AMReX_AlgMG.H

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#ifndef AMREX_ALGMG_H_
#define AMREX_ALGMG_H_
#include <AMReX_Config.H>
 
#include <AMReX_Algebra.H>
#include <AMReX_Enum.H>
#include <AMReX_BiCGStab_MV.H>
#include <AMReX_BLProfiler.H>
#include <AMReX_CArena.H>
#include <AMReX_GMRES_MV.H>
#include <AMReX_PCG_MV.H>
#include <AMReX_ParallelReduce.H>
#include <AMReX_Scan.H>
#include <AMReX_Smoother_MV.H>
#include <AMReX_Vector.H>
 
#include <cmath>
#include <cstdint>
#include <iomanip>
#include <limits>
#include <numeric>
#include <memory>
#include <stdexcept>
#include <string>
#include <type_traits>
#include <utility>
 
namespace amrex {
 
namespace detail {

template <typename TC, typename Index, typename T>
AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE
bool pmis_blocked (CsrView<TC const,Index> const& csr, Long row, T wi, Long ig,
                   T const* AMREX_RESTRICT w, int const* AMREX_RESTRICT state,
                   Long const* AMREX_RESTRICT gmap, Long goff, int undecided)
{
    for (Long idx = csr.row_offset[row]; idx < csr.row_offset[row+1]; ++idx) {
        Long j = csr.col_index[idx];
        Long jg = gmap ? gmap[j] : j + goff;
        if (jg != ig && state[j] == undecided &&
            (w[j] > wi || (w[j] == wi && jg > ig))) {
            return true;
        }
    }
    return false;
}

template <typename T>
AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE
T pmis_tiebreak (Long ig)
{
    auto z = static_cast<std::uint64_t>(ig) + 0x9E3779B97F4A7C15ULL;
    z = (z ^ (z >> 30)) * 0xBF58476D1CE4E5B9ULL;
    z = (z ^ (z >> 27)) * 0x94D049BB133111EBULL;
    z ^= (z >> 31);
    constexpr int digits = std::numeric_limits<T>::digits;
    static_assert(digits < 64, "pmis_tiebreak: T must be float or double");
    return T(z >> (64 - digits)) / T(std::uint64_t(1) << digits);
}

template <typename T, typename Index>
AMREX_GPU_HOST_DEVICE AMREX_FORCE_INLINE
bool csr_row_has_state (CsrView<T const,Index> const& csr, Long row,
                        int const* AMREX_RESTRICT state, int c)
{
    for (Long idx = csr.row_offset[row]; idx < csr.row_offset[row+1]; ++idx) {
        if (state[csr.col_index[idx]] == c) { return true; }
    }
    return false;
}

inline std::pair<std::size_t,std::size_t> arena_bytes ()
{
    if (auto* carena = dynamic_cast<CArena*>(The_Arena())) {
        return {carena->heap_space_actually_used(), carena->heap_space_used()};
    }
    return {0, 0};
}

template <typename T, typename TS>
struct DirectInterpRow
{
    ParCsr<T const> a;
    ParCsr<TS const> s;
    int const* AMREX_RESTRICT state;    // local rows
    Long const* AMREX_RESTRICT cidx;    // local coarse index, local rows
    int const* AMREX_RESTRICT state_r;  // A's remote columns
    int nlocal_c;                        // number of local coarse columns
    int coarse;

    template <typename F>
    AMREX_GPU_HOST_DEVICE void walk (Long i, T& diag, F const& f) const
    {
        diag = T(0);
        {
            auto const& ac = a.csr0;
            auto const& sc = s.csr0;
            Long sp = sc.row_offset[i];
            Long const se = sc.row_offset[i+1];
            for (Long idx = ac.row_offset[i]; idx < ac.row_offset[i+1]; ++idx) {
                Long const j = ac.col_index[idx];
                T const v = ac.mat[idx];
                if (j == i) { diag = v; continue; }
                while (sp < se && sc.col_index[sp] < j) { ++sp; }
                bool const strong_c = (sp < se && sc.col_index[sp] == j
                                       && state[j] == coarse);
                f(int(cidx[j]), v, strong_c);
            }
        }
        if (has_remote_row(a, i)) {
            auto const& ac = a.csr1;
            auto const& sc = s.csr1;
            Long const ii = a.row_map[i];
            Long sp = 0, se = 0;
            if (has_remote_row(s, i)) {
                Long const jj = s.row_map[i];
                sp = sc.row_offset[jj];
                se = sc.row_offset[jj+1];
            }
            for (Long idx = ac.row_offset[ii]; idx < ac.row_offset[ii+1]; ++idx) {
                Long const j = ac.col_index[idx];
                Long const jg = a.col_map[j];
                T const v = ac.mat[idx];
                while (sp < se && s.col_map[sc.col_index[sp]] < jg) { ++sp; }
                bool const strong_c = (sp < se && s.col_map[sc.col_index[sp]] == jg
                                       && state_r[j] == coarse);
                f(int(nlocal_c + j), v, strong_c);
            }
        }
    }
};

template <typename T, typename TS>
struct MMInterpRow
{
    ParCsr<TS const> s;
    ParCsr<TS const> st;
    int const* AMREX_RESTRICT state;
    T const* AMREX_RESTRICT dbeta;
    Long const* AMREX_RESTRICT cidx;    // local coarse index, local rows
    int const* AMREX_RESTRICT state_r;  // S's remote columns
    T const* AMREX_RESTRICT dbeta_r;
    int nlocal_f;                        // local fine columns
    int nlocal_c;                        // local coarse columns

    template <typename F>
    AMREX_GPU_HOST_DEVICE void walk (Long i, F const& f) const
    {
        {
            auto const& sc = s.csr0;
            auto const& tc = st.csr0;
            Long tp = tc.row_offset[i];
            Long const te = tc.row_offset[i+1];
            for (Long idx = sc.row_offset[i]; idx < sc.row_offset[i+1]; ++idx) {
                Long const j = sc.col_index[idx];
                while (tp < te && tc.col_index[tp] < j) { ++tp; }
                T const aji = (tp < te && tc.col_index[tp] == j) ? T(tc.mat[tp]) : T(0);
                f(int(j), int(cidx[j]), T(sc.mat[idx]), aji, state[j], dbeta[j]);
            }
        }
        if (has_remote_row(s, i)) {
            auto const& sc = s.csr1;
            auto const& tc = st.csr1;
            Long const ii = s.row_map[i];
            Long tp = 0, te = 0;
            if (has_remote_row(st, i)) {
                Long const jj = st.row_map[i];
                tp = tc.row_offset[jj];
                te = tc.row_offset[jj+1];
            }
            for (Long idx = sc.row_offset[ii]; idx < sc.row_offset[ii+1]; ++idx) {
                Long const j = sc.col_index[idx];
                Long const jg = s.col_map[j];
                while (tp < te && st.col_map[tc.col_index[tp]] < jg) { ++tp; }
                T const aji = (tp < te && st.col_map[tc.col_index[tp]] == jg)
                    ? T(tc.mat[tp]) : T(0);
                f(int(nlocal_f + j), int(nlocal_c + j), T(sc.mat[idx]), aji,
                  state_r[j], dbeta_r[j]);
            }
        }
    }
};
 
}
 

AMREX_ENUM(AlgMGBottomSolver, jacobi, bicgstab, gmres, direct);

AMREX_ENUM(AlgMGKrylovSolver, none, bicgstab, gmres, pcg);

AMREX_ENUM(AlgMGInterpType, direct, mm_ext, mm_ext_i);

AMREX_ENUM(AlgMGSmoother, jacobi, l1_jacobi, chebyshev, l1_gauss_seidel);
template <typename T>
class AlgMG
{
public:
    using TS = std::conditional_t<std::is_same_v<T,double>, float, T>;
 
    using BottomSolver = AlgMGBottomSolver;
    using KrylovSolver = AlgMGKrylovSolver;
    using InterpType   = AlgMGInterpType;
    using Smoother     = AlgMGSmoother;
 
    AlgMG () = default;
    explicit AlgMG (SpMatrix<T>& a_mat);

    void define (SpMatrix<T>& a_mat);

    bool updateMatrix (SpMatrix<T>& a_mat);

    void solve (AlgVector<T>& a_sol, AlgVector<T> const& a_rhs);
 
    void setVerbose (int v) { m_verbose = v; }
    void setPrintIndentation (std::string s) { m_print_ident = std::move(s); }
    void setMaxIter (int n) { m_maxiter = n; }
    void setFixedIter (int n) { m_fixediter = n; }
    void setRelTol (T t) { m_reltol = t; }
    void setAbsTol (T t) { m_abstol = t; }
    void setSmoother (Smoother sm) { if (m_smoother_type != sm) { m_smoother_type = sm; m_smoothers_stale = true; } }
    void setRelaxWeight (T w) { if (m_relax_weight != w) { m_relax_weight = w; m_smoothers_stale = true; } }
    void setChebyshevDegree (int d) { if (m_cheby_degree != d) { m_cheby_degree = d; m_smoothers_stale = true; } }
    void setChebyshevRatio (T r) {
        AMREX_ALWAYS_ASSERT_WITH_MESSAGE(r > T(1), "AlgMG::setChebyshevRatio: the ratio must exceed 1");
        if (m_cheby_ratio != r) { m_cheby_ratio = r; m_smoothers_stale = true; }
    }

    void setPreSmooth (int nu) { m_nu1 = nu; }
    void setPostSmooth (int nu) { m_nu2 = nu; }
    void setBottomSmooth (int nu) { m_nu_bottom = nu; }
    void setStrongThreshold (T theta) { if (m_strong_threshold != theta) { m_strong_threshold = theta; m_prepared = false; } }
    void setMaxLevels (int n) { if (m_max_levels != n) { m_max_levels = n; m_prepared = false; } }
    void setMaxCoarseSize (Long n) { if (m_max_coarse_size != n) { m_max_coarse_size = n; m_prepared = false; } }
    void setAggressiveNumLevels (int n) { if (m_aggressive_num_levels != n) { m_aggressive_num_levels = n; m_prepared = false; } }
    void setAggressiveDirectInterp (bool b) { if (m_aggressive_direct_interp != b) { m_aggressive_direct_interp = b; m_prepared = false; } }
    void setSingular (bool b) {
        if (m_singular != b) { m_singular = b; m_prepared = false; }
    }

    void setInterpType (InterpType it) { if (m_interp_type != it) { m_interp_type = it; m_prepared = false; } }
    void setPMaxElmts (int n) {
        AMREX_ALWAYS_ASSERT_WITH_MESSAGE(n <= max_p_elmts, "AlgMG::setPMaxElmts: at most 32");
        int const p = (n < 0) ? default_p_max_elmts : n;
        if (m_p_max_elmts != p) { m_p_max_elmts = p; m_prepared = false; }
    }
    static constexpr int max_p_elmts = 32;
    static constexpr int default_p_max_elmts = 4;
    void setTruncFactor (T f) { if (m_trunc_factor != f) { m_trunc_factor = f; m_prepared = false; } }
    void setKrylovSolver (KrylovSolver a) {
        if (m_krylov == a) { return; }
        m_krylov = a;
        m_krylov_bicgstab.reset();
        m_krylov_gmres.reset();
        m_krylov_pcg.reset();
    }

    void precond (AlgVector<T>& x, AlgVector<T> const& b);
    void setBottomSolver (BottomSolver bs) {
        if (m_bottom_solver == bs) { return; }
        m_bottom_solver = bs;
        m_gmres.reset();
        m_bicgstab.reset();
        m_bottom_n = -1;
    }
    void setBottomTol (T t) { m_bottom_reltol = t; }
    void setBottomMaxIter (int n) { m_bottom_maxiter = n; }
    void setBottomVerbose (int v) { m_bottom_verbose = v; }
    void setThrowException (bool b) { m_throw_exception = b; }
 
