solve_KKT.cpp

#include <Epetra_RowMatrixTransposer.h>

#include <deal.II/base/conditional_ostream.h>

#include <deal.II/distributed/tria.h>
#include <deal.II/grid/grid_generator.h>
#include <deal.II/grid/grid_tools.h>

#include <deal.II/numerics/vector_tools.h>

#include "physics/physics_factory.h"
#include "parameters/all_parameters.h"
#include "dg/dg_factory.hpp"
#include "functional/functional.h"

#include <deal.II/lac/full_matrix.h>

#include <deal.II/lac/solver_bicgstab.h>
#include <deal.II/lac/solver_cg.h>
#include <deal.II/lac/solver_gmres.h>
#include <deal.II/lac/solver_minres.h>

#include <deal.II/lac/block_sparsity_pattern.h>

#include <deal.II/lac/precondition.h>

#include <deal.II/lac/trilinos_block_sparse_matrix.h>
#include <deal.II/lac/trilinos_precondition.h>
#include <deal.II/lac/trilinos_solver.h>
#include <deal.II/lac/trilinos_sparsity_pattern.h>
#include <deal.II/lac/trilinos_sparse_matrix.h>

#include <deal.II/lac/trilinos_parallel_block_vector.h>
#include <deal.II/lac/la_parallel_block_vector.h>


#include <deal.II/lac/packaged_operation.h>
#include <deal.II/lac/trilinos_linear_operator.h>

//#include <deal.II/lac/block_linear_operator.h>
#include <deal.II/lac/solver_control.h>


#include <deal.II/lac/read_write_vector.h>

const double STEPSIZE = 1e-7;
const double TOLERANCE = 1e-6;

#if PHILIP_DIM==1
    using Triangulation = dealii::Triangulation<PHILIP_DIM>;
#else
    using Triangulation = dealii::parallel::distributed::Triangulation<PHILIP_DIM>;
#endif

std::pair<unsigned int, double>
solve_linear (
    dealii::TrilinosWrappers::BlockSparseMatrix &matrix_A,
    dealii::LinearAlgebra::distributed::BlockVector<double> &right_hand_side,
    dealii::LinearAlgebra::distributed::BlockVector<double> &solution,
    const PHiLiP::Parameters::LinearSolverParam &)//param)
{

    //const dealii::BlockLinearOperator kkt_operator = dealii::block_operator (matrix_A);
    using block_vec = dealii::LinearAlgebra::distributed::BlockVector<double>;
    const auto kkt_operator = dealii::block_operator<block_vec> (matrix_A);
    dealii::SolverControl solver_control(100000, 1.e-15, true, true);
    const unsigned int  max_n_tmp_vectors = 2000;
    const bool  right_preconditioning = false;
    const bool  use_default_residual = true;
    const bool  force_re_orthogonalization = false;
    dealii::SolverGMRES<block_vec>::AdditionalData add_data( max_n_tmp_vectors, right_preconditioning, use_default_residual, force_re_orthogonalization);
    dealii::SolverGMRES<block_vec> solver_gmres(solver_control, add_data);
    auto kkt_inv = inverse_operator(kkt_operator, solver_gmres, dealii::PreconditionIdentity());
    solution = kkt_inv * right_hand_side;

    return {-1.0, -1.0};
}

template <int dim, int nstate, typename real>
class L2_Norm_Functional : public PHiLiP::Functional<dim, nstate, real>
{

    using FadType = Sacado::Fad::DFad<real>; 
    using FadFadType = Sacado::Fad::DFad<FadType>; 
public:
    L2_Norm_Functional(
        std::shared_ptr<PHiLiP::DGBase<dim,real>> dg_input,
        const bool uses_solution_values = true,
        const bool uses_solution_gradient = false)
    : PHiLiP::Functional<dim,nstate,real>(dg_input,uses_solution_values,uses_solution_gradient)
    {}

