3}), the formula produces a system of linear equations

(\sum_{a,b=-1}^1 stencil(a,b) * u_{(i+a,j+b} = -f_k h^2), on any inner node with a neighborhood of inner nodes

On any node, where neighbor is on the border, the neighbor is replaced with a (-stencil(a,b) * u_{i+a,j+b}) and added to the right hand side vector. For example a node with a neighborhood of only edge nodes may look like this

[ \sum_{a,b=-1}^(1,0) stencil(a,b) * u_{(i+a,j+b} = -f_k h^2 - \sum_{a=-1}^1 stencil(a,1) * u_{(i+a,j+1} ]

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,y) = 6x + 6y) (making the solution (u(x,y) = x^3

y^3)), but that can be changed in the main function. Also the stencil values for the core, the faces, the edge and the corners can be changed when passing additional parameters.

The intention of this is to show how generation of stencil values and the right hand side vector changes when increasing the dimension.

About the example

The commented program

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,y) = 6x + 6y` (making the solution `u(x,y) = x^3
+ y^3`), but that can be changed in the `main` function. Also the stencil values
for the core, the faces, the edge and the corners can be changed when passing
additional parameters.
 
The intention of this is to show how generation of stencil values and the right
hand side vector changes when increasing the dimension.
*****************************<DESCRIPTION>********************************** /
 
#include <array>
#include <chrono>
#include <iostream>
#include <map>
#include <string>
#include <vector>
 
#include <ginkgo/ginkgo.hpp>

Stencil values. Ordering can be seen in the main function Can also be changed by passing additional parameter when executing

constexpr double default_alpha = 10.0 / 3.0;
constexpr double default_beta = -2.0 / 3.0;
constexpr double default_gamma = -1.0 / 6.0;
 
/ * Possible alternative default values are
 * default_alpha = 8.0;
 * default_beta = -1.0;
 * default_gamma = -1.0;
 * /

Creates a stencil matrix in CSR format for the given number of discretization points.

template <typename ValueType, typename IndexType>
void generate_stencil_matrix(IndexType dp, IndexType* row_ptrs,
                             IndexType* col_idxs, ValueType* values,
                             ValueType* coefs)
{
    IndexType pos = 0;
    const size_t dp_2 = dp * dp;
    row_ptrs[0] = pos;
    for (IndexType k = 0; k < dp; ++k) {
        for (IndexType i = 0; i < dp; ++i) {
            const size_t index = i + k * dp;
            for (IndexType j = -1; j <= 1; ++j) {
                for (IndexType l = -1; l <= 1; ++l) {
                    const IndexType offset = l + 1 + 3 * (j + 1);
                    if ((k + j) >= 0 && (k + j) < dp && (i + l) >= 0 &&
                        (i + l) < dp) {
                        values[pos] = coefs[offset];
                        col_idxs[pos] = index + l + dp * j;
                        ++pos;
                    }
                }
            }
            row_ptrs[index + 1] = pos;
        }
    }
}

Generates the RHS vector given f and the boundary conditions.

template <typename Closure, typename ClosureT, typename ValueType,
          typename IndexType>
void generate_rhs(IndexType dp, Closure f, ClosureT u, ValueType* rhs,
                  ValueType* coefs)
{
    const size_t dp_2 = dp * dp;
    const ValueType h = 1.0 / (dp + 1.0);
    for (IndexType i = 0; i < dp; ++i) {
        const auto yi = ValueType(i + 1) * h;
        for (IndexType j = 0; j < dp; ++j) {
            const auto xi = ValueType(j + 1) * h;
            const auto index = i * dp + j;
            rhs[index] = -f(xi, yi) * h * h;
        }
    }