    [[nodiscard]] int getNumIters () const { return m_niters; }
    [[nodiscard]] int getNumSetups () const { return m_num_setups; }
    [[nodiscard]] T getResidualNorm () const { return m_rnorm; }
    [[nodiscard]] int numLevels () const { return int(m_a.size()); }
    [[nodiscard]] std::size_t getSetupPeakBytes () const { return m_setup_peak_bytes; }
    [[nodiscard]] Long numGlobalRows (int lev) const { return m_a[lev]->numGlobalRows(); }
    [[nodiscard]] SpMatrix<T> const& getMatrix (int lev) const { return *m_a[lev]; }
    [[nodiscard]] SpMatrix<T> const& getInterp (int lev) const { return m_P[lev]; }
 
private:
 
    enum PointType : int { undecided = 0, fine = 1, coarse = 2 };
 
    bool m_prepared = false;
    int m_num_setups = 0;
    bool m_smoothers_stale = false;
    bool m_throw_exception = false;
    int m_verbose = 0;
    int m_maxiter = 200;
    int m_fixediter = 0;
    int m_niters = 0;
 
    Smoother m_smoother_type = Smoother::chebyshev;
    T m_relax_weight = T(-1);
    int m_cheby_degree = 2;
    T m_cheby_ratio = T(6);
    int m_nu1 = -1;
    int m_nu2 = -1;
    int m_nu_bottom = -1;
 
    [[nodiscard]] int numPreSmooth () const {
        return (m_nu1 >= 0) ? m_nu1 : ((m_smoother_type == Smoother::chebyshev) ? 1 : 2);
    }
    [[nodiscard]] int numPostSmooth () const {
        return (m_nu2 >= 0) ? m_nu2 : ((m_smoother_type == Smoother::chebyshev) ? 1 : 2);
    }
    [[nodiscard]] int numBottomSmooth () const {
        return (m_nu_bottom >= 0) ? m_nu_bottom : ((m_smoother_type == Smoother::chebyshev) ? 4 : 8);
    }
 
    int m_max_levels = 25;
    Long m_max_coarse_size = 9;
    int m_aggressive_num_levels = 0;
    bool m_aggressive_direct_interp = false;
    bool m_singular = false;
    std::unique_ptr<AlgVector<T>> m_bproj; // rhs with the mean removed
    T m_strong_threshold = T(0.25);
 
    InterpType m_interp_type = InterpType::mm_ext_i;
    int m_p_max_elmts = default_p_max_elmts;
    T m_trunc_factor = T(0);
    BottomSolver m_bottom_solver = BottomSolver::direct;
    // Safety limit for the dense LU on every process; the coarsest level
    // normally has a few rows (max_coarse_size). Beyond it smoother sweeps
    // are used, which keeps the cycle linear for PCG and GMRES.
    static constexpr Long max_direct_size = 1024;
    Long m_bottom_n = -1;          // rows of the factored coarsest matrix; -1: not built, 0: not applicable
    Vector<double> m_bottom_lu;    // dense LU, row-major, pivoted
    Vector<int> m_bottom_piv;
    T m_bottom_reltol = T(1.e-4);
    int m_bottom_maxiter = 200;
    int m_bottom_verbose = 0;
 
    T m_reltol = std::is_same_v<T,float> ? T(1.e-4) : T(1.e-8);
    T m_abstol = T(0);
    T m_init_rnorm = std::numeric_limits<T>::max();
    T m_rnorm = std::numeric_limits<T>::max();
    T m_bnorm = std::numeric_limits<T>::max();
 
    std::string m_print_ident;
    std::size_t m_setup_peak_bytes = 0;

    void note_memory (int lev, char const* stage)
    {
        auto const [bytes, reserved] = detail::arena_bytes();
        m_setup_peak_bytes = std::max(m_setup_peak_bytes, bytes);
        if (m_verbose >= 2 && bytes > 0) {
            amrex::Print() << m_print_ident << "AlgMG: level " << lev << ": after " << stage
                           << ": arena in use " << double(bytes)/double(1<<20)
                           << " MB, reserved " << double(reserved)/double(1<<20) << " MB"
#ifdef AMREX_USE_GPU
                           << ", device free " << double(Gpu::Device::freeMemAvailable())/double(1<<20) << " MB"
#endif
                           << "\n";
        }
    }
 
    SpMatrix<T>* m_mat = nullptr;
    Vector<SpMatrix<T>*> m_a;                        // matrix on each level
    Vector<std::unique_ptr<SpMatrix<T>>> m_crsemat;  // owns levels 1..n-1
    Vector<SpMatrix<T>> m_P;   // m_P[lev] interpolates lev+1 -> lev
    Vector<SpMatrix<T>> m_R;   // m_R[lev] restricts lev -> lev+1
 
    Vector<AlgVector<T>> m_res;    // rhs - A(sol)
    Vector<AlgVector<T>> m_cor;    // cor = A^-1 (res)
    Vector<AlgVector<T>> m_rescor; // res - A(cor)
 
    Vector<std::unique_ptr<JacobiSmoother<T>>> m_smoother;
    Vector<std::unique_ptr<ChebyshevSmoother<T>>> m_cheby;
    Vector<std::unique_ptr<L1GaussSeidelSmoother<T>>> m_l1gs;
    std::unique_ptr<GMRES_MV<T>> m_gmres;
    std::unique_ptr<BiCGStab_MV<T>> m_bicgstab;
 
    KrylovSolver m_krylov = KrylovSolver::none;
    std::unique_ptr<BiCGStab_MV<T>> m_krylov_bicgstab;
    std::unique_ptr<GMRES_MV<T>> m_krylov_gmres;
    std::unique_ptr<PCG_MV<T>> m_krylov_pcg;
 
    void clear_levels ();
    void build_smoothers ();
    void build_direct_bottom ();
    void direct_bottom_solve ();
    // In-place all-gather of a row-distributed dense array (stride doubles
    // per row): each rank's rows already sit at their global offset in buf.
    static void allgather_rows (double* buf, AlgPartition const& part, Long stride);
    void prepare ()
    {
        if (!m_prepared) { setup(); }
        else if (m_smoothers_stale) { build_smoothers(); }
    }
    void vcycle (AlgVector<T>& x, AlgVector<T> const& b);
    void blew_up () const;
    void solve_krylov (AlgVector<T>& a_sol, AlgVector<T> const& a_rhs, T res_target);
    void smooth (int lev, AlgVector<T>& xvec, AlgVector<T> const& bvec, int nu,
                 bool with_initial_guess, bool backward = false);
    void bottom_solve ();
 
    static std::pair<T,T> compute_rbnorms (SpMatrix<T> const& mat,
                                           AlgVector<T> const& xvec,
                                           AlgVector<T> const& bvec,
                                           AlgVector<T>& rvec);
 
    static T compute_rnorm (SpMatrix<T> const& mat, AlgVector<T> const& xvec,
                            AlgVector<T> const& bvec, AlgVector<T>& rvec);
 
//Public for CUDA
public: // NOLINT
 
    void setup ();

    bool coarsen_level (int lev, SpMatrix<T>& A, SpMatrix<T>& P, AlgPartition& cpart,
                        bool& aggressive, Long& ncoarse, bool& truncated,
                        Long& nnz_untruncated);

    [[nodiscard]] SpMatrix<TS> create_soc (SpMatrix<T> const& A, T scale) const;

    static T strength_scale (SpMatrix<T> const& A);

    void pmis (SpMatrix<TS>& S, SpMatrix<TS>& ST, Gpu::DeviceVector<int>& state,
               bool isolated_as_coarse = false) const;

    static AlgPartition coarse_numbering (Gpu::DeviceVector<int> const& state,
                                         Gpu::DeviceVector<Long>& cidx,
                                         int which = PointType::coarse);

    void project_out (AlgVector<T>& x)
    {
        if (!m_singular) { return; }
        T const mean = x.sum() / T(x.partition().numGlobalRows());
        auto* AMREX_RESTRICT px = x.data();
        ParallelForOMP(x.numLocalRows(), [=] AMREX_GPU_DEVICE (Long i) noexcept { px[i] -= mean; });
        Gpu::streamSynchronize();
    }

    static SpMatrix<T> build_interp (SpMatrix<T>& A, SpMatrix<TS> const& S,
                                     Gpu::DeviceVector<int> const& state,
                                     Gpu::DeviceVector<Long> const& cidx,
                                     AlgPartition const& cpart);

    static SpMatrix<T> truncate_interp (SpMatrix<T>& P, AlgPartition const& cpart,
                                        int max_elmts, T trunc_factor);
    static void normalize_rows (SpMatrix<T>& P);

    static SpMatrix<TS> second_pass_strength (SpMatrix<TS>& S,
                                             Gpu::DeviceVector<int> const& state,
                                             Gpu::DeviceVector<Long> const& cidx,
                                             AlgPartition const& cpart);

    static SpMatrix<T> select_coarse_rows (SpMatrix<T> const& P,
                                           Gpu::DeviceVector<int> const& state,
                                           Gpu::DeviceVector<Long> const& cidx,
                                           AlgPartition const& cpart);

