The custom-matrix-format program

The custom matrix format example.

This example depends on simple-solver, poisson-solver, three-pt-stencil-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:

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:

For an equidistant grid with K "inner" discretization points and step size , the formula produces a system of linear equations

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 is set to (making the solution ), but that can be changed in the main function.

The intention of this example is to show how a custom linear operator can be created and integrated into Ginkgo to achieve performance benefits.

About the example

The commented program

#include <iostream>
#include <map>
#include <string>


#include <omp.h>
#include <ginkgo/ginkgo.hpp>

A CUDA kernel implementing the stencil, which will be used if running on the CUDA executor. Unfortunately, NVCC has serious problems interpreting some parts of Ginkgo's code, so the kernel has to be compiled separately.

extern void stencil_kernel(std::size_t size, const double *coefs,
                           const double *b, double *x);

A stencil matrix class representing the 3pt stencil linear operator. We include the gko::EnableLinOp mixin which implements the entire LinOp interface, except the two apply_impl methods, which get called inside the default implementation of apply (after argument verification) to perform the actual application of the linear operator. In addition, it includes the implementation of the entire PolymorphicObject interface.

It also includes the gko::EnableCreateMethod mixin which provides a default implementation of the static create method. This method will forward all its arguments to the constructor to create the object, and return an std::unique_ptr to the created object.

class StencilMatrix : public gko::EnableLinOp<StencilMatrix>,
                      public gko::EnableCreateMethod<StencilMatrix> {
public:

This constructor will be called by the create method. Here we initialize the coefficients of the stencil.

    StencilMatrix(std::shared_ptr<const gko::Executor> exec,
                  gko::size_type size = 0, double left = -1.0,
                  double center = 2.0, double right = -1.0)
        : gko::EnableLinOp<StencilMatrix>(exec, gko::dim<2>{size}),
          coefficients(exec, {left, center, right})
    {}

protected:
    using vec = gko::matrix::Dense<>;
    using coef_type = gko::Array<double>;

Here we implement the application of the linear operator, x = A * b. apply_impl will be called by the apply method, after the arguments have been moved to the correct executor and the operators checked for conforming sizes.

For simplicity, we assume that there is always only one right hand side and the stride of consecutive elements in the vectors is 1 (both of these are always true in this example).

void apply_impl(const gko::LinOp *b, gko::LinOp *x) const override
{

we only implement the operator for dense RHS. gko::as will throw an exception if its argument is not Dense.

auto dense_b = gko::as<vec>(b);
auto dense_x = gko::as<vec>(x);

we need separate implementations depending on the executor, so we create an operation which maps the call to the correct implementation

struct stencil_operation : gko::Operation {
    stencil_operation(const coef_type &coefficients, const vec *b,
                      vec *x)
        : coefficients{coefficients}, b{b}, x{x}
    {}

OpenMP implementation

            void run(std::shared_ptr<const gko::OmpExecutor>) const override
            {
                auto b_values = b->get_const_values();
                auto x_values = x->get_values();
#pragma omp parallel for
                for (std::size_t i = 0; i < x->get_size()[0]; ++i) {
                    auto coefs = coefficients.get_const_data();
                    auto result = coefs[1] * b_values[i];
                    if (i > 0) {
                        result += coefs[0] * b_values[i - 1];
                    }
                    if (i < x->get_size()[0] - 1) {
                        result += coefs[2] * b_values[i + 1];
                    }
                    x_values[i] = result;
                }
            }

CUDA implementation

void run(std::shared_ptr<const gko::CudaExecutor>) const override
{
    stencil_kernel(x->get_size()[0], coefficients.get_const_data(),
                   b->get_const_values(), x->get_values());
}

We do not provide an implementation for reference executor. If not provided, Ginkgo will use the implementation for the OpenMP executor when calling it in the reference executor.

        const coef_type &coefficients;
        const vec *b;
        vec *x;
    };
    this->get_executor()->run(
        stencil_operation(coefficients, dense_b, dense_x));
}

