Ginkgo: The poisson-solver program

| Ginkgo Generated from branch based on main. Ginkgo version 1.11.0 A numerical linear algebra library targeting many-core architectures

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The poisson-solver program

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:

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 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

row_ptrs[i + 1] = pos;
}
}

Generates the RHS vector given f and the boundary conditions.

template <typename Closure, typename ValueType>
void generate_rhs(Closure f, ValueType u0, ValueType u1,
gko::matrix::Dense<ValueType>* rhs)
{
const auto discretization_points = rhs->get_size()[0];
auto values = rhs->get_values();
const ValueType h = 1.0 / static_cast<ValueType>(discretization_points + 1);
for (gko::size_type i = 0; i < discretization_points; ++i) {
const auto xi = static_cast<ValueType>(i + 1) * h;
values[i] = -f(xi) * h * h;
}
values[0] += u0;
values[discretization_points - 1] += u1;
}
gko::matrix::Dense
Dense is a matrix format which explicitly stores all values of the matrix.
Definition dense.hpp:120
gko::size_type
std::size_t size_type
Integral type used for allocation quantities.
Definition types.hpp:90

Prints the solution u.

template <typename Closure, typename ValueType>
void print_solution(ValueType u0, ValueType u1,
const gko::matrix::Dense<ValueType>* 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;
}
gko::matrix::Dense::get_const_values
const value_type * get_const_values() const noexcept
Returns a pointer to the array of values of the matrix.
Definition dense.hpp:870

Computes the 1-norm of the error given the computed u and the correct solution function correct_u.

template <typename Closure, typename ValueType>
gko::remove_complex<ValueType> calculate_error(
int discretization_points, const gko::matrix::Dense<ValueType>* u,
Closure correct_u)
{
const ValueType h = 1.0 / static_cast<ValueType>(discretization_points + 1);
gko::remove_complex<ValueType> error = 0.0;
for (int i = 0; i < discretization_points; ++i) {
using std::abs;
const auto xi = static_cast<ValueType>(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[])
{
gko::remove_complex
typename detail::remove_complex_s< T >::type remove_complex
Obtain the type which removed the complex of complex/scalar type or the template parameter of class b...
Definition math.hpp:264

Some shortcuts

using ValueType = double;
using IndexType = int;
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>;
gko::matrix::Csr
CSR is a matrix format which stores only the nonzero coefficients by compressing each row of the matr...
Definition csr.hpp:126
gko::preconditioner::Jacobi
A block-Jacobi preconditioner is a block-diagonal linear operator, obtained by inverting the diagonal...
Definition jacobi.hpp:190
gko::solver::Cg
CG or the conjugate gradient method is an iterative type Krylov subspace method which is suitable for...
Definition cg.hpp:50

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);
}
gko::version_info::get
static const version_info & get()
Returns an instance of version_info.
Definition version.hpp:139

Get number of discretization points

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

Get the executor string

const auto executor_string = argc >= 2 ? argv[1] : "reference";

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(); }}};
gko::CudaExecutor::create
static std::shared_ptr< CudaExecutor > create(int device_id, std::shared_ptr< Executor > master, bool device_reset, allocation_mode alloc_mode=default_cuda_alloc_mode, CUstream_st *stream=nullptr)
Creates a new CudaExecutor.
gko::DpcppExecutor::create
static std::shared_ptr< DpcppExecutor > create(int device_id, std::shared_ptr< Executor > master, std::string device_type="all", dpcpp_queue_property property=dpcpp_queue_property::in_order)
Creates a new DpcppExecutor.
gko::HipExecutor::create
static std::shared_ptr< HipExecutor > create(int device_id, std::shared_ptr< Executor > master, bool device_reset, allocation_mode alloc_mode=default_hip_alloc_mode, CUstream_st *stream=nullptr)
Creates a new HipExecutor.
gko::OmpExecutor::create
static std::shared_ptr< OmpExecutor > create(std::shared_ptr< CpuAllocatorBase > alloc=std::make_shared< CpuAllocator >())
Creates a new OmpExecutor.
Definition executor.hpp:1396

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->get_master();

Set up the problem: define the exact solution, the right hand side and the Dirichlet boundary condition.