    SpMatrix<T> interpolation (SpMatrix<T>& A, SpMatrix<TS>& S, SpMatrix<TS> const& ST,
                               Gpu::DeviceVector<int> const& state,
                               Gpu::DeviceVector<Long> const& cidx,
                               AlgPartition const& cpart, T scale) const;

    static SpMatrix<T> build_interp_mm (SpMatrix<T>& A, SpMatrix<TS>& S,
                                        SpMatrix<TS> const& ST,
                                        Gpu::DeviceVector<int> const& state,
                                        Gpu::DeviceVector<Long> const& cidx,
                                        AlgPartition const& cpart, T scale,
                                        bool plus_i, int const* row_state = nullptr,
                                        int max_elmts = 0, T trunc_factor = T(0),
                                        Long* nnz_untruncated = nullptr);
 
#ifndef AMREX_USE_GPU
    static SpMatrix<T> build_interp_mm_host (SpMatrix<T>& A, SpMatrix<TS>& S,
                                             SpMatrix<TS> const& ST,
                                             Gpu::DeviceVector<int> const& state,
                                             Gpu::DeviceVector<Long> const& cidx,
                                             AlgPartition const& cpart, T scale, bool plus_i,
                                             int max_elmts, T trunc_factor,
                                             Long& nnz_untruncated);
#endif
};
 
template <typename T>
AlgMG<T>::AlgMG (SpMatrix<T>& a_mat)
{
    define(a_mat);
}
 
template <typename T>
void AlgMG<T>::define (SpMatrix<T>& a_mat)
{
    m_mat = &a_mat;
    m_prepared = false;
    clear_levels();
    m_bproj.reset();
    m_krylov_bicgstab.reset();
    m_krylov_gmres.reset();
    m_krylov_pcg.reset();
}
 
template <typename T>
bool AlgMG<T>::updateMatrix (SpMatrix<T>& a_mat)
{
    if (!m_prepared || m_a.empty()) {
        define(a_mat);
        return false;
    }
    AMREX_ALWAYS_ASSERT_WITH_MESSAGE(a_mat.partition() == m_mat->partition(),
        "AlgMG::updateMatrix: the matrix must have the same row partition");
    m_mat = &a_mat;
    m_mat->setColumnPartition(m_mat->partition()); // as in setup
    m_a[0] = m_mat;
    // The Krylov solvers hold the matrix
    m_krylov_bicgstab.reset();
    m_krylov_gmres.reset();
    m_krylov_pcg.reset();
    if (numLevels() == 1) { // the finest level is the bottom
        m_bottom_n = -1;
        m_bottom_lu.clear();
        m_bottom_piv.clear();
    }
    m_smoothers_stale = true;
    return true;
}
 
template <typename T>
void AlgMG<T>::clear_levels ()
{
    m_a.clear();
    m_crsemat.clear();
    m_P.clear();
    m_R.clear();
    m_res.clear();
    m_cor.clear();
    m_rescor.clear();
    m_smoother.clear();
    m_cheby.clear();
    m_l1gs.clear();
    m_gmres.reset();
    m_bicgstab.reset();
    m_bottom_n = -1;
    m_bottom_lu.clear();
    m_bottom_piv.clear();
}
 
template <typename T>
void AlgMG<T>::solve (AlgVector<T>& a_sol, AlgVector<T> const& a_rhs)
{
    BL_PROFILE("AlgMG::solve");
 
    prepare();
 
    // Singular: iterate on the rhs with its mean removed.
    AlgVector<T> const* prhs = &a_rhs;
    if (m_singular) {
        if (!m_bproj) { m_bproj = std::make_unique<AlgVector<T>>(m_mat->partition()); }
        m_bproj->copy(a_rhs);
        project_out(*m_bproj);
        prhs = m_bproj.get();
    }
    AlgVector<T> const& rhs = *prhs;
 
    auto norms = compute_rbnorms(*m_mat, a_sol, rhs, m_res[0]);
    m_bnorm = norms.first;
    m_init_rnorm = m_rnorm = norms.second;
    m_niters = 0;
 
    if (m_bnorm == 0) {
        a_sol.setVal(0); // also the zero-mean solution of a singular system
        m_rnorm = 0;
        return;
    }
 
    if (m_verbose >= 1) {
        amrex::Print() << m_print_ident << "AlgMG: Initial rhs      2-norm = " << m_bnorm << "\n"
                       << m_print_ident << "AlgMG: Initial residual 2-norm = " << m_rnorm << "\n";
    }
 
    T res_target = std::max(m_bnorm * m_reltol, m_abstol);
    if (m_fixediter <= 0 && m_rnorm <= res_target) {
        if (m_verbose >= 1) {
            amrex::Print() << m_print_ident << "AlgMG: No iterations needed\n";
        }
        project_out(a_sol);
        return;
    }
 
    if (m_krylov != KrylovSolver::none) {
        solve_krylov(a_sol, rhs, res_target);
        project_out(a_sol);
        return;
    }
 
    int const niters = (m_fixediter > 0) ? m_fixediter : m_maxiter;
    for (int iter = 0; iter < niters; ++iter)
    {
        vcycle(m_cor[0], m_res[0]);
        a_sol.plusAsync(m_cor[0]);
 
        m_rnorm = compute_rnorm(*m_mat, a_sol, rhs, m_res[0]);
        m_niters = iter+1;
 
        if (m_verbose >= 2) {
            amrex::Print() << m_print_ident << "AlgMG: Iteration " << std::setw(3)
                           << iter+1 << ": rnorm, rnorm/bnorm = "
                           << m_rnorm << ", " << m_rnorm/m_bnorm << "\n";
        }
 
        bool converged = (m_fixediter <= 0) && (m_rnorm <= res_target);
        if (converged) {
            if (m_verbose >= 1) {
                amrex::Print() << m_print_ident << "AlgMG: Final Iter. " << iter+1
                               << ": rnorm, rnorm/bnorm = "
                               << m_rnorm << ", " << m_rnorm/m_bnorm << "\n";
            }
            break;
        } else if (m_rnorm > T(1.e20) * m_init_rnorm || amrex::isnan(m_rnorm)) {
            if (m_verbose > 0) {
                amrex::Print() << m_print_ident << "AlgMG: Failing to converge after " << iter+1 << " iterations."
                               << " rnorm, rnorm/bnorm = "
                               << m_rnorm << ", " << m_rnorm/m_bnorm << "\n";
            }
            blew_up();
        }
    }
 
    if (m_verbose >= 1 && m_fixediter <= 0 && m_rnorm > res_target) {
        amrex::Print() << m_print_ident << "AlgMG: Not converged after " << m_niters
                       << " iterations. rnorm/bnorm = " << m_rnorm/m_bnorm << "\n";
    }
    project_out(a_sol);
}
 
template <typename T>
void AlgMG<T>::precond (AlgVector<T>& x, AlgVector<T> const& b)
{
    prepare();
    vcycle(x, b);
    project_out(x);
}
 
template <typename T>
void AlgMG<T>::blew_up () const
{
    if (m_throw_exception) {
        throw std::runtime_error("AlgMG blew up.");
    } else {
        amrex::Abort("AlgMG failing so lets stop here");
    }
}
 
template <typename T>
void AlgMG<T>::solve_krylov (AlgVector<T>& a_sol, AlgVector<T> const& a_rhs, T res_target)
{
    BL_PROFILE("AlgMG::solve_krylov");
 
    // Bound on every solve so that a moved AlgMG object stays usable.
    auto pc = [this] (AlgVector<T>& x, AlgVector<T> const& b) { this->precond(x, b); };
    int const niters = (m_fixediter > 0) ? m_fixediter : m_maxiter;
    T const atol = (m_fixediter > 0) ? T(0) : res_target;
    int const verbose = std::max(0, m_verbose-1);
    T krylov_rnorm = T(0);
 
    // Converge on the absolute target reltol*|b|, as the V-cycle iteration.
    // Restart if the recurrence residual converged before the true one.
    auto run = [&] (auto& s) {
        s.setVerbose(verbose);
        s.setInitialGuessNonzero(true);
        m_niters = 0;
        while (true) {
            s.setMaxIters(niters - m_niters);
            s.solve(a_sol, a_rhs, T(0), atol);
            m_niters += s.getNumIters();
            krylov_rnorm = s.getResidualNorm();
            m_rnorm = compute_rnorm(*m_mat, a_sol, a_rhs, m_res[0]);
            if (m_fixediter > 0 || m_rnorm <= res_target || krylov_rnorm > res_target ||
                m_niters >= niters || s.getNumIters() == 0 || amrex::isnan(m_rnorm)) {
                break;
            }
            if (m_verbose >= 2) {
                amrex::Print() << m_print_ident << "AlgMG: restarting Krylov solver at iteration "
                               << m_niters << ", rnorm/bnorm = " << m_rnorm/m_bnorm << "\n";
            }
        }
    };
 
    char const* name = "";
    if (m_krylov == KrylovSolver::bicgstab) {
        name = "BiCGStab";
        if (!m_krylov_bicgstab) {
            m_krylov_bicgstab = std::make_unique<BiCGStab_MV<T>>(m_mat);
        }
        m_krylov_bicgstab->setPrecond(pc);
        run(m_krylov_bicgstab->getSolver());
    } else if (m_krylov == KrylovSolver::gmres) {
        name = "GMRES";
        if (!m_krylov_gmres) {
            m_krylov_gmres = std::make_unique<GMRES_MV<T>>(m_mat);
        }
        m_krylov_gmres->setPrecond(pc);
        run(m_krylov_gmres->getSolver());
    } else {
        name = "PCG";
        if (!m_krylov_pcg) {
            m_krylov_pcg = std::make_unique<PCG_MV<T>>(m_mat);
        }
        m_krylov_pcg->setPrecond(pc);
        run(m_krylov_pcg->getSolver());
    }
 