There is also a version of the apply function which does the operation x = alpha * A * b + beta * x. This function is commonly used and can often be better optimized than implementing it using x = A * b. However, for simplicity, we will implement it exactly like that in this example.

    void apply_impl(const gko::LinOp *alpha, const gko::LinOp *b,
                    const gko::LinOp *beta, gko::LinOp *x) const override
    {
        auto dense_b = gko::as<vec>(b);
        auto dense_x = gko::as<vec>(x);
        auto tmp_x = dense_x->clone();
        this->apply_impl(b, lend(tmp_x));
        dense_x->scale(beta);
        dense_x->add_scaled(alpha, lend(tmp_x));
    }

private:
    coef_type coefficients;
};

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>;

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]) : 100u;
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 vectors

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))
    .on(exec)

notice how our custom StencilMatrix can be used in the same way as any built-in type

        ->generate(
            StencilMatrix::create(exec, discretization_points, -1, 2, -1))
        ->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

Comments about programming and debugging

The plain program

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#include <iostream>
#include <map>
#include <string>


#include <omp.h>
#include <ginkgo/ginkgo.hpp>


extern void stencil_kernel(std::size_t size, const double *coefs,
                           const double *b, double *x);


class StencilMatrix : public gko::EnableLinOp<StencilMatrix>,
                      public gko::EnableCreateMethod<StencilMatrix> {
public:
    StencilMatrix(std::shared_ptr<const gko::Executor> exec,
                  gko::size_type size = 0, double left = -1.0,
                  double center = 2.0, double right = -1.0)
        : gko::EnableLinOp<StencilMatrix>(exec, gko::dim<2>{size}),
          coefficients(exec, {left, center, right})
    {}

protected:
    using vec = gko::matrix::Dense<>;
    using coef_type = gko::Array<double>;

    void apply_impl(const gko::LinOp *b, gko::LinOp *x) const override
    {
        auto dense_b = gko::as<vec>(b);
        auto dense_x = gko::as<vec>(x);

        struct stencil_operation : gko::Operation {
            stencil_operation(const coef_type &coefficients, const vec *b,
                              vec *x)
                : coefficients{coefficients}, b{b}, x{x}
            {}

            void run(std::shared_ptr<const gko::OmpExecutor>) const override
            {
                auto b_values = b->get_const_values();
                auto x_values = x->get_values();
#pragma omp parallel for
                for (std::size_t i = 0; i < x->get_size()[0]; ++i) {
                    auto coefs = coefficients.get_const_data();
                    auto result = coefs[1] * b_values[i];
                    if (i > 0) {
                        result += coefs[0] * b_values[i - 1];
                    }
                    if (i < x->get_size()[0] - 1) {
                        result += coefs[2] * b_values[i + 1];
                    }
                    x_values[i] = result;
                }
            }

            void run(std::shared_ptr<const gko::CudaExecutor>) const override
            {
                stencil_kernel(x->get_size()[0], coefficients.get_const_data(),
                               b->get_const_values(), x->get_values());
            }


            const coef_type &coefficients;
            const vec *b;
            vec *x;
        };
        this->get_executor()->run(
            stencil_operation(coefficients, dense_b, dense_x));
    }

    void apply_impl(const gko::LinOp *alpha, const gko::LinOp *b,
                    const gko::LinOp *beta, gko::LinOp *x) const override
    {
        auto dense_b = gko::as<vec>(b);
        auto dense_x = gko::as<vec>(x);
        auto tmp_x = dense_x->clone();
        this->apply_impl(b, lend(tmp_x));
        dense_x->scale(beta);
        dense_x->add_scaled(alpha, lend(tmp_x));
    }

private:
    coef_type coefficients;
};


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>;

    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]) : 100u;
    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 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))
        .on(exec)
        ->generate(
            StencilMatrix::create(exec, discretization_points, -1, 2, -1))
        ->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;
}