    template <typename real2>
    real2 evaluate_volume_integrand(
                const PHiLiP::Physics::PhysicsBase<dim,nstate,real2> &physics,
                const dealii::Point<dim,real2> &phys_coord,
                const std::array<real2,nstate> &soln_at_q,
                const std::array<dealii::Tensor<1,dim,real2>,nstate> &/*soln_grad_at_q*/) const
    {
        real2 l2error = 0;
        for (int istate=0; istate<nstate; ++istate) {
            const real2 uexact = physics.manufactured_solution_function->value(phys_coord, istate);
            l2error += std::pow(soln_at_q[istate] - uexact, 2);
        }
        return l2error;
    }


    template<typename real2>
    real2 evaluate_boundary_integrand(
        const PHiLiP::Physics::PhysicsBase<dim,nstate,real2> &physics,
        const unsigned int /*boundary_id*/,
        const dealii::Point<dim,real2> &phys_coord,
        const dealii::Tensor<1,dim,real2> &/*normal*/,
        const std::array<real2,nstate> &soln_at_q,
        const std::array<dealii::Tensor<1,dim,real2>,nstate> &/*soln_grad_at_q*/) const
    {
        real2 l1error = 0;
        for (int istate=0; istate<nstate; ++istate) {
            const real2 uexact = physics.manufactured_solution_function->value(phys_coord, istate);
            l1error += std::abs(soln_at_q[istate] - uexact);
        }
        return l1error;
    }


    virtual real evaluate_boundary_integrand(
        const PHiLiP::Physics::PhysicsBase<dim,nstate,real> &physics,
        const unsigned int boundary_id,
        const dealii::Point<dim,real> &phys_coord,
        const dealii::Tensor<1,dim,real> &normal,
        const std::array<real,nstate> &soln_at_q,
        const std::array<dealii::Tensor<1,dim,real>,nstate> &soln_grad_at_q) const override
    {
        return evaluate_boundary_integrand<real>(
            physics,
            boundary_id,
            phys_coord,
            normal,
            soln_at_q,
            soln_grad_at_q);
    }

    virtual FadFadType evaluate_boundary_integrand(
        const PHiLiP::Physics::PhysicsBase<dim,nstate,FadFadType> &physics,
        const unsigned int boundary_id,
        const dealii::Point<dim,FadFadType> &phys_coord,
        const dealii::Tensor<1,dim,FadFadType> &normal,
        const std::array<FadFadType,nstate> &soln_at_q,
        const std::array<dealii::Tensor<1,dim,FadFadType>,nstate> &soln_grad_at_q) const override
    {
        return evaluate_boundary_integrand<FadFadType>(
            physics,
            boundary_id,
            phys_coord,
            normal,
            soln_at_q,
            soln_grad_at_q);
    }



    real evaluate_volume_integrand(
            const PHiLiP::Physics::PhysicsBase<dim,nstate,real> &physics,
            const dealii::Point<dim,real> &phys_coord,
            const std::array<real,nstate> &soln_at_q,
            const std::array<dealii::Tensor<1,dim,real>,nstate> &soln_grad_at_q) const override
    {
        return evaluate_volume_integrand<>(physics, phys_coord, soln_at_q, soln_grad_at_q);
    }
    FadFadType evaluate_volume_integrand(
            const PHiLiP::Physics::PhysicsBase<dim,nstate,FadFadType> &physics,
            const dealii::Point<dim,FadFadType> &phys_coord,
            const std::array<FadFadType,nstate> &soln_at_q,
            const std::array<dealii::Tensor<1,dim,FadFadType>,nstate> &soln_grad_at_q) const override
    {
        return evaluate_volume_integrand<>(physics, phys_coord, soln_at_q, soln_grad_at_q);
    }

};


template <int dim, int nstate>
void initialize_perturbed_solution(PHiLiP::DGBase<dim,double> &dg, const PHiLiP::Physics::PhysicsBase<dim,nstate,double> &physics)
{
    dealii::LinearAlgebra::distributed::Vector<double> solution_no_ghost;
    solution_no_ghost.reinit(dg.locally_owned_dofs, MPI_COMM_WORLD);
    dealii::VectorTools::interpolate(dg.dof_handler, *physics.manufactured_solution_function, solution_no_ghost);
    dg.solution = solution_no_ghost;
}

int main(int argc, char *argv[])
{

    const int dim = PHILIP_DIM;
    const int nstate = 1;
    int fail_bool = false;