    if (amrex::isnan(m_rnorm) || m_rnorm > T(1.e20) * m_init_rnorm) { blew_up(); }
 
    if (m_verbose >= 1) {
        amrex::Print() << m_print_ident << "AlgMG(" << name << "): Final Iter. " << m_niters
                       << ": rnorm, rnorm/bnorm = " << m_rnorm << ", " << m_rnorm/m_bnorm;
        if (m_verbose >= 2) {
            amrex::Print() << " (Krylov recurrence " << krylov_rnorm << ")";
        }
        amrex::Print() << "\n";
        if (m_fixediter <= 0 && m_rnorm > res_target) {
            amrex::Print() << m_print_ident << "AlgMG(" << name << "): Not converged after "
                           << m_niters << " iterations.\n";
        }
    }
}
 
template <typename T>
std::pair<T,T> AlgMG<T>::compute_rbnorms (SpMatrix<T> const& mat,
                                        AlgVector<T> const& xvec,
                                        AlgVector<T> const& bvec,
                                        AlgVector<T>& rvec)
{
    auto bnorm = bvec.norm2(true);
 
    amrex::computeResidual(rvec, mat, xvec, bvec);
    auto rnorm = rvec.norm2(true);
 
    if (ParallelContext::NProcsSub() > 1) {
        bnorm = Math::powi<2>(bnorm);
        rnorm = Math::powi<2>(rnorm);
        ParallelAllReduce::Sum<T>({bnorm,rnorm}, ParallelContext::CommunicatorSub());
        bnorm = std::sqrt(bnorm);
        rnorm = std::sqrt(rnorm);
    }
 
    return {bnorm, rnorm};
}
 
template <typename T>
T AlgMG<T>::compute_rnorm (SpMatrix<T> const& mat, AlgVector<T> const& xvec,
                         AlgVector<T> const& bvec, AlgVector<T>& rvec)
{
    amrex::computeResidual(rvec, mat, xvec, bvec);
    return rvec.norm2();
}
 
template <typename T>
void AlgMG<T>::vcycle (AlgVector<T>& x, AlgVector<T> const& b)
{
    BL_PROFILE("AlgMG::vcycle");
 
    int const nlev = numLevels();
    int const bottom = nlev-1;
 
    if (nlev == 1) { // bottom_solve works on the level vectors
        if (&b != &m_res[0]) { m_res[0].copyAsync(b); }
        bottom_solve();
        if (&x != &m_cor[0]) { x.copyAsync(m_cor[0]); }
        return;
    }
 
    // Level 0 uses the caller's vectors, the coarse levels their own.
    auto cor = [&] (int lev) -> AlgVector<T>& { return (lev == 0) ? x : m_cor[lev]; };
    auto res = [&] (int lev) -> AlgVector<T> const& { return (lev == 0) ? b : m_res[lev]; };
 
    smooth(0, x, b, numPreSmooth(), false);
 
    for (int lev = 0; lev < bottom; ++lev) {
        computeResidual(m_rescor[lev], *m_a[lev], cor(lev), res(lev));
        SpMV(m_res[lev+1], m_R[lev], m_rescor[lev]);
        if (lev+1 == bottom) {
            bottom_solve();
        } else {
            smooth(lev+1, m_cor[lev+1], m_res[lev+1], numPreSmooth(), false);
        }
    }
 
    for (int lev = bottom-1; lev >= 0; --lev) {
        SpMV(m_rescor[lev], m_P[lev], m_cor[lev+1]);
        cor(lev).plusAsync(m_rescor[lev]);
        smooth(lev, cor(lev), res(lev), numPostSmooth(), true, true);
    }
}
 
template <typename T>
void AlgMG<T>::smooth (int lev, AlgVector<T>& xvec, AlgVector<T> const& bvec, int nu,
                     bool with_initial_guess, bool backward)
{
    if (m_smoother_type == Smoother::l1_gauss_seidel) {
        auto& smoother = *m_l1gs[lev];
        smoother.setNumIters(nu);
        smoother(xvec, bvec, with_initial_guess, backward);
    } else if (m_smoother_type == Smoother::chebyshev) {
        auto& smoother = *m_cheby[lev];
        smoother.setNumIters(nu);
        smoother(xvec, bvec, with_initial_guess);
    } else {
        auto& smoother = *m_smoother[lev];
        smoother.setNumIters(nu);
        smoother(xvec, bvec, with_initial_guess);
    }
}
 
template <typename T>
void AlgMG<T>::bottom_solve ()
{
    BL_PROFILE("AlgMG::bottom_solve");
 
    int const lev = numLevels()-1;
    project_out(m_res[lev]); // keep the coarsest solve consistent
    auto smooth_bottom = [&] () {
        if (m_smoother_type == Smoother::l1_gauss_seidel) {
            // Forward then backward sweeps keep the cycle symmetric.
            int const nu = (numBottomSmooth()+1)/2;
            smooth(lev, m_cor[lev], m_res[lev], nu, false);
            smooth(lev, m_cor[lev], m_res[lev], nu, true, true);
        } else {
            smooth(lev, m_cor[lev], m_res[lev], numBottomSmooth(), false);
        }
    };
    // As in MLMG, a Krylov solve that does not reduce the residual is
    // discarded; its iterate can be far worse than no correction. The
    // solver starts from zero unless told otherwise.
    auto krylov_bottom = [&] (auto& krylov) {
        krylov.setVerbose(m_bottom_verbose);
        krylov.getSolver().setMaxIters(m_bottom_maxiter);
        krylov.solve(m_cor[lev], m_res[lev], m_bottom_reltol, T(0));
        auto const& solver = krylov.getSolver();
        T const rnorm0 = solver.getInitialResidualNorm();
        if (rnorm0 > T(0) && !(solver.getResidualNorm() < rnorm0)) {
            if (m_verbose >= 2) {
                amrex::Print() << m_print_ident << "AlgMG: bottom solve failed, smoothing instead\n";
            }
            smooth_bottom();
        }
    };
    auto bottom_solver = m_bottom_solver;
    if (bottom_solver == BottomSolver::direct) {
        if (m_bottom_n < 0) { build_direct_bottom(); }
        if (m_bottom_n == 0) { bottom_solver = BottomSolver::jacobi; }
    }
    switch (bottom_solver) {
    case BottomSolver::direct:
        direct_bottom_solve();
        break;
    case BottomSolver::jacobi:
        smooth_bottom();
        break;
    case BottomSolver::gmres:
        if (!m_gmres) {
            m_gmres = std::make_unique<GMRES_MV<T>>(m_a[lev]);
            m_gmres->setPrecond(JacobiSmoother<T>(m_a[lev], m_smoother_type != Smoother::jacobi));
        }
        krylov_bottom(*m_gmres);
        break;
    case BottomSolver::bicgstab:
        if (!m_bicgstab) {
            m_bicgstab = std::make_unique<BiCGStab_MV<T>>(m_a[lev]);
            m_bicgstab->setPrecond(JacobiSmoother<T>(m_a[lev], m_smoother_type != Smoother::jacobi));
        }
        krylov_bottom(*m_bicgstab);
        break;
    }
    project_out(m_cor[lev]);
}
 
template <typename T>
void AlgMG<T>::setup ()
{
    BL_PROFILE("AlgMG::setup");
 
    ++m_num_setups;
    clear_levels();
    m_a.push_back(m_mat);
    m_mat->setColumnPartition(m_mat->partition()); // square: local column indexing
    m_setup_peak_bytes = 0;
    note_memory(0, "start");
 
    // The coarsening follows De Sterck, Yang & Heys [2].
 
    for (int lev = 0; lev+1 < m_max_levels; ++lev)
    {
        auto& A = *m_a[lev];
        if (A.numGlobalRows() <= m_max_coarse_size) { break; }
 
        SpMatrix<T> P;
        AlgPartition cpart;
        bool aggressive = false;
        Long ncoarse = 0;
        bool truncated = false;
        Long nnz0 = 0;
        if (!coarsen_level(lev, A, P, cpart, aggressive, ncoarse, truncated, nnz0)) { break; }
        note_memory(lev, "strength release");
        if (m_p_max_elmts > 0 || m_trunc_factor > T(0)) {
            if (!truncated) {
                nnz0 = P.numLocalNonZeros();
                P = truncate_interp(P, cpart, m_p_max_elmts, m_trunc_factor);
                note_memory(lev, "truncation");
            }
            if (m_verbose >= 2) {
                Long nnz[2] = {nnz0, P.numLocalNonZeros()};
                ParallelAllReduce::Sum(nnz, 2, ParallelContext::CommunicatorSub());
                amrex::Print() << m_print_ident << "AlgMG: level " << lev
                               << ": P truncated from " << nnz[0] << " to "
                               << nnz[1] << " nonzeros\n";
            }
        }
        // P 1 = 1 only to single precision, since the weights come from the
        // strength matrix; singular problems need it exactly.
        if (m_singular) { normalize_rows(P); }
        if (m_verbose >= 2 && aggressive) {
            amrex::Print() << m_print_ident << "AlgMG: level " << lev
                           << ": aggressive coarsening, " << ncoarse << " -> "
                           << cpart.numGlobalRows() << " C-points\n";
        }
        if (m_verbose >= 3) {
            // P should reproduce constants: row sums close to one.
            AlgVector<T> ones(cpart), rs(A.partition());
            ones.setVal(T(1));
            SpMV(rs, P, ones);
            Long nzero = Reduce::Sum<Long>(rs.numLocalRows(),
                [q = rs.data()] AMREX_GPU_DEVICE (Long i) -> Long { return q[i] == T(0); });
            T rmin = Reduce::Min<T>(rs.numLocalRows(),
                [q = rs.data()] AMREX_GPU_DEVICE (Long i) -> T { return q[i]; });
            T rmax = Reduce::Max<T>(rs.numLocalRows(),
                [q = rs.data()] AMREX_GPU_DEVICE (Long i) -> T { return q[i]; });
            ParallelAllReduce::Sum(nzero, ParallelContext::CommunicatorSub());
            ParallelAllReduce::Min(rmin, ParallelContext::CommunicatorSub());
            ParallelAllReduce::Max(rmax, ParallelContext::CommunicatorSub());
            amrex::Print() << m_print_ident << "AlgMG: level " << lev << ": P row sums in ["
                           << rmin << ", " << rmax << "], " << nzero << " empty rows\n";
        }
        auto R  = amrex::transpose(P, cpart);
        note_memory(lev, "R");
        SpMatrix<T> Ac;
#ifndef AMREX_USE_GPU
        Ac = amrex::RAP(R, A, P, cpart);
#else
        {
            // R (A P): measured smaller and faster than (R A) P, whose
            // intermediate gathers several fine rows per coarse row.
            auto AP = amrex::SpGEMM(A, P, cpart);
            note_memory(lev, "A*P");
            Ac = amrex::SpGEMM(R, AP, cpart);
        }
#endif
        note_memory(lev, "coarse operator");
 
