The ir-ilu-preconditioned-solver program

The IR-ILU preconditioned solver example..

This example depends on ilu-preconditioned-solver, iterative-refinement.

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

Introduction

About the example

This example shows how to combine iterative refinement with the adaptive precision block-Jacobi preconditioner in order to approximately solve the triangular systems occurring in ILU preconditioning. Using an adaptive precision block-Jacobi preconditioner matrix as inner solver for the iterative refinement method is equivalent to doing adaptive precision block-Jacobi relaxation in the triangular solves. This example roughly approximates the triangular solves with five adaptive precision block-Jacobi sweeps with a maximum block size of 16.

This example is motivated by "Multiprecision block-Jacobi for Iterative Triangular Solves" (Göbel, Anzt, Cojean, Flegar, Quintana-Ortí, Euro-Par 2020). The theory and a detailed analysis can be found there.

The commented program

    std::cerr << "Usage: " << argv[0] << " [executor]" << std::endl;
    std::exit(-1);
}
 
const auto executor_string = argc >= 2 ? argv[1] : "reference";
const unsigned int sweeps = argc == 3 ? std::atoi(argv[2]) : 5u;
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

Read data

auto A = gko::share(gko::read<mtx>(std::ifstream("data/A.mtx"), exec));

Create RHS and initial guess as 1

gko::size_type num_rows = A->get_size()[0];
auto host_x = vec::create(exec->get_master(), gko::dim<2>(num_rows, 1));
for (gko::size_type i = 0; i < num_rows; i++) {
    host_x->at(i, 0) = 1.;
}
auto x = gko::clone(exec, host_x);
auto b = gko::clone(exec, host_x);
auto clone_x = gko::clone(exec, x);

Generate incomplete factors using ParILU

auto par_ilu_fact =
    gko::factorization::ParIlu<ValueType, IndexType>::build().on(exec);

Generate concrete factorization for input matrix

auto par_ilu = gko::share(par_ilu_fact->generate(A));

Generate an iterative refinement factory to be used as a triangular solver in the preconditioner application. The generated method is equivalent to doing five block-Jacobi sweeps with a maximum block size of 16.

auto bj_factory = gko::share(
    bj::build()
        .with_max_block_size(16u)
        .with_storage_optimization(gko::precision_reduction::autodetect())
        .on(exec));
 
auto trisolve_factory =
    ir::build()
        .with_solver(bj_factory)
        .with_criteria(gko::stop::Iteration::build().with_max_iters(sweeps))
        .on(exec);

Generate an ILU preconditioner factory by setting lower and upper triangular solver - in this case the previously defined iterative refinement method.

auto ilu_pre_factory = gko::preconditioner::Ilu<ir, ir>::build()
                           .with_l_solver(gko::clone(trisolve_factory))
                           .with_u_solver(gko::clone(trisolve_factory))
                           .on(exec);

Use incomplete factors to generate ILU preconditioner

auto ilu_preconditioner = gko::share(ilu_pre_factory->generate(par_ilu));

Create stopping criteria for Gmres

const RealValueType reduction_factor{1e-12};
auto iter_stop = gko::share(
    gko::stop::Iteration::build().with_max_iters(1000u).on(exec));
auto tol_stop = gko::share(gko::stop::ResidualNorm<ValueType>::build()
                               .with_reduction_factor(reduction_factor)
                               .on(exec));

Use preconditioner inside GMRES solver factory Generating a solver factory tied to a specific preconditioner makes sense if there are several very similar systems to solve, and the same solver+preconditioner combination is expected to be effective.

auto ilu_gmres_factory =
    gmres::build()
        .with_criteria(iter_stop, tol_stop)
        .with_generated_preconditioner(ilu_preconditioner)
        .on(exec);

Generate preconditioned solver for a specific target system

auto ilu_gmres = ilu_gmres_factory->generate(A);

Add logger

std::shared_ptr<const gko::log::Convergence<ValueType>> logger =
    gko::log::Convergence<ValueType>::create();
ilu_gmres->add_logger(logger);

Warmup run

ilu_gmres->apply(b, x);

Solve system 100 times and take the average time.

std::chrono::nanoseconds time(0);
for (int i = 0; i < 100; i++) {
    x->copy_from(clone_x);
    auto tic = std::chrono::high_resolution_clock::now();
    ilu_gmres->apply(b, x);
    auto toc = std::chrono::high_resolution_clock::now();
    time += std::chrono::duration_cast<std::chrono::nanoseconds>(toc - tic);
}
 
std::cout << "Using " << sweeps << " block-Jacobi sweeps.\n";

Print solution

std::cout << "Solution (x):\n";
write(std::cout, x);

Get residual

    auto res = gko::as<vec>(logger->get_residual_norm());
 
    std::cout << "GMRES iteration count:     " << logger->get_num_iterations()
              << "\n";
    std::cout << "GMRES execution time [ms]: "
              << static_cast<double>(time.count()) / 100000000.0 << "\n";
    std::cout << "Residual norm sqrt(r^T r):\n";
    write(std::cout, res);
}

