NLProblem.cpp

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#include "NLProblem.hpp"
 
#include <polyfem/io/OBJWriter.hpp>
#include <polyfem/utils/MatrixUtils.hpp>
#include <polyfem/utils/MatrixUtils.hpp>
 
#include <polyfem/utils/Logger.hpp>
 
#include <polysolve/linear/Solver.hpp>
#ifdef POLYSOLVE_WITH_SPQR
#include <Eigen/SPQRSupport>
#include <SuiteSparseQR.hpp>
#endif
 
#ifdef POLYFEM_WITH_ARMADILLO
#include <armadillo>
#endif
 
#include <igl/cat.h>
#include <igl/Timer.h>
 
#include <set>
 
/*
m \frac{\partial^2 u}{\partial t^2} = \psi = \text{div}(\sigma[u])\newline
u^{t+1} = u(t+\Delta t)\approx u(t) + \Delta t \dot u + \frac{\Delta t^2} 2 \ddot u \newline
= u(t) + \Delta t \dot u + \frac{\Delta t^2}{2} \psi\newline
M u^{t+1}_h \approx M u^t_h + \Delta t M v^t_h + \frac{\Delta t^2} {2} A u^{t+1}_h \newline
%
M (u^{t+1}_h - (u^t_h + \Delta t v^t_h)) - \frac{\Delta t^2} {2} A u^{t+1}_h
*/
// mü = ψ = div(σ[u])
// uᵗ⁺¹ = u(t + Δt) ≈ u(t) + Δtu̇ + ½Δt²ü = u(t) + Δtu̇ + ½Δt²ψ
// Muₕᵗ⁺¹ ≈ Muₕᵗ + ΔtMvₕᵗ ½Δt²Auₕᵗ⁺¹
// Root-finding form:
// M(uₕᵗ⁺¹ - (uₕᵗ + Δtvₕᵗ)) - ½Δt²Auₕᵗ⁺¹ = 0
 
namespace polyfem::solver
{
 
    namespace
    {
#ifdef POLYSOLVE_WITH_SPQR
        void fill_cholmod(Eigen::SparseMatrix<double, Eigen::ColMajor, long> &mat, cholmod_sparse &cmat)
        {
            long *p = mat.outerIndexPtr();
 
            cmat.nzmax = mat.nonZeros();
            cmat.nrow = mat.rows();
            cmat.ncol = mat.cols();
            cmat.p = p;
            cmat.i = mat.innerIndexPtr();
            cmat.x = mat.valuePtr();
            cmat.z = 0;
            cmat.sorted = 1;
            cmat.packed = 1;
            cmat.nz = 0;
            cmat.dtype = 0;
            cmat.stype = -1;
            cmat.xtype = CHOLMOD_REAL;
            cmat.dtype = CHOLMOD_DOUBLE;
            cmat.stype = 0;
            cmat.itype = CHOLMOD_LONG;
        }
#endif
 
#ifdef POLYFEM_WITH_ARMADILLO
        arma::sp_mat fill_arma(const StiffnessMatrix &mat)
        {
            std::vector<unsigned long long> rowind_vect(mat.innerIndexPtr(), mat.innerIndexPtr() + mat.nonZeros());
            std::vector<unsigned long long> colptr_vect(mat.outerIndexPtr(), mat.outerIndexPtr() + mat.outerSize() + 1);
            std::vector<double> values_vect(mat.valuePtr(), mat.valuePtr() + mat.nonZeros());
 
            arma::dvec values(values_vect.data(), values_vect.size(), false);
            arma::uvec rowind(rowind_vect.data(), rowind_vect.size(), false);
            arma::uvec colptr(colptr_vect.data(), colptr_vect.size(), false);
 
            arma::sp_mat amat(rowind, colptr, values, mat.rows(), mat.cols(), false);
 
            return amat;
        }
 
        StiffnessMatrix fill_eigen(const arma::sp_mat &mat)
        {
            // convert to eigen sparse
            std::vector<long> outerIndexPtr(mat.row_indices, mat.row_indices + mat.n_nonzero);
            std::vector<long> innerIndexPtr(mat.col_ptrs, mat.col_ptrs + mat.n_cols + 1);
 
            const StiffnessMatrix out = Eigen::Map<const Eigen::SparseMatrix<double, Eigen::ColMajor, long>>(
                mat.n_rows, mat.n_cols, mat.n_nonzero, innerIndexPtr.data(), outerIndexPtr.data(), mat.values);
            return out;
        }
#endif
 