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);

initialize matrix and vectors

auto matrix = mtx::create(app_exec, gko::dim<2>(discretization_points),
3 * discretization_points - 2);
generate_stencil_matrix(matrix.get());
auto rhs = vec::create(app_exec, gko::dim<2>(discretization_points, 1));
generate_rhs(f, u0, u1, rhs.get());
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;
}
const gko::remove_complex<ValueType> reduction_factor = 1e-7;
gko::matrix::Dense::get_values
value_type * get_values() noexcept
Returns a pointer to the array of values of the matrix.
Definition dense.hpp:861
gko::dim
A type representing the dimensions of a multidimensional object.
Definition dim.hpp:26

Generate solver and solve the system

cg::build()
.with_criteria(
gko::stop::Iteration::build().with_max_iters(discretization_points),
gko::stop::ResidualNorm<ValueType>::build().with_reduction_factor(
reduction_factor))
.with_preconditioner(bj::build())
.on(exec)
->generate(clone(exec, matrix)) // copy the matrix to the executor
->apply(rhs, u);

Uncomment to print the solution print_solution<ValueType>(u0, u1, u.get());

std::cout << "Solve complete.\nThe average relative error is "
<< calculate_error(discretization_points, u.get(), correct_u) /
static_cast<gko::remove_complex<ValueType>>(
discretization_points)
<< std::endl;
}

# Results

This is the expected output:

Solve complete.
The average relative error is 2.52236e-11

### Comments about programming and debugging

# The plain program

#include <iostream>
#include <map>
#include <string>
#include <vector>
#include <ginkgo/ginkgo.hpp>
template <typename ValueType, typename IndexType>
void generate_stencil_matrix(gko::matrix::Csr<ValueType, IndexType>* 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 ValueType 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, typename ValueType>
void generate_rhs(Closure f, ValueType u0, ValueType u1,
gko::matrix::Dense<ValueType>* rhs)
{
const auto discretization_points = rhs->get_size()[0];
auto values = rhs->get_values();
const ValueType h = 1.0 / static_cast<ValueType>(discretization_points + 1);
for (gko::size_type i = 0; i < discretization_points; ++i) {
const auto xi = static_cast<ValueType>(i + 1) * h;
values[i] = -f(xi) * h * h;
}
values[0] += u0;
values[discretization_points - 1] += u1;
}
template <typename Closure, typename ValueType>
void print_solution(ValueType u0, ValueType u1,
const gko::matrix::Dense<ValueType>* 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, typename ValueType>
gko::remove_complex<ValueType> calculate_error(
int discretization_points, const gko::matrix::Dense<ValueType>* u,
Closure correct_u)
{
const ValueType h = 1.0 / static_cast<ValueType>(discretization_points + 1);
gko::remove_complex<ValueType> error = 0.0;
for (int i = 0; i < discretization_points; ++i) {
using std::abs;
const auto xi = static_cast<ValueType>(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 ValueType = double;
using IndexType = int;
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>;
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 unsigned int discretization_points =
argc >= 3 ? std::atoi(argv[2]) : 100;
const auto executor_string = argc >= 2 ? argv[1] : "reference";
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 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);
auto matrix = mtx::create(app_exec, gko::dim<2>(discretization_points),
3 * discretization_points - 2);
generate_stencil_matrix(matrix.get());
auto rhs = vec::create(app_exec, gko::dim<2>(discretization_points, 1));
generate_rhs(f, u0, u1, rhs.get());
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;
}
const gko::remove_complex<ValueType> reduction_factor = 1e-7;
cg::build()
.with_criteria(
gko::stop::Iteration::build().with_max_iters(discretization_points),
gko::stop::ResidualNorm<ValueType>::build().with_reduction_factor(
reduction_factor))
.with_preconditioner(bj::build())
.on(exec)
->generate(clone(exec, matrix)) // copy the matrix to the executor
->apply(rhs, u);
std::cout << "Solve complete.\nThe average relative error is "
<< calculate_error(discretization_points, u.get(), correct_u) /
static_cast<gko::remove_complex<ValueType>>(
discretization_points)
<< std::endl;
}
gko::matrix::Csr::get_row_ptrs
index_type * get_row_ptrs() noexcept
Returns the row pointers of the matrix.
Definition csr.hpp:1264
gko::matrix::Csr::get_values
value_type * get_values() noexcept
Returns the values of the matrix.
Definition csr.hpp:1214
gko::matrix::Csr::get_col_idxs
index_type * get_col_idxs() noexcept
Returns the column indexes of the matrix.
Definition csr.hpp:1245
gko::solver::initial_guess_mode::rhs
@ rhs
the input is right hand side
Definition solver_base.hpp:41
gko::abs
constexpr std::enable_if_t<!is_complex_s< T >::value, T > abs(const T &x)
Returns the absolute value of the object.
Definition math.hpp:962
gko::clone
detail::cloned_type< Pointer > clone(const Pointer &p)
Creates a unique clone of the object pointed to by p.
Definition utils_helper.hpp:173