Results

This is the expected output:

Using 5 block-Jacobi sweeps.
Solution (x):
%%MatrixMarket matrix array real general
19 1
0.252218
0.108645
0.0662811
0.0630433
0.0384088
0.0396536
0.0402648
0.0338935
0.0193098
0.0234653
0.0211499
0.0196413
0.0199151
0.0181674
0.0162722
0.0150714
0.0107016
0.0121141
0.0123025
GMRES iteration count:     8
GMRES execution time [ms]: 0.377673
Residual norm sqrt(r^T r):
%%MatrixMarket matrix array real general
1 1
1.65303e-12

Comments about programming and debugging

The plain program

#include <cstdlib>
#include <fstream>
#include <iostream>
#include <map>
#include <string>
 
#include <ginkgo/ginkgo.hpp>
 
 
int main(int argc, char* argv[])
{
    using ValueType = double;
    using RealValueType = gko::remove_complex<ValueType>;
    using IndexType = int;
 
    using vec = gko::matrix::Dense<ValueType>;
    using real_vec = gko::matrix::Dense<RealValueType>;
    using mtx = gko::matrix::Csr<ValueType, IndexType>;
    using gmres = gko::solver::Gmres<ValueType>;
    using ir = gko::solver::Ir<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]" << std::endl;
        std::exit(-1);
    }
 
    const auto executor_string = argc >= 2 ? argv[1] : "reference";
    const unsigned int sweeps = argc == 3 ? std::atoi(argv[2]) : 5u;
    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
 
    auto A = gko::share(gko::read<mtx>(std::ifstream("data/A.mtx"), exec));
    gko::size_type num_rows = A->get_size()[0];
    auto host_x = vec::create(exec->get_master(), gko::dim<2>(num_rows, 1));
    for (gko::size_type i = 0; i < num_rows; i++) {
        host_x->at(i, 0) = 1.;
    }
    auto x = gko::clone(exec, host_x);
    auto b = gko::clone(exec, host_x);
    auto clone_x = gko::clone(exec, x);
 
    auto par_ilu_fact =
        gko::factorization::ParIlu<ValueType, IndexType>::build().on(exec);
    auto par_ilu = gko::share(par_ilu_fact->generate(A));
 
    auto bj_factory = gko::share(
        bj::build()
            .with_max_block_size(16u)
            .with_storage_optimization(gko::precision_reduction::autodetect())
            .on(exec));
 
    auto trisolve_factory =
        ir::build()
            .with_solver(bj_factory)
            .with_criteria(gko::stop::Iteration::build().with_max_iters(sweeps))
            .on(exec);
 
    auto ilu_pre_factory = gko::preconditioner::Ilu<ir, ir>::build()
                               .with_l_solver(gko::clone(trisolve_factory))
                               .with_u_solver(gko::clone(trisolve_factory))
                               .on(exec);
 
    auto ilu_preconditioner = gko::share(ilu_pre_factory->generate(par_ilu));
 
    const RealValueType reduction_factor{1e-12};
    auto iter_stop = gko::share(
        gko::stop::Iteration::build().with_max_iters(1000u).on(exec));
    auto tol_stop = gko::share(gko::stop::ResidualNorm<ValueType>::build()
                                   .with_reduction_factor(reduction_factor)
                                   .on(exec));
 
    auto ilu_gmres_factory =
        gmres::build()
            .with_criteria(iter_stop, tol_stop)
            .with_generated_preconditioner(ilu_preconditioner)
            .on(exec);
 
    auto ilu_gmres = ilu_gmres_factory->generate(A);
 
    std::shared_ptr<const gko::log::Convergence<ValueType>> logger =
        gko::log::Convergence<ValueType>::create();
    ilu_gmres->add_logger(logger);
 
    ilu_gmres->apply(b, x);
 
    std::chrono::nanoseconds time(0);
    for (int i = 0; i < 100; i++) {
        x->copy_from(clone_x);
        auto tic = std::chrono::high_resolution_clock::now();
        ilu_gmres->apply(b, x);
        auto toc = std::chrono::high_resolution_clock::now();
        time += std::chrono::duration_cast<std::chrono::nanoseconds>(toc - tic);
    }
 
    std::cout << "Using " << sweeps << " block-Jacobi sweeps.\n";
 
    std::cout << "Solution (x):\n";
    write(std::cout, x);
 
    auto res = gko::as<vec>(logger->get_residual_norm());
 
    std::cout << "GMRES iteration count:     " << logger->get_num_iterations()
              << "\n";
    std::cout << "GMRES execution time [ms]: "
              << static_cast<double>(time.count()) / 100000000.0 << "\n";
    std::cout << "Residual norm sqrt(r^T r):\n";
    write(std::cout, res);
}