    } // namespace
    NLProblem::NLProblem(
        const int full_size,
        const std::vector<std::shared_ptr<Form>> &forms,
        const std::vector<std::shared_ptr<AugmentedLagrangianForm>> &penalty_forms,
        const std::shared_ptr<polysolve::linear::Solver> &solver)
        : FullNLProblem(forms),
          full_size_(full_size),
          t_(0),
          penalty_forms_(penalty_forms),
          solver_(solver)
    {
        setup_constraints();
        use_reduced_size();
    }
 
    NLProblem::NLProblem(
        const int full_size,
        const std::shared_ptr<utils::PeriodicBoundary> &periodic_bc,
        const double t,
        const std::vector<std::shared_ptr<Form>> &forms,
        const std::vector<std::shared_ptr<AugmentedLagrangianForm>> &penalty_forms,
        const std::shared_ptr<polysolve::linear::Solver> &solver,
        const double char_length,
        const double char_force,
        StiffnessMatrix lumped_mass,
        const int dimension)
        : FullNLProblem(forms),
          full_size_(full_size),
          t_(t),
          penalty_forms_(penalty_forms),
          solver_(solver),
          L(char_length),
          F0(char_force),
          lumped_mass_(lumped_mass.diagonal().asDiagonal()),
          dim(dimension)
    {
        setup_constraints();
        use_reduced_size();
 
        double total_lumped_mass = 0;
        int num_nonzero_mass_entries = 0;
        for (int i = 0; i < lumped_mass_.diagonal().size(); i++)
        {
            if (lumped_mass_.diagonal()[i] > 0)
            {
                total_lumped_mass += lumped_mass_.diagonal()[i];
                num_nonzero_mass_entries++;
            }
        }
        const double avg_lumped_mass = total_lumped_mass / num_nonzero_mass_entries;
        for (int i = 0; i < lumped_mass_.diagonal().size(); i++)
        {
            if (lumped_mass_.diagonal()[i] == 0)
            {
                lumped_mass_.diagonal()[i] = avg_lumped_mass;
            }
        }
    }
 
    double NLProblem::normalize_forms()
    {
        return 1;
 
        // double total_weight = 0;
        // for (const auto &f : forms_)
        //  total_weight += f->weight();
        // if (full_size() == current_size())
        // {
        //  for (const auto &f : penalty_forms_)
        //      total_weight += f->weight() * f->lagrangian_weight();
        // }
 
        // logger().debug("Normalizing forms with scale: {}", total_weight);
 
        // for (auto &f : forms_)
        //  f->set_scale(total_weight);
        // for (auto &f : penalty_forms_)
        //  f->set_scale(total_weight);
 
        // return total_weight;
    }
 
    double NLProblem::grad_norm_rescaling(const polysolve::nonlinear::NormType norm_type) const
    {
        switch (norm_type)
        {
        case polysolve::nonlinear::NormType::EUCLIDEAN:
            return 1;
        case polysolve::nonlinear::NormType::L2:
            return F0 * (dim == 2 ? L : std::pow(L, 1.5));
        case polysolve::nonlinear::NormType::Linf:
            return F0;
        }
        log_and_throw_error("Unrecognized norm type!");
    }
 
    double NLProblem::step_norm_rescaling(const polysolve::nonlinear::NormType norm_type) const
    {
        switch (norm_type)
        {
        case polysolve::nonlinear::NormType::EUCLIDEAN:
            return 1;
        case polysolve::nonlinear::NormType::L2:
            return dim == 2 ? L * L : std::pow(L, 2.5);
        case polysolve::nonlinear::NormType::Linf:
            return L;
        }
        log_and_throw_error("Unrecognized norm type!");
    }
 