    // Initializing MPI
    dealii::Utilities::MPI::MPI_InitFinalize mpi_initialization(argc, argv, 1);
    const int this_mpi_process = dealii::Utilities::MPI::this_mpi_process(MPI_COMM_WORLD);
    dealii::ConditionalOStream pcout(std::cout, this_mpi_process==0);

    // Initializing parameter handling
    dealii::ParameterHandler parameter_handler;
    PHiLiP::Parameters::AllParameters::declare_parameters(parameter_handler);
    PHiLiP::Parameters::AllParameters all_parameters;
    all_parameters.parse_parameters(parameter_handler);

    // polynomial order and mesh size
    const unsigned poly_degree = 1;

    // creating the grid
    std::shared_ptr<Triangulation> grid = std::make_shared<Triangulation>(
#if PHILIP_DIM!=1
        MPI_COMM_WORLD,
#endif
        typename dealii::Triangulation<dim>::MeshSmoothing(
            dealii::Triangulation<dim>::smoothing_on_refinement |
            dealii::Triangulation<dim>::smoothing_on_coarsening));

    const unsigned int n_refinements = 2;
    double left = 0.0;
    double right = 2.0;
    const bool colorize = true;

    dealii::GridGenerator::hyper_cube(*grid, left, right, colorize);
    grid->refine_global(n_refinements);
    const double random_factor = 0.2;
    const bool keep_boundary = false;
    if (random_factor > 0.0) dealii::GridTools::distort_random (random_factor, *grid, keep_boundary);

    pcout << "Grid generated and refined" << std::endl;

    // creating the dg
    std::shared_ptr < PHiLiP::DGBase<dim, double> > dg = PHiLiP::DGFactory<dim,double>::create_discontinuous_galerkin(&all_parameters, poly_degree, grid);
    pcout << "dg created" << std::endl;

    dg->allocate_system();
    pcout << "dg allocated" << std::endl;

    const int n_refine = 2;
    for (int i=0; i<n_refine;i++) {
        dg->high_order_grid->prepare_for_coarsening_and_refinement();
        grid->prepare_coarsening_and_refinement();
        unsigned int icell = 0;
        for (auto cell = grid->begin_active(); cell!=grid->end(); ++cell) {
            icell++;
            if (!cell->is_locally_owned()) continue;
            if (icell < grid->n_global_active_cells()/2) {
                cell->set_refine_flag();
            }
        }
        grid->execute_coarsening_and_refinement();
        bool mesh_out = (i==n_refine-1);
        dg->high_order_grid->execute_coarsening_and_refinement(mesh_out);
    }
    dg->allocate_system ();

    // manufactured solution function
    std::shared_ptr <PHiLiP::Physics::PhysicsBase<dim,nstate,double>> physics_double = PHiLiP::Physics::PhysicsFactory<dim, nstate, double>::create_Physics(&all_parameters);
    pcout << "Physics created" << std::endl;
 
    // performing the interpolation for the intial conditions
    initialize_perturbed_solution(*dg, *physics_double);
    pcout << "solution initialized" << std::endl;

    // evaluating the derivative (using SACADO)
    pcout << std::endl << "Starting Hessian AD... " << std::endl;
    L2_Norm_Functional<dim,nstate,double> functional(dg,true,false);
    const bool compute_dIdW = false, compute_dIdX = false, compute_d2I = true;
    double functional_value = functional.evaluate_functional(compute_dIdW, compute_dIdX, compute_d2I);
    (void) functional_value;

    // Evaluate residual Hessians
    bool compute_dRdW, compute_dRdX, compute_d2R;

    pcout << "Evaluating RHS only to use as dual variables..." << std::endl;
    compute_dRdW = false; compute_dRdX = false, compute_d2R = false;
    dg->assemble_residual(compute_dRdW, compute_dRdX, compute_d2R);
    dealii::LinearAlgebra::distributed::Vector<double> dummy_dual(dg->right_hand_side);
    dg->set_dual(dummy_dual);