Iterating over the edges to add boundary values and adding the overlapping 3x1 to the rhs

for (size_t i = 0; i < dp; ++i) {
    const auto xi = ValueType(i + 1) * h;
    const auto index_top = i;
    const auto index_bot = i + dp * (dp - 1);
 
    rhs[index_top] -= u(xi - h, 0.0) * coefs[0];
    rhs[index_top] -= u(xi, 0.0) * coefs[1];
    rhs[index_top] -= u(xi + h, 0.0) * coefs[2];
 
    rhs[index_bot] -= u(xi - h, 1.0) * coefs[6];
    rhs[index_bot] -= u(xi, 1.0) * coefs[7];
    rhs[index_bot] -= u(xi + h, 1.0) * coefs[8];
}
for (size_t i = 0; i < dp; ++i) {
    const auto yi = ValueType(i + 1) * h;
    const auto index_left = i * dp;
    const auto index_right = i * dp + (dp - 1);
 
    rhs[index_left] -= u(0.0, yi - h) * coefs[0];
    rhs[index_left] -= u(0.0, yi) * coefs[3];
    rhs[index_left] -= u(0.0, yi + h) * coefs[6];
 
    rhs[index_right] -= u(1.0, yi - h) * coefs[2];
    rhs[index_right] -= u(1.0, yi) * coefs[5];
    rhs[index_right] -= u(1.0, yi + h) * coefs[8];
}

remove the double corner values

    rhs[0] += u(0.0, 0.0) * coefs[0];
    rhs[(dp - 1)] += u(1.0, 0.0) * coefs[2];
    rhs[(dp - 1) * dp] += u(0.0, 1.0) * coefs[6];
    rhs[dp * dp - 1] += u(1.0, 1.0) * coefs[8];
}

Prints the solution u.

template <typename ValueType, typename IndexType>
void print_solution(IndexType dp, const ValueType* u)
{
    for (IndexType i = 0; i < dp; ++i) {
        for (IndexType j = 0; j < dp; ++j) {
            std::cout << u[i * dp + j] << ' ';
        }
        std::cout << '\n';
    }
    std::cout << 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 dp, const ValueType* u,
                                               Closure correct_u)
{
    const ValueType h = 1.0 / (dp + 1);
    gko::remove_complex<ValueType> error = 0.0;
    for (IndexType j = 0; j < dp; ++j) {
        const auto xi = ValueType(j + 1) * h;
        for (IndexType i = 0; i < dp; ++i) {
            using std::abs;
            const auto yi = ValueType(i + 1) * h;
            error +=
                abs(u[i * dp + j] - correct_u(xi, yi)) / abs(correct_u(xi, yi));
        }
    }
    return error;
}
 
 
template <typename ValueType, typename IndexType>
void solve_system(const std::string& executor_string,
                  unsigned int 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;
const gko::size_type dp_2 = dp * dp;

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_2),
    val_array::view(app_exec, (3 * dp - 2) * (3 * dp - 2), values),
    idx_array::view(app_exec, (3 * dp - 2) * (3 * dp - 2), col_idxs),
    idx_array::view(app_exec, dp_2 + 1, row_ptrs));

RHS: similar to matrix

auto b = vec::create(exec, gko::dim<2>(dp_2, 1),

val_array::view(app_exec, dp_2, 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_2, 1),

val_array::view(app_exec, dp_2, u), 1);

Generate solver

auto solver_gen =
    cg::build()
        .with_criteria(gko::stop::Iteration::build().with_max_iters(dp_2),
                       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] [alpha] [beta] [gamma]"
        << 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;
const ValueType alpha_c = argc >= 4 ? std::atof(argv[3]) : default_alpha;
const ValueType beta_c = argc >= 5 ? std::atof(argv[4]) : default_beta;
const ValueType gamma_c = argc >= 6 ? std::atof(argv[5]) : default_gamma;

clang-format off

std::array<ValueType, 9> coefs{
    gamma_c, beta_c, gamma_c,
        beta_c, alpha_c, beta_c,
    gamma_c, beta_c, gamma_c};

clang-format on

const auto dp = discretization_points;

const size_t dp_2 = dp * dp;

problem:

auto correct_u = [](ValueType x, ValueType y) {
    return x * x * x + y * y * y;
};
auto f = [](ValueType x, ValueType y) {
    return ValueType(6) * x + ValueType(6) * y;
};

matrix

std::vector<IndexType> row_ptrs(dp_2 + 1);
std::vector<IndexType> col_idxs((3 * dp - 2) * (3 * dp - 2));
std::vector<ValueType> values((3 * dp - 2) * (3 * dp - 2));