    double NLProblem::energy_norm_rescaling(const polysolve::nonlinear::NormType norm_type) const
    {
        const double density_scale = dim == 2 ? L * L : L * L * L;
        return F0 * density_scale * L;
    }
 
    double NLProblem::grad_norm(const TVector &grad, const polysolve::nonlinear::NormType norm_type) const
    {
        switch (norm_type)
        {
        case polysolve::nonlinear::NormType::EUCLIDEAN:
            return grad.norm();
        case polysolve::nonlinear::NormType::L2:
            return sqrt(grad.transpose() * current_lumped_mass().inverse() * grad);
        case polysolve::nonlinear::NormType::Linf:
            return (current_lumped_mass().inverse() * grad).cwiseAbs().maxCoeff();
        }
        log_and_throw_error("Unrecognized norm type!");
    }
 
    double NLProblem::step_norm(const TVector &x, const polysolve::nonlinear::NormType norm_type) const
    {
        switch (norm_type)
        {
        case polysolve::nonlinear::NormType::EUCLIDEAN:
            return x.norm();
        case polysolve::nonlinear::NormType::L2:
            return sqrt(x.transpose() * current_lumped_mass() * x);
        case polysolve::nonlinear::NormType::Linf:
            return x.cwiseAbs().maxCoeff();
        }
        log_and_throw_error("Unrecognized norm type!");
    }
 
    void NLProblem::setup_constraints()
    {
        if (penalty_forms_.empty())
        {
            reduced_size_ = full_size_;
            return;
        }
        igl::Timer timer;
 
        if (penalty_forms_.size() == 1 && penalty_forms_.front()->has_projection())
        {
            Q2_ = penalty_forms_.front()->constraint_projection_matrix();
            Q2t_ = Q2_.transpose();
 
            reduced_size_ = Q2_.cols();
            if (reduced_size_ == 0)
                return;
            num_penalty_constraints_ = full_size_ - reduced_size_;
 
            timer.start();
            StiffnessMatrix Q2tQ2 = Q2t_ * Q2_;
            solver_->analyze_pattern(Q2tQ2, Q2tQ2.rows());
            solver_->factorize(Q2tQ2);
            timer.stop();
            logger().debug("Factorization and computation of Q2tQ2 took: {}", timer.getElapsedTime());
 
            std::vector<std::shared_ptr<Form>> tmp;
            tmp.insert(tmp.end(), penalty_forms_.begin(), penalty_forms_.end());
            penalty_problem_ = std::make_shared<FullNLProblem>(tmp);
 
            update_constraint_values();
 
            return;
        }
 
        std::vector<Eigen::Triplet<double>> Ae;
        int index = 0;
        for (const auto &f : penalty_forms_)
        {
            const auto &tmp = f->constraint_matrix();
            for (int i = 0; i < tmp.outerSize(); i++)
            {
                for (typename StiffnessMatrix::InnerIterator it(tmp, i); it; ++it)
                {
                    Ae.emplace_back(index + it.row(), it.col(), it.value());
                }
            }
            index += tmp.rows();
        }
        StiffnessMatrix A(index, full_size_);
        A.setFromTriplets(Ae.begin(), Ae.end());
        A.makeCompressed();
 
        int constraint_size = A.rows();
        num_penalty_constraints_ = A.rows();
        Eigen::SparseMatrix<double, Eigen::ColMajor, long> At = A.transpose();
        At.makeCompressed();
 
        logger().debug("Constraint size: {} x {}", A.rows(), A.cols());
 