        m_P.push_back(std::move(P));
        m_R.push_back(std::move(R));
        m_crsemat.push_back(std::make_unique<SpMatrix<T>>(std::move(Ac)));
        m_a.push_back(m_crsemat.back().get());
    }
 
    int const nlev = numLevels();
    for (int lev = 0; lev < nlev; ++lev) {
        m_res.emplace_back(m_a[lev]->partition());
        m_cor.emplace_back(m_a[lev]->partition());
        if (lev < nlev-1) {
            m_rescor.emplace_back(m_a[lev]->partition());
        }
    }
    build_smoothers();
 
    if (m_verbose >= 1) {
        Long nnz0 = 0, nnz_total = 0;
        for (int lev = 0; lev < nlev; ++lev) {
            Long nnz = m_a[lev]->numLocalNonZeros();
            ParallelAllReduce::Sum(nnz, ParallelContext::CommunicatorSub());
            if (lev == 0) { nnz0 = nnz; }
            nnz_total += nnz;
            amrex::Print() << m_print_ident << "AlgMG: level " << lev << ": "
                           << m_a[lev]->numGlobalRows() << " rows, "
                           << nnz << " nonzeros\n";
        }
        amrex::Print() << m_print_ident << "AlgMG: operator complexity = "
                       << double(nnz_total)/double(std::max(nnz0,Long(1))) << "\n";
        if (m_setup_peak_bytes > 0) {
            auto peak = m_setup_peak_bytes;
            ParallelAllReduce::Max(peak, ParallelContext::CommunicatorSub());
            amrex::Print() << m_print_ident << "AlgMG: setup peak arena use = "
                           << double(peak)/double(1<<20) << " MB ("
                           << double(peak)/double(m_mat->numGlobalRows())
                           << " B per row)\n";
        }
    }
 
    m_prepared = true;
}
 
template <typename T>
void AlgMG<T>::build_direct_bottom ()
{
    BL_PROFILE("AlgMG::build_direct_bottom");
    int const lev = numLevels()-1;
    auto const& A = *m_a[lev];
    Long const n = A.numGlobalRows();
    m_bottom_n = 0;
    m_bottom_lu.clear();
    m_bottom_piv.clear();
    if (n < 1 || n > max_direct_size) {
        if (m_verbose >= 1 && n > max_direct_size) {
            amrex::Print() << m_print_ident << "AlgMG: coarsest level has " << n
                           << " rows, using smoother sweeps instead of the direct bottom solver\n";
        }
        return;
    }
 
    // Dense copy of the local rows at their global offsets; gathered below.
    Vector<double> dense(n*n, 0.0);
    auto const pc = A.const_parcsr();
    auto to_host = [] (auto const* dptr, Long cnt) {
        Vector<std::remove_const_t<std::remove_pointer_t<decltype(dptr)>>> h(cnt);
        if (cnt > 0) { Gpu::copy(Gpu::deviceToHost, dptr, dptr+cnt, h.begin()); }
        return h;
    };
    {
        auto const& c = pc.csr0;
        auto ro = to_host(c.row_offset, (c.nrows > 0) ? c.nrows+1 : 0);
        auto ci = to_host(c.col_index, c.nnz);
        auto va = to_host(c.mat, c.nnz);
        for (Long i = 0; i < c.nrows; ++i) {
            Long const ig = pc.row_begin + i;
            for (int idx = ro[i]; idx < ro[i+1]; ++idx) {
                dense[ig*n + pc.col_begin + ci[idx]] += double(va[idx]);
            }
        }
    }
    if (pc.csr1.nrows > 0 && pc.csr1.nnz > 0) {
        auto const& c = pc.csr1;
        auto ro = to_host(c.row_offset, c.nrows+1);
        auto ci = to_host(c.col_index, c.nnz);
        auto va = to_host(c.mat, c.nnz);
        auto rmap = to_host(pc.row_map, pc.csr0.nrows); // local row -> csr1 row or -1
        int ncols = 0;
        for (auto j : ci) { ncols = std::max(ncols, j+1); }
        auto cmap = to_host(pc.col_map, ncols);
        for (Long i = 0; i < pc.csr0.nrows; ++i) {
            Long const r = rmap[i];
            if (r < 0) { continue; }
            Long const ig = pc.row_begin + i;
            for (int idx = ro[r]; idx < ro[r+1]; ++idx) {
                dense[ig*n + cmap[ci[idx]]] += double(va[idx]);
            }
        }
    }
    allgather_rows(dense.data(), A.partition(), n);
 
    // Singular: the constant vector is in the null space. Adding (1/n) 1 1^T
    // makes the matrix regular; the coarsest residual and correction have
    // their mean removed, so the added term never acts on them.
    if (m_singular) {
        double dsum = 0.0;
        for (Long i = 0; i < n; ++i) { dsum += std::abs(dense[i*n+i]); }
        double const w = (dsum/double(n))/double(n);
        for (auto& v : dense) { v += w; }
    }
 
    // LU with partial pivoting, in place (Doolittle). A pivot at roundoff
    // level relative to the matrix means a singular matrix.
    double amat = 0.0;
    for (auto v : dense) { amat = std::max(amat, std::abs(v)); }
    double const pivot_tol = double(n) * std::numeric_limits<double>::epsilon() * amat;
    Vector<int> piv(n);
    std::iota(piv.begin(), piv.end(), 0);
    for (Long i = 0; i < n; ++i) {
        Long imax = i;
        double amax = std::abs(dense[i*n+i]);
        for (Long k = i+1; k < n; ++k) {
            double const a = std::abs(dense[k*n+i]);
            if (a > amax) { amax = a; imax = k; }
        }
        if (amax <= pivot_tol) {
            if (m_verbose >= 1) {
                amrex::Print() << m_print_ident << "AlgMG: coarsest matrix is singular, "
                               << "using smoother sweeps instead of the direct bottom solver\n";
            }
            return;
        }
        if (imax != i) {
            std::swap(piv[i], piv[imax]);
            for (Long j = 0; j < n; ++j) { std::swap(dense[i*n+j], dense[imax*n+j]); }
        }
        double const dinv = 1.0/dense[i*n+i];
        for (Long j = i+1; j < n; ++j) {
            double const f = dense[j*n+i]*dinv;
            dense[j*n+i] = f;
            for (Long k = i+1; k < n; ++k) { dense[j*n+k] -= f*dense[i*n+k]; }
        }
    }
    m_bottom_lu = std::move(dense);
    m_bottom_piv = std::move(piv);
    m_bottom_n = n;
}
 
template <typename T>
void AlgMG<T>::allgather_rows (double* buf, AlgPartition const& part, Long stride)
{
#ifdef AMREX_USE_MPI
    int const nprocs = ParallelContext::NProcsSub();
    if (nprocs == 1) { return; }
    Vector<int> counts(nprocs), displs(nprocs);
    for (int p = 0; p < nprocs; ++p) {
        displs[p] = int(part[p]*stride);
        counts[p] = int((part[p+1]-part[p])*stride);
    }
    BL_MPI_REQUIRE(MPI_Allgatherv(MPI_IN_PLACE, 0, MPI_DATATYPE_NULL,
                                  buf, counts.data(), displs.data(), MPI_DOUBLE,
                                  ParallelContext::CommunicatorSub()));
#else
    amrex::ignore_unused(buf, part, stride);
#endif
}
 
template <typename T>
void AlgMG<T>::direct_bottom_solve ()
{
    int const lev = numLevels()-1;
    auto const& r = m_res[lev];
    auto& x = m_cor[lev];
    Long const n = m_bottom_n;
    Long const nl = r.numLocalRows();
    Long const b0 = r.globalBegin();
 
    // Gather the residual.
    PODVector<double> rhs(n); // uninitialized; every entry is written below
    Gpu::PinnedVector<T> loc(nl);
    if (nl > 0) {
        Gpu::copy(Gpu::deviceToHost, r.data(), r.data()+nl, loc.begin());
        for (Long i = 0; i < nl; ++i) { rhs[b0+i] = double(loc[i]); }
    }
    allgather_rows(rhs.data(), r.partition(), 1);
 
    // P A = L U: forward substitution with the unit lower part, then back.
    Vector<double> y(n);
    auto const& lu = m_bottom_lu;
    for (Long i = 0; i < n; ++i) {
        double v = rhs[m_bottom_piv[i]];
        for (Long k = 0; k < i; ++k) { v -= lu[i*n+k]*y[k]; }
        y[i] = v;
    }
    for (Long i = n-1; i >= 0; --i) {
        double v = y[i];
        for (Long k = i+1; k < n; ++k) { v -= lu[i*n+k]*y[k]; }
        y[i] = v/lu[i*n+i];
    }
 
    if (nl > 0) {
        for (Long i = 0; i < nl; ++i) { loc[i] = T(y[b0+i]); }
        Gpu::copy(Gpu::hostToDevice, loc.begin(), loc.end(), x.data());
    }
}
 
template <typename T>
void AlgMG<T>::build_smoothers ()
{
    m_smoother.clear();
    m_cheby.clear();
    m_l1gs.clear();
    m_gmres.reset();    // their preconditioner depends on the smoother type
    m_bicgstab.reset();
    int const nlev = numLevels();
    for (int lev = 0; lev < nlev; ++lev) {
        if (m_smoother_type == Smoother::l1_gauss_seidel) {
#ifdef AMREX_USE_GPU
            amrex::Abort("AlgMG: L1GaussSeidel smoother is only available in CPU builds");
#endif
            m_l1gs.push_back(std::make_unique<L1GaussSeidelSmoother<T>>(m_a[lev]));
        } else if (m_smoother_type == Smoother::chebyshev) {
            m_cheby.push_back(std::make_unique<ChebyshevSmoother<T>>(m_a[lev], true));
            m_cheby.back()->setDegree(m_cheby_degree);
            m_cheby.back()->setEigRatio(m_cheby_ratio);
        } else {
            bool const l1 = (m_smoother_type == Smoother::l1_jacobi);
            m_smoother.push_back(std::make_unique<JacobiSmoother<T>>(m_a[lev], l1));
            T const w = (m_relax_weight >= T(0)) ? m_relax_weight : (l1 ? T(4./3.) : T(2./3.));
            m_smoother.back()->setWeight(w);
        }
    }
    m_smoothers_stale = false;
}
 
template <typename T>
bool AlgMG<T>::coarsen_level (int lev, SpMatrix<T>& A, SpMatrix<T>& P, AlgPartition& cpart,
                            bool& aggressive, Long& ncoarse, bool& truncated,
                            Long& nnz_untruncated)
{
#ifdef AMREX_USE_GPU
    amrex::ignore_unused(truncated, nnz_untruncated); // set by the host paths only
#endif
    T const sscale = strength_scale(A);
    auto S = create_soc(A, sscale);
    S.setColumnPartition(S.partition());
    note_memory(lev, "S");
    auto ST = amrex::transpose(S, S.partition());
    ST.setColumnPartition(ST.partition());
 
    note_memory(lev, "strength");
    Gpu::DeviceVector<int> state;
    // Singular: an isolated point becomes C, so every row of P sums to one.
    pmis(S, ST, state, m_singular);
    note_memory(lev, "PMIS");
 