right hand side

std::vector<ValueType> rhs(dp_2);

solution

std::vector<ValueType> u(dp_2, 0.0);
 
generate_stencil_matrix(dp, row_ptrs.data(), col_idxs.data(), values.data(),
                        coefs.data());

looking for solution u = x^3: f = 6x, u(0) = 0, u(1) = 1

generate_rhs(dp, f, correct_u, rhs.data(), coefs.data());
 
const gko::remove_complex<ValueType> reduction_factor = 1e-7;
 
auto start_time = std::chrono::steady_clock::now();
solve_system(executor_string, dp, row_ptrs.data(), col_idxs.data(),
             values.data(), rhs.data(), u.data(), reduction_factor);
auto stop_time = std::chrono::steady_clock::now();
auto runtime_duration =
    static_cast<double>(
        std::chrono::duration_cast<std::chrono::nanoseconds>(stop_time -
                                                             start_time)
            .count()) *
    1e-6;

Uncomment to print the solution print_solution(dp, u.data());

    std::cout << "The average relative error is "
              << calculate_error(dp, u.data(), correct_u) /
                     static_cast<gko::remove_complex<ValueType>>(dp_2)
              << std::endl;
    std::cout << "The runtime is " << std::to_string(runtime_duration) << " ms"
              << std::endl;
}

Results

The expected output should be

The average relative error is 6.35715e-06

The runtime is 167.320520 ms

Comments about programming and debugging

The plain program

 
/*****************************<DESCRIPTION>***********************************
This example solves a 2D Poisson equation:
 
    \Omega = (0,1)^2
    \Omega_b = [0,1]^2   (with boundary)
    \partial\Omega = \Omega_b \backslash \Omega
    u : \Omega_b -> R
    u'' = f in \Omega
    u = u_D on \partial\Omega
 
using a finite difference method on an equidistant grid with `K` discretization
points (`K` can be controlled with a command line parameter). The discretization
may be done by any order Taylor polynomial.
For an equidistant grid with K "inner" discretization points (x1,y1), ...,
(xk,y1),(x1,y2), ..., (xk,yk) step size h = 1 / (K + 1) and a stencil \in
\R^{3 x 3}, the formula produces a system of linear equations
 
\sum_{a,b=-1}^1 stencil(a,b) * u_{(i+a,j+b} = -f_k h^2,  on any inner node with
a neighborhood of inner nodes
 
On any node, where neighbor is on the border, the neighbor is replaced with a
'-stencil(a,b) * u_{i+a,j+b}' and added to the right hand side vector. For
example a node with a neighborhood of only edge nodes may look like this
 
\sum_{a,b=-1}^(1,0) stencil(a,b) * u_{(i+a,j+b} = -f_k h^2 - \sum_{a=-1}^1
stencil(a,1) * u_{(i+a,j+1}
 
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,y) = 6x + 6y` (making the solution `u(x,y) = x^3
+ y^3`), but that can be changed in the `main` function. Also the stencil values
for the core, the faces, the edge and the corners can be changed when passing
additional parameters.
 
The intention of this is to show how generation of stencil values and the right
hand side vector changes when increasing the dimension.
*****************************<DESCRIPTION>**********************************/
 
#include <array>
#include <chrono>
#include <iostream>
#include <map>
#include <string>
#include <vector>
 
#include <ginkgo/ginkgo.hpp>
 
constexpr double default_alpha = 10.0 / 3.0;
constexpr double default_beta = -2.0 / 3.0;
constexpr double default_gamma = -1.0 / 6.0;
 