#ifdef POLYSOLVE_WITH_SPQR
        timer.start();
 
        cholmod_common cc;
        cholmod_l_start(&cc); // start CHOLMOD
 
        // const int ordering = 0; // all, except 3:given treated as 0:fixed
        const int ordering = SPQR_ORDERING_DEFAULT; // all, except 3:given treated as 0:fixed
        const double tol = SPQR_DEFAULT_TOL;
        SuiteSparse_long econ = At.rows();
        SuiteSparse_long *E; // permutation of 0:n-1, NULL if identity
 
        cholmod_sparse Ac, *Qc, *Rc;
 
        fill_cholmod(At, Ac);
 
        const auto rank = SuiteSparseQR<double>(ordering, tol, econ, &Ac,
                                                // outputs
                                                &Qc, &Rc, &E, &cc);
 
        if (!Rc)
            log_and_throw_error("Failed to factorize constraints matrix");
 
        const auto n = Rc->ncol;
        P_.resize(n);
        if (E)
        {
            for (long j = 0; j < n; j++)
                P_.indices()(j) = E[j];
        }
        else
            P_.setIdentity();
 
        if (Qc->stype != 0 || Qc->sorted != 1 || Qc->packed != 1 || Rc->stype != 0 || Rc->sorted != 1 || Rc->packed != 1)
            log_and_throw_error("Q and R must be unsymmetric sorted and packed");
 
        const StiffnessMatrix Q = Eigen::Map<Eigen::SparseMatrix<double, Eigen::ColMajor, long>>(
            Qc->nrow, Qc->ncol, Qc->nzmax,
            static_cast<long *>(Qc->p), static_cast<long *>(Qc->i), static_cast<double *>(Qc->x));
 
        const StiffnessMatrix R = Eigen::Map<Eigen::SparseMatrix<double, Eigen::ColMajor, long>>(
            Rc->nrow, Rc->ncol, Rc->nzmax,
            static_cast<long *>(Rc->p), static_cast<long *>(Rc->i), static_cast<double *>(Rc->x));
 
        cholmod_l_free_sparse(&Qc, &cc);
        cholmod_l_free_sparse(&Rc, &cc);
        std::free(E);
        cholmod_l_finish(&cc);
 
        timer.stop();
        logger().debug("QR took: {}", timer.getElapsedTime());
#else
        timer.start();
 
        // Eigen::SparseQR<StiffnessMatrix, Eigen::NaturalOrdering<int>> QR(At);
        Eigen::SparseQR<StiffnessMatrix, Eigen::COLAMDOrdering<int>> QR(At);
 
        timer.stop();
        logger().debug("QR took: {}", timer.getElapsedTime());
 
        if (QR.info() != Eigen::Success)
            log_and_throw_error("Failed to factorize constraints matrix");
 
        timer.start();
        StiffnessMatrix Q;
        Q = QR.matrixQ();
        timer.stop();
        logger().debug("Computation of Q took: {}", timer.getElapsedTime());
 
        const Eigen::SparseMatrix<double, Eigen::RowMajor> R = QR.matrixR();
 
        P_ = QR.colsPermutation();
#endif
 
        for (; constraint_size >= 0; --constraint_size)
        {
            const StiffnessMatrix tmp = R.row(constraint_size);
            if (tmp.nonZeros() != 0)
            {
                constraint_size++;
                break;
            }
        }
        if (constraint_size != num_penalty_constraints_)
            logger().warn("Matrix A is not full rank, constraint size: {} instead of {}", constraint_size, num_penalty_constraints_);
 
        reduced_size_ = full_size_ - constraint_size;
 
        timer.start();
 
        Q1_ = Q.leftCols(constraint_size);
        assert(Q1_.rows() == full_size_);
        assert(Q1_.cols() == constraint_size);
 
        Q2_ = Q.rightCols(reduced_size_);
        Q2t_ = Q2_.transpose();
 
        assert(Q2_.rows() == full_size_);
        assert(Q2_.cols() == reduced_size_);
 
        R1_ = R.topRows(constraint_size);
        assert(R1_.rows() == constraint_size);
        assert(R1_.cols() == num_penalty_constraints_);
 
        assert((Q1_.transpose() * Q2_).norm() < 1e-10);
 
        timer.stop();
        logger().debug("Getting Q1 Q2, R1 took: {}", timer.getElapsedTime());
 
        timer.start();
 