    Gpu::DeviceVector<Long> cidx;
    cpart = coarse_numbering(state, cidx);
    ncoarse = cpart.numGlobalRows();
    if (ncoarse == 0 || ncoarse == A.numGlobalRows()) { return false; }
#ifndef AMREX_USE_GPU
    bool const host_interp = m_interp_type != InterpType::direct
        && detail::spmat_comm_is_local(A.partition(), cpart);
#endif
 
    if (lev < m_aggressive_num_levels) {
        // Second PMIS pass on the C-points connected at distance two.
        auto S2 = second_pass_strength(S, state, cidx, cpart);
        S2.setColumnPartition(cpart);
        auto S2T = amrex::transpose(S2, cpart);
        S2T.setColumnPartition(cpart);
        note_memory(lev, "second-pass strength");
        Gpu::DeviceVector<int> state2;
        pmis(S2, S2T, state2, true); // isolated C-points stay coarse
        Gpu::DeviceVector<Long> cidx2;
        auto cpart2 = coarse_numbering(state2, cidx2);
        Long const nc2 = cpart2.numGlobalRows();
        if (nc2 > 0 && nc2 < ncoarse) {
            // C/F split of the fine points with respect to C2.
            Long const nrows = A.numLocalRows();
            Gpu::DeviceVector<int> statef(nrows);
            Gpu::DeviceVector<Long> cidxf(nrows);
            auto const* p1 = state.data();
            auto const* c1 = cidx.data();
            auto const* p2 = state2.data();
            auto const* c2 = cidx2.data();
            auto* pf = statef.data();
            auto* cf = cidxf.data();
            ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
            {
                bool const c = (p1[i] == PointType::coarse) &&
                               (p2[c1[i]] == PointType::coarse);
                pf[i] = c ? PointType::coarse : PointType::fine;
                cf[i] = (p1[i] == PointType::coarse) ? c2[c1[i]] : Long(0);
            });
            Gpu::streamSynchronize();
 
            bool const trunc = (m_p_max_elmts > 0 || m_trunc_factor > T(0));
            SpMatrix<T> P1;
#ifndef AMREX_USE_GPU
            if (!m_aggressive_direct_interp && host_interp) {
                Long unused;
                P1 = build_interp_mm_host(A, S, ST, state, cidx, cpart, sscale,
                                          m_interp_type == InterpType::mm_ext_i,
                                          m_p_max_elmts, m_trunc_factor, unused);
            } else if (!m_aggressive_direct_interp && m_interp_type != InterpType::direct) {
                P1 = build_interp_mm(A, S, ST, state, cidx, cpart, sscale,
                                     m_interp_type == InterpType::mm_ext_i, nullptr,
                                     m_p_max_elmts, m_trunc_factor);
            } else
#endif
            {
                P1 = m_aggressive_direct_interp ? build_interp(A, S, state, cidx, cpart)
                                                : interpolation(A, S, ST, state, cidx, cpart, sscale);
                if (trunc) { P1 = truncate_interp(P1, cpart, m_p_max_elmts, m_trunc_factor); }
            }
            note_memory(lev, "P1");
            // P2 needs only the rows of the first-pass C-points.
#ifndef AMREX_USE_GPU
            auto Pf = build_interp_mm(A, S, ST, statef, cidxf, cpart2, sscale,
                                      m_interp_type != InterpType::mm_ext,
                                      state.data(), m_p_max_elmts, m_trunc_factor);
#else
            auto Pf = build_interp_mm(A, S, ST, statef, cidxf, cpart2, sscale,
                                      m_interp_type != InterpType::mm_ext,
                                      state.data());
            if (trunc) { Pf = truncate_interp(Pf, cpart2, m_p_max_elmts, m_trunc_factor); }
#endif
            auto P2 = select_coarse_rows(Pf, state, cidx, cpart);
            note_memory(lev, "P2");
            P = amrex::SpGEMM(P1, P2, cpart2);
            cpart = cpart2;
            aggressive = true;
        }
    }
    if (!aggressive) {
#ifndef AMREX_USE_GPU
        if (host_interp) {
            P = build_interp_mm_host(A, S, ST, state, cidx, cpart, sscale,
                                     m_interp_type == InterpType::mm_ext_i,
                                     m_p_max_elmts, m_trunc_factor, nnz_untruncated);
            truncated = true;
        } else if (m_interp_type != InterpType::direct) {
            P = build_interp_mm(A, S, ST, state, cidx, cpart, sscale,
                                m_interp_type == InterpType::mm_ext_i, nullptr,
                                m_p_max_elmts, m_trunc_factor, &nnz_untruncated);
            truncated = true;
        } else
#endif
        {
            P = interpolation(A, S, ST, state, cidx, cpart, sscale);
        }
    }
    note_memory(lev, "interpolation");
    return true;
}
 
template <typename T>
void AlgMG<T>::normalize_rows (SpMatrix<T>& P)
{
    auto const rs = P.rowSum();
    auto const* AMREX_RESTRICT prs = rs.data();
    Long nzero = Reduce::Sum<Long>(rs.numLocalRows(),
        [=] AMREX_GPU_DEVICE (Long i) -> Long { return prs[i] == T(0); });
    ParallelAllReduce::Sum(nzero, ParallelContext::CommunicatorSub());
    if (nzero > 0) {
        amrex::Abort("AlgMG: singular mode needs P 1 = 1, but " + std::to_string(nzero)
                     + " rows of P are empty");
    }
    auto a = P.parcsr();
    ParallelForOMP(rs.numLocalRows(), [=] AMREX_GPU_DEVICE (Long i) noexcept
    {
        T const f = T(1) / prs[i];
        for (Long idx = a.csr0.row_offset[i]; idx < a.csr0.row_offset[i+1]; ++idx) {
            a.csr0.mat[idx] *= f;
        }
        if (detail::has_remote_row(a, i)) {
            Long const ii = a.row_map[i];
            for (Long idx = a.csr1.row_offset[ii]; idx < a.csr1.row_offset[ii+1]; ++idx) {
                a.csr1.mat[idx] *= f;
            }
        }
    });
    Gpu::streamSynchronize();
}
 
template <typename T>
T AlgMG<T>::strength_scale (SpMatrix<T> const& A)
{
    auto const& Ad = A.diagonalVector();
    auto const* AMREX_RESTRICT p = Ad.data();
    T dmax = Reduce::Max<T>(Ad.numLocalRows(),
        [=] AMREX_GPU_DEVICE (Long i) -> T { return std::abs(p[i]); });
    ParallelAllReduce::Max(dmax, ParallelContext::CommunicatorSub());
    return (dmax > T(0)) ? dmax : T(1);
}
 
template <typename T>
SpMatrix<typename AlgMG<T>::TS> AlgMG<T>::create_soc (SpMatrix<T> const& A, T scale) const
{
    BL_PROFILE("AlgMG::create_soc");
 
    auto const& a = A.const_parcsr();
    Long const nrows = a.csr0.nrows;
    int const nlocal = int(A.numLocalRows());
 
    auto const& Ad = A.diagonalVector();
    auto const* AMREX_RESTRICT A_ii = Ad.data();
    auto const threshold = m_strong_threshold;
 
    // Threshold of row i: theta times the strongest connection, relaxed
    // by a few ulps so that entries nominally on the threshold are strong
    // in every row [1]. Rows dominated by their diagonal (row sum above
    // 0.9 |a_ii|) have no strong connection.
    T const tol = T(64) * std::numeric_limits<T>::epsilon();
 
#ifndef AMREX_USE_GPU
    {
        // One pass per row on the host.
        auto const& c0 = a.csr0;
        auto const& c1 = a.csr1;
        auto rlen1 = [&] (Long i) -> Long {
            if (!detail::has_remote_row(a, i)) { return 0; }
            Long const ii = a.row_map[i];
            return c1.row_offset[ii+1] - c1.row_offset[ii];
        };
        auto csr = detail::csr_from_rows_cpu<TS,SpMatrix<TS>::template container_type,int>
            (nrows, [&] (Long i) { return c0.row_offset[i+1] - c0.row_offset[i] + rlen1(i); },
             [&] (int) {
            return [&, col = Vector<int>(), val = Vector<TS>()]
                (Long i, int const*& rc, TS const*& rv) mutable -> Long
            {
                Long const b = c0.row_offset[i];
                Long const e = c0.row_offset[i+1];
                Long const n1 = rlen1(i);
                Long const b1 = (n1 > 0) ? Long(c1.row_offset[a.row_map[i]]) : 0;
                T const sgn = std::copysign(T(1), A_ii[i]);
                T rs = T(0);
                T amin = std::numeric_limits<T>::max();
                for (Long idx = b; idx < e; ++idx) {
                    rs += c0.mat[idx];
                    if (c0.col_index[idx] != i) { amin = std::min(amin, sgn*c0.mat[idx]); }
                }
                for (Long idx = b1; idx < b1+n1; ++idx) {
                    rs += c1.mat[idx];
                    amin = std::min(amin, sgn*c1.mat[idx]);
                }
                Long n = 0;
                if (!(std::abs(rs) > T(0.9)*std::abs(A_ii[i])) && amin < T(0)) {
                    T const thr = amin * threshold * (T(1) - tol);
                    if (Long(col.size()) < e-b+n1) {
                        col.resize(e-b+n1);
                        val.resize(e-b+n1);
                    }
                    for (Long idx = b; idx < e; ++idx) {
                        if (c0.col_index[idx] != i && sgn*c0.mat[idx] <= thr) {
                            col[n] = c0.col_index[idx];
                            val[n] = TS(c0.mat[idx]/scale);
                            ++n;
                        }
                    }
                    for (Long idx = b1; idx < b1+n1; ++idx) {
                        if (sgn*c1.mat[idx] <= thr) {
                            col[n] = nlocal + c1.col_index[idx];
                            val[n] = TS(c1.mat[idx]/scale);
                            ++n;
                        }
                    }
                }
                rc = col.data();
                rv = val.data();
                return n;
            };
        });
        SpMatrix<TS> S;
        S.define_split(A.partition(), A.partition(), std::move(csr), nlocal, a.col_map, 0);
        return S;
    }
#else
 