/* Possible alternative default values are
 * default_alpha = 8.0;
 * default_beta = -1.0;
 * default_gamma = -1.0;
 */
 
template <typename ValueType, typename IndexType>
void generate_stencil_matrix(IndexType dp, IndexType* row_ptrs,
                             IndexType* col_idxs, ValueType* values,
                             ValueType* coefs)
{
    IndexType pos = 0;
    const size_t dp_2 = dp * dp;
    row_ptrs[0] = pos;
    for (IndexType k = 0; k < dp; ++k) {
        for (IndexType i = 0; i < dp; ++i) {
            const size_t index = i + k * dp;
            for (IndexType j = -1; j <= 1; ++j) {
                for (IndexType l = -1; l <= 1; ++l) {
                    const IndexType offset = l + 1 + 3 * (j + 1);
                    if ((k + j) >= 0 && (k + j) < dp && (i + l) >= 0 &&
                        (i + l) < dp) {
                        values[pos] = coefs[offset];
                        col_idxs[pos] = index + l + dp * j;
                        ++pos;
                    }
                }
            }
            row_ptrs[index + 1] = pos;
        }
    }
}
 
 
template <typename Closure, typename ClosureT, typename ValueType,
          typename IndexType>
void generate_rhs(IndexType dp, Closure f, ClosureT u, ValueType* rhs,
                  ValueType* coefs)
{
    const size_t dp_2 = dp * dp;
    const ValueType h = 1.0 / (dp + 1.0);
    for (IndexType i = 0; i < dp; ++i) {
        const auto yi = ValueType(i + 1) * h;
        for (IndexType j = 0; j < dp; ++j) {
            const auto xi = ValueType(j + 1) * h;
            const auto index = i * dp + j;
            rhs[index] = -f(xi, yi) * h * h;
        }
    }
 
    for (size_t i = 0; i < dp; ++i) {
        const auto xi = ValueType(i + 1) * h;
        const auto index_top = i;
        const auto index_bot = i + dp * (dp - 1);
 
        rhs[index_top] -= u(xi - h, 0.0) * coefs[0];
        rhs[index_top] -= u(xi, 0.0) * coefs[1];
        rhs[index_top] -= u(xi + h, 0.0) * coefs[2];
 
        rhs[index_bot] -= u(xi - h, 1.0) * coefs[6];
        rhs[index_bot] -= u(xi, 1.0) * coefs[7];
        rhs[index_bot] -= u(xi + h, 1.0) * coefs[8];
    }
    for (size_t i = 0; i < dp; ++i) {
        const auto yi = ValueType(i + 1) * h;
        const auto index_left = i * dp;
        const auto index_right = i * dp + (dp - 1);
 
        rhs[index_left] -= u(0.0, yi - h) * coefs[0];
        rhs[index_left] -= u(0.0, yi) * coefs[3];
        rhs[index_left] -= u(0.0, yi + h) * coefs[6];
 
        rhs[index_right] -= u(1.0, yi - h) * coefs[2];
        rhs[index_right] -= u(1.0, yi) * coefs[5];
        rhs[index_right] -= u(1.0, yi + h) * coefs[8];
    }
 
    rhs[0] += u(0.0, 0.0) * coefs[0];
    rhs[(dp - 1)] += u(1.0, 0.0) * coefs[2];
    rhs[(dp - 1) * dp] += u(0.0, 1.0) * coefs[6];
    rhs[dp * dp - 1] += u(1.0, 1.0) * coefs[8];
}
 
 
template <typename ValueType, typename IndexType>
void print_solution(IndexType dp, const ValueType* u)
{
    for (IndexType i = 0; i < dp; ++i) {
        for (IndexType j = 0; j < dp; ++j) {
            std::cout << u[i * dp + j] << ' ';
        }
        std::cout << '\n';
    }
    std::cout << std::endl;
}
 
 
template <typename Closure, typename ValueType, typename IndexType>
gko::remove_complex<ValueType> calculate_error(IndexType dp, const ValueType* u,
                                               Closure correct_u)
{
    const ValueType h = 1.0 / (dp + 1);
    gko::remove_complex<ValueType> error = 0.0;
    for (IndexType j = 0; j < dp; ++j) {
        const auto xi = ValueType(j + 1) * h;
        for (IndexType i = 0; i < dp; ++i) {
            using std::abs;
            const auto yi = ValueType(i + 1) * h;
            error +=
                abs(u[i * dp + j] - correct_u(xi, yi)) / abs(correct_u(xi, yi));
        }
    }
    return error;
}
 
 
template <typename ValueType, typename IndexType>
void solve_system(const std::string& executor_string,
                  unsigned int discretization_points, IndexType* row_ptrs,
                  IndexType* col_idxs, ValueType* values, ValueType* rhs,
                  ValueType* u, gko::remove_complex<ValueType> reduction_factor)
{
    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;
    const gko::size_type dp_2 = dp * dp;
 