    pcout << "Evaluating RHS with d2R..." << std::endl;
    compute_dRdW = true; compute_dRdX = false, compute_d2R = false;
    dg->assemble_residual(compute_dRdW, compute_dRdX, compute_d2R);
    compute_dRdW = false; compute_dRdX = true, compute_d2R = false;
    dg->assemble_residual(compute_dRdW, compute_dRdX, compute_d2R);
    compute_dRdW = false; compute_dRdX = false, compute_d2R = true;
    dg->assemble_residual(compute_dRdW, compute_dRdX, compute_d2R);
    dealii::LinearAlgebra::distributed::Vector<double> rhs_d2R(dg->right_hand_side);
    // pcout << "*******************************************************************************" << std::endl;

    dealii::TrilinosWrappers::SparseMatrix dRdW_transpose;
    {
        Epetra_CrsMatrix *transpose_CrsMatrix;
        Epetra_RowMatrixTransposer epmt(const_cast<Epetra_CrsMatrix *>(&dg->system_matrix.trilinos_matrix()));
        epmt.CreateTranspose(false, transpose_CrsMatrix);
        dRdW_transpose.reinit(*transpose_CrsMatrix);
    }

    dealii::TrilinosWrappers::SparseMatrix dRdX_transpose;
    {
        Epetra_CrsMatrix *transpose_CrsMatrix;
        Epetra_RowMatrixTransposer epmt(const_cast<Epetra_CrsMatrix *>(&dg->dRdXv.trilinos_matrix()));
        epmt.CreateTranspose(false, transpose_CrsMatrix);
        dRdX_transpose.reinit(*transpose_CrsMatrix);
    }

    dealii::TrilinosWrappers::SparseMatrix d2RdXdW;
    {
        Epetra_CrsMatrix *transpose_CrsMatrix;
        Epetra_RowMatrixTransposer epmt(const_cast<Epetra_CrsMatrix *>(&dg->d2RdWdX.trilinos_matrix()));
        epmt.CreateTranspose(false, transpose_CrsMatrix);
        d2RdXdW.reinit(*transpose_CrsMatrix);
    }
    dealii::TrilinosWrappers::SparseMatrix d2IdXdW;
    {
        Epetra_CrsMatrix *transpose_CrsMatrix;
        Epetra_RowMatrixTransposer epmt(const_cast<Epetra_CrsMatrix *>(&functional.d2IdWdX->trilinos_matrix()));
        epmt.CreateTranspose(false, transpose_CrsMatrix);
        d2IdXdW.reinit(*transpose_CrsMatrix);
    }

    // Form Lagrangian Hessian
    functional.d2IdWdW->add(1.0,dg->d2RdWdW);
    functional.d2IdWdX->add(1.0,dg->d2RdWdX);
    d2IdXdW.add(1.0,d2RdXdW);
    functional.d2IdXdX->add(1.0,dg->d2RdXdX);


    // Block Sparsity pattern
    // L_ww  L_wx  R_w^T
    // L_xw  L_xx  R_x^T
    // R_w   R_x   0
    // const unsigned int n_constraints = rhs_only.size();
    // const unsigned int n_flow_var    = dg->dof_handler.n_dofs();
    // const unsigned int n_geom_var    = dg->high_order_grid->dof_handler_grid.n_dofs();
    //BlockDynamicSparsityPattern dsp(3, 3);
    //dsp.block(0, 0).reinit(n_flow_var, n_flow_var);
    //dsp.block(0, 1).reinit(n_flow_var, n_geom_var);
    //dsp.block(0, 2).reinit(n_flow_var, n_constraints);

    //dsp.block(1, 0).reinit(n_geom_var, n_flow_var);
    //dsp.block(1, 1).reinit(n_geom_var, n_geom_var);
    //dsp.block(1, 2).reinit(n_geom_var, n_constraints);