        // arma::sp_mat q2a = fill_arma(Q2_);
        // arma::sp_mat q2tq2 = q2a.t() * q2a;
        // const StiffnessMatrix Q2tQ2 = fill_eigen(q2tq2);
        StiffnessMatrix Q2tQ2 = Q2t_ * Q2_;
        timer.stop();
        logger().debug("Getting Q2'*Q2, took: {}", timer.getElapsedTime());
 
        timer.start();
        solver_->analyze_pattern(Q2tQ2, Q2tQ2.rows());
        solver_->factorize(Q2tQ2);
        timer.stop();
        logger().debug("Factorization of Q2'*Q2 took: {}", timer.getElapsedTime());
 
#ifndef NDEBUG
        StiffnessMatrix test = R.bottomRows(reduced_size_);
        assert(test.nonZeros() == 0);
 
        StiffnessMatrix test1 = R1_.row(R1_.rows() - 1);
        assert(test1.nonZeros() != 0);
#endif
 
        // assert((Q1_ * R1_ - At * P_).norm() < 1e-10);
 
        std::vector<std::shared_ptr<Form>> tmp;
        tmp.insert(tmp.end(), penalty_forms_.begin(), penalty_forms_.end());
        penalty_problem_ = std::make_shared<FullNLProblem>(tmp);
 
        update_constraint_values();
    }
 
    void NLProblem::update_constraint_values()
    {
        if (penalty_forms_.size() == 1 && penalty_forms_.front()->has_projection())
        {
            Q1R1iTb_ = penalty_forms_.front()->constraint_projection_vector();
            return;
        }
 
        igl::Timer timer;
        timer.start();
        // x =  Q1 * R1^(-T) * P^T b  +  Q2 * y
        int index = 0;
        TVector constraint_values(num_penalty_constraints_);
        for (const auto &f : penalty_forms_)
        {
            constraint_values.segment(index, f->constraint_value().rows()) = f->constraint_value();
            index += f->constraint_value().rows();
        }
        constraint_values = P_.transpose() * constraint_values;
 
        Eigen::VectorXd sol;
 
        if (R1_.rows() == R1_.cols())
        {
            sol = R1_.transpose().triangularView<Eigen::Lower>().solve(constraint_values);
        }
        else
        {
 
#ifdef POLYSOLVE_WITH_SPQR
            Eigen::SparseMatrix<double, Eigen::ColMajor, long> R1t = R1_.transpose();
            cholmod_common cc;
            cholmod_l_start(&cc); // start CHOLMOD
            cholmod_sparse R1tc;
            fill_cholmod(R1t, R1tc);
 
            cholmod_dense b;
            b.nrow = constraint_values.size();
            b.ncol = 1;
            b.nzmax = constraint_values.size();
            b.d = constraint_values.size();
            b.x = constraint_values.data();
            b.z = 0;
            b.xtype = CHOLMOD_REAL;
            b.dtype = 0;
 
            const int ordering = SPQR_ORDERING_DEFAULT; // all, except 3:given treated as 0:fixed
            const double tol = SPQR_DEFAULT_TOL;
 
            cholmod_dense *solc = SuiteSparseQR<double>(ordering, tol, &R1tc, &b, &cc);
 
            sol = Eigen::Map<Eigen::VectorXd>(static_cast<double *>(solc->x), solc->nrow);
 
            cholmod_l_free_dense(&solc, &cc);
            cholmod_l_finish(&cc);
#else
            Eigen::SparseQR<StiffnessMatrix, Eigen::COLAMDOrdering<int>> solver;
            solver.compute(R1_.transpose());
            if (solver.info() != Eigen::Success)
            {
                log_and_throw_error("Failed to factorize R1^T");
            }
            sol = solver.solve(constraint_values);
#endif
        }
 
        assert((R1_.transpose() * sol - constraint_values).norm() < 1e-10);
 