    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(); }}};
 
    const auto exec = exec_map.at(executor_string)();  // throws if not valid
    const auto app_exec = exec->get_master();
 
 
    auto matrix = mtx::create(
        exec, gko::dim<2>(dp_2),
        val_array::view(app_exec, (3 * dp - 2) * (3 * dp - 2), values),
        idx_array::view(app_exec, (3 * dp - 2) * (3 * dp - 2), col_idxs),
        idx_array::view(app_exec, dp_2 + 1, row_ptrs));
 
    auto b = vec::create(exec, gko::dim<2>(dp_2, 1),
                         val_array::view(app_exec, dp_2, rhs), 1);
 
    auto x = vec::create(app_exec, gko::dim<2>(dp_2, 1),
                         val_array::view(app_exec, dp_2, u), 1);
 
    auto solver_gen =
        cg::build()
            .with_criteria(gko::stop::Iteration::build().with_max_iters(dp_2),
                           gko::stop::ResidualNorm<ValueType>::build()
                               .with_reduction_factor(reduction_factor))
            .with_preconditioner(bj::build())
            .on(exec);
    auto solver = solver_gen->generate(gko::give(matrix));
 
    solver->apply(b, x);
}
 
 
int main(int argc, char* argv[])
{
    using ValueType = double;
    using IndexType = int;
 
    std::cout << gko::version_info::get() << std::endl;
 
    if (argc == 2 && std::string(argv[1]) == "--help") {
        std::cerr
            << "Usage: " << argv[0]
            << " [executor] [DISCRETIZATION_POINTS] [alpha] [beta] [gamma]"
            << 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;
    const ValueType alpha_c = argc >= 4 ? std::atof(argv[3]) : default_alpha;
    const ValueType beta_c = argc >= 5 ? std::atof(argv[4]) : default_beta;
    const ValueType gamma_c = argc >= 6 ? std::atof(argv[5]) : default_gamma;
 
    std::array<ValueType, 9> coefs{
        gamma_c, beta_c, gamma_c,
        beta_c, alpha_c, beta_c,
        gamma_c, beta_c, gamma_c};
 
    const auto dp = discretization_points;
    const size_t dp_2 = dp * dp;
 
    auto correct_u = [](ValueType x, ValueType y) {
        return x * x * x + y * y * y;
    };
    auto f = [](ValueType x, ValueType y) {
        return ValueType(6) * x + ValueType(6) * y;
    };
 
    std::vector<IndexType> row_ptrs(dp_2 + 1);
    std::vector<IndexType> col_idxs((3 * dp - 2) * (3 * dp - 2));
    std::vector<ValueType> values((3 * dp - 2) * (3 * dp - 2));
    std::vector<ValueType> rhs(dp_2);
    std::vector<ValueType> u(dp_2, 0.0);
 
    generate_stencil_matrix(dp, row_ptrs.data(), col_idxs.data(), values.data(),
                            coefs.data());
    generate_rhs(dp, f, correct_u, rhs.data(), coefs.data());
 
    const gko::remove_complex<ValueType> reduction_factor = 1e-7;
 
    auto start_time = std::chrono::steady_clock::now();
    solve_system(executor_string, dp, row_ptrs.data(), col_idxs.data(),
                 values.data(), rhs.data(), u.data(), reduction_factor);
    auto stop_time = std::chrono::steady_clock::now();
    auto runtime_duration =
        static_cast<double>(
            std::chrono::duration_cast<std::chrono::nanoseconds>(stop_time -
                                                                 start_time)
                .count()) *
        1e-6;
 
    std::cout << "The average relative error is "
              << calculate_error(dp, u.data(), correct_u) /
                     static_cast<gko::remove_complex<ValueType>>(dp_2)
              << std::endl;
    std::cout << "The runtime is " << std::to_string(runtime_duration) << " ms"
              << std::endl;
}