    //dsp.block(2, 0).reinit(n_constraints, n_flow_var);
    //dsp.block(2, 1).reinit(n_constraints, n_geom_var);
    //dsp.block(2, 2).reinit(n_constraints, n_constraints);

    dealii::TrilinosWrappers::BlockSparseMatrix kkt_hessian;
    kkt_hessian.reinit(3,3);
    kkt_hessian.block(0, 0).copy_from( *functional.d2IdWdW);
    kkt_hessian.block(0, 1).copy_from( *functional.d2IdWdX);
    kkt_hessian.block(0, 2).copy_from( dRdW_transpose);

    kkt_hessian.block(1, 0).copy_from( d2IdXdW);
    kkt_hessian.block(1, 1).copy_from( *functional.d2IdXdX);
    kkt_hessian.block(1, 2).copy_from( dRdX_transpose);

    kkt_hessian.block(2, 0).copy_from( dg->system_matrix);
    kkt_hessian.block(2, 1).copy_from( dg->dRdXv);
    dealii::TrilinosWrappers::SparsityPattern zero_sparsity_pattern(dg->locally_owned_dofs, MPI_COMM_WORLD, 0);
    //dealii::TrilinosWrappers::SparseMatrix zero_block;//(n_constraints, n_constraints, 0);
    //zero_block.reinit(dg->locally_owned_dofs, zero_sparsity_pattern, MPI_COMM_WORLD);
    // zero_block.reinit(dg->locally_owned_dofs);
    // dealii::TrilinosWrappers::SparseMatrix zero_block(dg->locally_owned_dofs);
    // kkt_hessian.block(2, 2).copy_from( zero_block );
    zero_sparsity_pattern.compress();
    kkt_hessian.block(2, 2).reinit(zero_sparsity_pattern);

    kkt_hessian.collect_sizes();

    pcout << "kkt_hessian.frobenius_norm()  " << kkt_hessian.frobenius_norm() << std::endl;

    // const int n_mpi_processes = dealii::Utilities::MPI::n_mpi_processes(MPI_COMM_WORLD);
    // if (n_mpi_processes == 1) {
    //     dealii::FullMatrix<double> fullA(kkt_hessian.m());
    //     fullA.copy_from(kkt_hessian);
    //     pcout<<"d2IdWdW:"<<std::endl;
    //     if (pcout.is_active()) fullA.print_formatted(pcout.get_stream(), 3, true, 10, "0", 1., 0.);
    // }

    dealii::LinearAlgebra::distributed::BlockVector<double> block_vector(3);
    block_vector.block(0) = dg->solution;
    block_vector.block(1) = dg->high_order_grid->volume_nodes;
    block_vector.block(2) = dummy_dual;
    dealii::LinearAlgebra::distributed::BlockVector<double> Hv(3);
    dealii::LinearAlgebra::distributed::BlockVector<double> Htv(3);
    Hv.reinit(block_vector);
    Htv.reinit(block_vector);

    kkt_hessian.vmult(Hv,block_vector);
    kkt_hessian.Tvmult(Htv,block_vector);

    Htv.sadd(-1.0, Hv);

    const double vector_norm = Hv.l2_norm();
    const double vector_abs_diff = Htv.l2_norm();
    const double vector_rel_diff = vector_abs_diff / vector_norm;

    const double tol = 1e-11;
    pcout << "Error: "
                    << " vector_abs_diff: " << vector_abs_diff
                    << " vector_rel_diff: " << vector_rel_diff
                    << std::endl
                    << " vector_abs_diff: " << vector_abs_diff
                    << " vector_rel_diff: " << vector_rel_diff
                    << std::endl;
    if (vector_abs_diff > tol && vector_rel_diff > tol) fail_bool = true;

    const int n_mpi_processes = dealii::Utilities::MPI::n_mpi_processes(MPI_COMM_WORLD);
    if (n_mpi_processes == 1) {
        dealii::FullMatrix<double> fullA(kkt_hessian.m());
        fullA.copy_from(kkt_hessian);
        const int n_digits = 8;
        if (pcout.is_active()) fullA.print_formatted(pcout.get_stream(), n_digits, true, n_digits+7, "0", 1., 0.);
    }

    dealii::deallog.depth_console(3);
    solve_linear (
        kkt_hessian,
        Hv, // b
        Htv, // x
        all_parameters.linear_solver_param);

    dealii::LinearAlgebra::distributed::BlockVector<double> diff(3);
    diff.reinit(block_vector);
    kkt_hessian.vmult(diff,Htv);
    diff -= Hv;
    double diff_norm = diff.l2_norm();
    std::cout
        << " diff_norm "
        << diff_norm
        << std::endl;

    return fail_bool;
}