    auto const rsA = A.rowSum();
    auto const* AMREX_RESTRICT prs = rsA.data();
    auto row_thresh = [=] AMREX_GPU_DEVICE (Long i) -> T {
        if (std::abs(prs[i]) > T(0.9)*std::abs(A_ii[i])) {
            return std::numeric_limits<T>::lowest();
        }
        auto const& c0 = a.csr0;
        T const sgn = std::copysign(T(1), A_ii[i]);
        T amin = std::numeric_limits<T>::max();
        for (Long idx = c0.row_offset[i]; idx < c0.row_offset[i+1]; ++idx) {
            if (c0.col_index[idx] != i) {
                amin = std::min(amin, sgn*c0.mat[idx]);
            }
        }
        if (detail::has_remote_row(a, i)) {
            auto const& c1 = a.csr1;
            Long const ii = a.row_map[i];
            for (Long idx = c1.row_offset[ii]; idx < c1.row_offset[ii+1]; ++idx) {
                amin = std::min(amin, sgn*c1.mat[idx]);
            }
        }
        if (amin >= T(0)) { return std::numeric_limits<T>::lowest(); }
        return amin * threshold * (T(1) - tol);
    };
    Gpu::DeviceVector<T> rthr(nrows);
    auto* AMREX_RESTRICT pthr = rthr.data();
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept { pthr[i] = row_thresh(i); });
 
    using local_csr_type = typename SpMatrix<TS>::local_csr_type;
    local_csr_type csr;
    csr.row_offset.resize(nrows+1);
    auto* AMREX_RESTRICT psrow = csr.row_offset.data();
    Long const nnz = Scan::PrefixSum<Long>
        (nrows+1,
         [=] AMREX_GPU_DEVICE (Long i) -> Long {
             if (i >= nrows) { return 0; }
             T const thr = pthr[i];
             T const sgn = std::copysign(T(1), A_ii[i]);
             Long n = 0;
             auto const& c0 = a.csr0;
             for (Long idx = c0.row_offset[i]; idx < c0.row_offset[i+1]; ++idx) {
                 if (c0.col_index[idx] != i && sgn*c0.mat[idx] <= thr) { ++n; }
             }
             if (detail::has_remote_row(a, i)) {
                 auto const& c1 = a.csr1;
                 Long const ii = a.row_map[i];
                 for (Long idx = c1.row_offset[ii]; idx < c1.row_offset[ii+1]; ++idx) {
                     if (sgn*c1.mat[idx] <= thr) { ++n; }
                 }
             }
             return n;
         },
         [=] AMREX_GPU_DEVICE (Long i, Long const& x) { psrow[i] = x; },
         Scan::Type::exclusive, Scan::retSum);
 
    csr.resize(nrows, nnz);
    auto* AMREX_RESTRICT ps = csr.mat.data();
    auto* AMREX_RESTRICT pc = csr.col_index.data();
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
    {
        T const thr = pthr[i];
        T const sgn = std::copysign(T(1), A_ii[i]);
        Long p = psrow[i];
        auto const& c0 = a.csr0;
        for (Long idx = c0.row_offset[i]; idx < c0.row_offset[i+1]; ++idx) {
            if (c0.col_index[idx] != i && sgn*c0.mat[idx] <= thr) {
                ps[p] = TS(c0.mat[idx]/scale);
                pc[p] = c0.col_index[idx];
                ++p;
            }
        }
        if (detail::has_remote_row(a, i)) {
            auto const& c1 = a.csr1;
            Long const ii = a.row_map[i];
            for (Long idx = c1.row_offset[ii]; idx < c1.row_offset[ii+1]; ++idx) {
                if (sgn*c1.mat[idx] <= thr) {
                    ps[p] = TS(c1.mat[idx]/scale);
                    pc[p] = nlocal + c1.col_index[idx];
                    ++p;
                }
            }
        }
    });
    Gpu::streamSynchronize();
 
    SpMatrix<TS> S;
    S.define_split(A.partition(), A.partition(), std::move(csr), nlocal, a.col_map, 0);
    return S;
#endif
}
 
template <typename T>
void AlgMG<T>::pmis (SpMatrix<TS>& S, SpMatrix<TS>& ST, Gpu::DeviceVector<int>& state,
                   bool isolated_as_coarse) const
{
    BL_PROFILE("AlgMG::pmis");
 
    Long const nrows = S.numLocalRows();
    int const U = PointType::undecided;
    int const C = PointType::coarse;
 
    // Measure: number of points that strongly depend on i. A negative
    // measure marks a point with no strong connection at all.
    auto const& s = S.const_parcsr();
    auto const& st = ST.const_parcsr();
    Gpu::DeviceVector<Long> measure(nrows);
    auto* AMREX_RESTRICT pm = measure.data();
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
    {
        Long n = st.csr0.row_offset[i+1] - st.csr0.row_offset[i];
        if (detail::has_remote_row(st, i)) {
            Long const ii = st.row_map[i];
            n += st.csr1.row_offset[ii+1] - st.csr1.row_offset[ii];
        }
        Long nout = s.csr0.row_offset[i+1] - s.csr0.row_offset[i];
        if (detail::has_remote_row(s, i)) {
            Long const ii = s.row_map[i];
            nout += s.csr1.row_offset[ii+1] - s.csr1.row_offset[ii];
        }
        pm[i] = (n == 0 && nout == 0) ? Long(-1) : n;
    });
 
    Gpu::DeviceVector<T> weight(nrows);
    state.resize(nrows);
    Gpu::DeviceVector<int> state2(nrows);
    auto* AMREX_RESTRICT pw = weight.data();
    auto* AMREX_RESTRICT ps = state.data();
    auto* AMREX_RESTRICT ps2 = state2.data();
 
    Long const row_begin = s.row_begin;
 
    // A point nobody depends on gets a weight below one, so it becomes C
    // only if all its dependencies are F.
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
    {
        if (pm[i] < 0) {
            pw[i] = T(0);
            ps[i] = isolated_as_coarse ? PointType::coarse : PointType::fine;
        } else {
            pw[i] = T(pm[i]) + detail::pmis_tiebreak<T>(i + row_begin);
            ps[i] = PointType::undecided;
        }
    });
 
    // Ghost weights are fixed; ghost states are gathered every sweep.
    auto const wS  = S.gatherRemote(pw);
    auto const wST = ST.gatherRemote(pw);
    auto const* AMREX_RESTRICT pwS  = wS.data();
    auto const* AMREX_RESTRICT pwST = wST.data();
 
    Gpu::DeviceVector<int> sS, sST, s2S;
 
    while (true) {
        S.gatherRemote(ps, sS);
        ST.gatherRemote(ps, sST);
        auto const* AMREX_RESTRICT psS  = sS.data();
        auto const* AMREX_RESTRICT psST = sST.data();
 
        // Independent set in S union S^T: keep only own index writes.
        ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
        {
            int st_i = ps[i];
            if (st_i == U) {
                T const wi = pw[i];
                Long const ig = i + row_begin;
                bool blocked =
                    detail::pmis_blocked(s.csr0,  i, wi, ig, pw, ps, nullptr, s.col_begin,  U) ||
                    detail::pmis_blocked(st.csr0, i, wi, ig, pw, ps, nullptr, st.col_begin, U);
                if (!blocked && detail::has_remote_row(s, i)) {
                    blocked = detail::pmis_blocked(s.csr1, s.row_map[i], wi, ig,
                                                   pwS, psS, s.col_map, 0, U);
                }
                if (!blocked && detail::has_remote_row(st, i)) {
                    blocked = detail::pmis_blocked(st.csr1, st.row_map[i], wi, ig,
                                                   pwST, psST, st.col_map, 0, U);
                }
                if (!blocked) { st_i = C; }
            }
            ps2[i] = st_i;
        });
 
        S.gatherRemote(ps2, s2S);
        auto const* AMREX_RESTRICT ps2S = s2S.data();
 
        // Undecided points that strongly depend on a C-point become F.
        ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
        {
            int st_i = ps2[i];
            if (st_i == U) {
                if (detail::csr_row_has_state(s.csr0, i, ps2, C) ||
                    (detail::has_remote_row(s, i) &&
                     detail::csr_row_has_state(s.csr1, s.row_map[i], ps2S, C))) {
                    st_i = PointType::fine;
                }
            }
            ps[i] = st_i;
        });
 