        Q1R1iTb_ = Q1_ * sol;
 
        timer.stop();
        logger().debug("Computing Q1R1iTb took: {}", timer.getElapsedTime());
    }
 
    void NLProblem::init_lagging(const TVector &x)
    {
        FullNLProblem::init_lagging(reduced_to_full(x));
 
        if (penalty_problem_ && full_size() == current_size())
            penalty_problem_->init_lagging(x);
    }
 
    void NLProblem::update_lagging(const TVector &x, const int iter_num)
    {
        FullNLProblem::update_lagging(reduced_to_full(x), iter_num);
 
        if (penalty_problem_ && full_size() == current_size())
            penalty_problem_->update_lagging(x, iter_num);
    }
 
    void NLProblem::update_quantities(const double t, const TVector &x)
    {
        t_ = t;
        const TVector full = reduced_to_full(x);
        assert(full.size() == full_size_);
        for (auto &f : forms_)
            f->update_quantities(t, full);
 
        for (auto &f : penalty_forms_)
            f->update_quantities(t, x);
 
        update_constraint_values();
    }
 
    void NLProblem::line_search_begin(const TVector &x0, const TVector &x1)
    {
        FullNLProblem::line_search_begin(reduced_to_full(x0), reduced_to_full(x1));
 
        if (penalty_problem_ && full_size() == current_size())
            penalty_problem_->line_search_begin(x0, x1);
    }
 
    double NLProblem::max_step_size(const TVector &x0, const TVector &x1)
    {
        double max_step = FullNLProblem::max_step_size(reduced_to_full(x0), reduced_to_full(x1));
 
        if (penalty_problem_ && full_size() == current_size())
            max_step = std::min(max_step, penalty_problem_->max_step_size(x0, x1));
 
        return max_step;
    }
 
    bool NLProblem::is_step_valid(const TVector &x0, const TVector &x1)
    {
        bool valid = FullNLProblem::is_step_valid(reduced_to_full(x0), reduced_to_full(x1));
 
        if (penalty_problem_ && valid && full_size() == current_size())
            valid = penalty_problem_->is_step_valid(x0, x1);
 
        return valid;
    }
 
    bool NLProblem::is_step_collision_free(const TVector &x0, const TVector &x1)
    {
        bool free = FullNLProblem::is_step_collision_free(reduced_to_full(x0), reduced_to_full(x1));
 
        if (penalty_problem_ && free && full_size() == current_size())
            free = penalty_problem_->is_step_collision_free(x0, x1);
 
        return free;
    }
 
    double NLProblem::value(const TVector &x)
    {
        // TODO: removed fearure const bool only_elastic
        double res = FullNLProblem::value(reduced_to_full(x));
 
        if (penalty_problem_ && full_size() == current_size())
        {
            res += penalty_problem_->value(x);
        }
 
        return res;
    }
 
    void NLProblem::gradient(const TVector &x, TVector &grad)
    {
        FullNLProblem::gradient(reduced_to_full(x), grad);
 
        if (full_size() != current_size())
        {
            if (penalty_forms_.size() == 1 && penalty_forms_.front()->can_project())
                penalty_forms_.front()->project_gradient(grad);
            else
                grad = Q2t_ * grad;
        }
        else if (penalty_problem_)
        {
            TVector tmp;
            penalty_problem_->gradient(x, tmp);
            grad += tmp;
        }
    }
 
    void NLProblem::hessian(const TVector &x, THessian &hessian)
    {
        FullNLProblem::hessian(reduced_to_full(x), hessian);
 
        if (full_size() != current_size())
        {
            full_hessian_to_reduced_hessian(hessian);
        }
        else if (penalty_problem_)
        {
            THessian tmp;
            penalty_problem_->hessian(x, tmp);
            hessian += tmp;
        }
    }
 
    void NLProblem::solution_changed(const TVector &newX)
    {
        FullNLProblem::solution_changed(reduced_to_full(newX));
 
        if (penalty_problem_ && full_size() == current_size())
            penalty_problem_->solution_changed(newX);
    }
 
    void NLProblem::post_step(const polysolve::nonlinear::PostStepData &data)
    {
        FullNLProblem::post_step(polysolve::nonlinear::PostStepData(data.iter_num, data.solver_info, reduced_to_full(data.x), reduced_to_full(data.grad)));
 
        if (penalty_problem_ && full_size() == current_size())
            penalty_problem_->post_step(data);
 
        // TODO: add me back
        static int nsolves = 0;
        if (data.iter_num == 0)
            nsolves++;
        // if (state && state->args["output"]["advanced"]["save_nl_solve_sequence"])
        // {
        //  const Eigen::MatrixXd displacements = utils::unflatten(reduced_to_full(data.x), state->mesh->dimension());
        //  io::OBJWriter::write(
        //      state->resolve_output_path(fmt::format("nonlinear_solve{:04d}_iter{:04d}.obj", nsolves, data.iter_num)),
        //      state->collision_mesh.displace_vertices(displacements),
        //      state->collision_mesh.edges(), state->collision_mesh.faces());
        // }
    }
 
    NLProblem::TVector NLProblem::full_to_reduced(const TVector &full) const
    {
        // Reduced is already at the full size
        if (full_size() == current_size() || full.size() == current_size())
        {
            return full;
        }
 
        TVector reduced(reduced_size());
        const TVector k = full - Q1R1iTb_;
        const TVector rhs = Q2t_ * k;
        solver_->solve(rhs, reduced);
 
#ifndef NDEBUG
        StiffnessMatrix Q2tQ2 = Q2t_ * Q2_;
        // std::cout << "err " << (Q2tQ2 * reduced - rhs).norm() << std::endl;
        assert((Q2tQ2 * reduced - rhs).norm() < 1e-8);
#endif
 
        return reduced;
    }
 
    NLProblem::TVector NLProblem::full_to_reduced_grad(const TVector &full) const
    {
        TVector grad = full;
        if (penalty_forms_.size() == 1 && penalty_forms_.front()->can_project())
            penalty_forms_.front()->project_gradient(grad);
        else
            grad = Q2t_ * grad;
 
        return grad;
    }
 
    NLProblem::TVector NLProblem::full_to_reduced_diag(const TVector &full_diag) const
    {
        if (full_size() == current_size() || full_diag.size() == current_size())
        {
            return full_diag;
        }
 
        TVector diag = full_diag;
        if (penalty_forms_.size() == 1 && penalty_forms_.front()->can_project())
            penalty_forms_.front()->project_diag(diag);
        else
        {
            Eigen::SparseMatrix<double> reduced_mat = Q2t_ * diag.asDiagonal() * Q2_;
            diag = reduced_mat.diagonal();
        }
 
        return diag;
    }
 
    NLProblem::TVector NLProblem::reduced_to_full(const TVector &reduced) const
    {
        // Full is already at the reduced size
        if (full_size() == current_size() || full_size() == reduced.size())
        {
            return reduced;
        }
 
        // x =  Q1 * R1^(-T) * P^T b  +  Q2 * y
 
        const TVector full = Q1R1iTb_ + Q2_ * reduced;
 
#ifndef NDEBUG
        for (const auto &f : penalty_forms_)
        {
            // std::cout << f->compute_error(full) << std::endl;
            assert(f->compute_error(full) < 1e-8);
        }
#endif
 
        return full;
    }
 
    void NLProblem::full_hessian_to_reduced_hessian(StiffnessMatrix &hessian) const
    {
        if (penalty_forms_.size() == 1 && penalty_forms_.front()->can_project())
            penalty_forms_.front()->project_hessian(hessian);
        else
        {
            // arma::sp_mat q2a = fill_arma(Q2_);
            // arma::sp_mat ha = fill_arma(hessian);
            // arma::sp_mat q2thq2 = q2a.t() * ha * q2a;
            // hessian = fill_eigen(q2thq2);
            hessian = Q2t_ * hessian * Q2_;
            // remove numerical zeros
            hessian.prune([](const Eigen::Index &row, const Eigen::Index &col, const Scalar &value) {
                return std::abs(value) > 1e-10;
            });
        }
    }
} // namespace polyfem::solver