        Long nund = Reduce::Sum<Long>(nrows, [=] AMREX_GPU_DEVICE (Long i) -> Long {
            return (ps[i] == U) ? 1 : 0;
        });
        ParallelAllReduce::Sum(nund, ParallelContext::CommunicatorSub());
        if (nund == 0) { break; }
    }
 
    if (m_verbose >= 2) {
        S.gatherRemote(ps, sS);
        auto const* AMREX_RESTRICT psS = sS.data();
        Long nc = Reduce::Sum<Long>(nrows, [=] AMREX_GPU_DEVICE (Long i) -> Long {
            return (ps[i] == C) ? 1 : 0;
        });
        // C-points that strongly depend on another C-point. A point becomes
        // F only through its own dependencies, so a few one-directional
        // C-C connections remain where the strength is asymmetric.
        Long ncc = Reduce::Sum<Long>(nrows, [=] AMREX_GPU_DEVICE (Long i) -> Long {
            if (ps[i] != C) { return 0; }
            return (detail::csr_row_has_state(s.csr0, i, ps, C) ||
                    (detail::has_remote_row(s, i) &&
                     detail::csr_row_has_state(s.csr1, s.row_map[i], psS, C))) ? 1 : 0;
        });
        // F-points without a strong C neighbor get no interpolation.
        Long nf0 = Reduce::Sum<Long>(nrows, [=] AMREX_GPU_DEVICE (Long i) -> Long {
            if (ps[i] != PointType::fine) { return 0; }
            return (detail::csr_row_has_state(s.csr0, i, ps, C) ||
                    (detail::has_remote_row(s, i) &&
                     detail::csr_row_has_state(s.csr1, s.row_map[i], psS, C))) ? 0 : 1;
        });
        Long counts[3] = {nc, ncc, nf0};
        ParallelAllReduce::Sum(counts, 3, ParallelContext::CommunicatorSub());
        amrex::Print() << m_print_ident << "AlgMG: PMIS: " << counts[0] << " C-points of "
                       << S.numGlobalRows() << ", C-C strong pairs " << counts[1]
                       << ", F-points w/o C neighbor " << counts[2] << "\n";
    }
}
 
template <typename T>
AlgPartition AlgMG<T>::coarse_numbering (Gpu::DeviceVector<int> const& state,
                                       Gpu::DeviceVector<Long>& cidx, int which)
{
    Long const nrows = Long(state.size());
    auto const* AMREX_RESTRICT ps = state.data();
    cidx.resize(nrows);
    auto* AMREX_RESTRICT pc = cidx.data();
    Long const nc = Scan::PrefixSum<Long>
        (nrows,
         [=] AMREX_GPU_DEVICE (Long i) -> Long { return (ps[i] == which) ? 1 : 0; },
         [=] AMREX_GPU_DEVICE (Long i, Long const& x) { pc[i] = x; },
         Scan::Type::exclusive, Scan::retSum);
 
    int const nprocs = ParallelContext::NProcsSub();
    Vector<Long> counts(nprocs, nc);
#ifdef AMREX_USE_MPI
    // One value per rank: an all-gather, not a reduction of a P-vector.
    ParallelAllGather::AllGather(nc, counts.data(), ParallelContext::CommunicatorSub());
#endif
    Vector<Long> rows(nprocs+1, 0); // exclusive prefix sum of counts
    std::partial_sum(counts.begin(), counts.end(), rows.begin()+1);
    return AlgPartition(std::move(rows));
}
 
template <typename T>
SpMatrix<T> AlgMG<T>::build_interp (SpMatrix<T>& A, SpMatrix<TS> const& S,
                                  Gpu::DeviceVector<int> const& state,
                                  Gpu::DeviceVector<Long> const& cidx,
                                  AlgPartition const& cpart)
{
    BL_PROFILE("AlgMG::build_interp");
 
    // Classical direct interpolation [1].
 
    auto const& a = A.const_parcsr();
    Long const nrows = a.csr0.nrows;
    auto const* AMREX_RESTRICT ps = state.data();
    auto const* AMREX_RESTRICT pc = cidx.data();
    Long const cbegin = cpart.globalRowBegin();
    int const nlocal_c = int(cpart.numLocalRows());
 
    // Global coarse index of local rows, and ghost copies for A's remote
    // columns (the remote column list of P).
    Gpu::DeviceVector<Long> gcidx(nrows);
    auto* AMREX_RESTRICT pgc = gcidx.data();
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept { pgc[i] = pc[i] + cbegin; });
    auto const state_r = A.gatherRemote(ps);
    auto const gcidx_r = A.gatherRemote(pgc);
 
    detail::DirectInterpRow<T,TS> const w{a, S.const_parcsr(), ps, pc,
                                       state_r.data(), nlocal_c,
                                       int(PointType::coarse)};
 
    using local_csr_type = typename SpMatrix<T>::local_csr_type;
    local_csr_type csr;
    csr.row_offset.resize(nrows+1);
    auto* AMREX_RESTRICT prow = csr.row_offset.data();
 
    Long const nnz = Scan::PrefixSum<Long>
        (nrows+1,
         [=] AMREX_GPU_DEVICE (Long i) -> Long {
             if (i >= nrows) { return 0; }
             if (ps[i] == PointType::coarse) { return 1; }
             Long n = 0;
             T diag;
             w.walk(i, diag, [&] (int, T, bool strong_c) { if (strong_c) { ++n; } });
             return n;
         },
         [=] AMREX_GPU_DEVICE (Long i, Long const& x) { prow[i] = x; },
         Scan::Type::exclusive, Scan::retSum);
 
    csr.resize(nrows, nnz);
    auto* AMREX_RESTRICT pmat = csr.mat.data();
    auto* AMREX_RESTRICT pcol = csr.col_index.data();
 
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
    {
        Long p = prow[i];
        if (ps[i] == PointType::coarse) {
            pcol[p] = int(pc[i]);
            pmat[p] = T(1);
            return;
        }
 
        T sum_n_neg = T(0), sum_n_pos = T(0), sum_p_neg = T(0), sum_p_pos = T(0);
        T diag;
        w.walk(i, diag, [&] (int, T v, bool strong_c) {
            if (v < T(0)) {
                sum_n_neg += v;
                if (strong_c) { sum_p_neg += v; }
            } else {
                sum_n_pos += v;
                if (strong_c) { sum_p_pos += v; }
            }
        });
 
        T alfa = T(0), beta = T(0);
        if (sum_p_neg != T(0)) { alfa = sum_n_neg / sum_p_neg; } else { diag += sum_n_neg; }
        if (sum_p_pos != T(0)) { beta = sum_n_pos / sum_p_pos; } else { diag += sum_n_pos; }
        if (diag == T(0)) { diag = T(1); } // no interpolation weights anyway
 
        T unused;
        w.walk(i, unused, [&] (int cj, T v, bool strong_c) {
            if (strong_c) {
                pcol[p] = cj;
                pmat[p] = ((v < T(0)) ? -alfa*v : -beta*v) / diag;
                ++p;
            }
        });
    });
    Gpu::streamSynchronize();
 
    SpMatrix<T> P;
    P.define_split(A.partition(), cpart, std::move(csr), nlocal_c, gcidx_r.data(), 0);
    return P;
}
 
namespace detail {

template <typename T, int KMAX>
struct TruncRow
{
    ParCsr<T const> p;
    int max_elmts;
    T trunc_factor;
    int nlocal_cols;

    template <typename F>
    AMREX_GPU_HOST_DEVICE void walk (Long i, F const& f) const
    {
        auto const& c0 = p.csr0;
        for (Long idx = c0.row_offset[i]; idx < c0.row_offset[i+1]; ++idx) {
            f(c0.col_index[idx], c0.mat[idx]);
        }
        if (has_remote_row(p, i)) {
            auto const& c1 = p.csr1;
            Long const ii = p.row_map[i];
            for (Long idx = c1.row_offset[ii]; idx < c1.row_offset[ii+1]; ++idx) {
                f(nlocal_cols + c1.col_index[idx], c1.mat[idx]);
            }
        }
    }

    AMREX_GPU_HOST_DEVICE T threshold (Long i, T& sum_pos, T& sum_neg) const
    {
        T amax = T(0);
        sum_pos = T(0);
        sum_neg = T(0);
        walk(i, [&] (int, T v) {
            amax = std::max(amax, std::abs(v));
            if (v > T(0)) { sum_pos += v; } else { sum_neg += v; }
        });
        return trunc_factor * amax;
    }

    [[nodiscard]] AMREX_GPU_HOST_DEVICE int count (Long i, T thresh) const
    {
        int n = 0;
        walk(i, [&] (int, T v) { if (std::abs(v) >= thresh) { ++n; } });
        return n;
    }

    AMREX_GPU_HOST_DEVICE
    int select (Long i, T thresh, int* cols, T* vals) const
    {
        int const kmax = std::min(max_elmts, KMAX);
        int n = 0;
        walk(i, [&] (int col, T v) {
            T const av = std::abs(v);
            if (av < thresh) { return; }
            if (n == kmax && av <= std::abs(vals[n-1])) { return; }
            int pos = (n < kmax) ? n : n-1;
            while (pos > 0 && std::abs(vals[pos-1]) < av) {
                cols[pos] = cols[pos-1];
                vals[pos] = vals[pos-1];
                --pos;
            }
            cols[pos] = col;
            vals[pos] = v;
            if (n < kmax) { ++n; }
        });
        return n;
    }
};
 
#ifndef AMREX_USE_GPU
template <typename T, int KMAX>
Long truncate_row_host (int* col, T* val, Long n, int max_elmts, T trunc_factor, int nc)
{
    // The row as a one-row CSR for TruncRow.
    int ro[2] = {0, int(n)};
    ParCsr<T const> prow;
    prow.csr0 = CsrView<T const,int>{val, col, ro, n, 1};
    TruncRow<T,KMAX> const tr{prow, max_elmts, trunc_factor, nc};
    T sum_pos, sum_neg;
    T const thresh = tr.threshold(0, sum_pos, sum_neg);
    int tcol[KMAX];
    T tval[KMAX];
    Long nk = 0;
    T kept_pos = T(0), kept_neg = T(0);
    if (max_elmts > 0) {
        nk = tr.select(0, thresh, tcol, tval);
        for (Long k = 0; k < nk; ++k) {
            if (tval[k] > T(0)) { kept_pos += tval[k]; } else { kept_neg += tval[k]; }
        }
    } else {
        for (Long k = 0; k < n; ++k) {
            if (std::abs(val[k]) >= thresh) {
                if (val[k] > T(0)) { kept_pos += val[k]; } else { kept_neg += val[k]; }
            }
        }
    }
    T const fpos = (kept_pos != T(0)) ? sum_pos/kept_pos : T(1);
    T const fneg = (kept_neg != T(0)) ? sum_neg/kept_neg : T(1);
    if (max_elmts > 0) {
        // Kept by decreasing magnitude; back to column order.
        for (Long k = 0; k < nk; ++k) {
            Long m = k;
            for (; m > 0 && col[m-1] > tcol[k]; --m) {
                col[m] = col[m-1];
                val[m] = val[m-1];
            }
            col[m] = tcol[k];
            val[m] = tval[k] * ((tval[k] > T(0)) ? fpos : fneg);
        }
    } else {
        for (Long k = 0; k < n; ++k) {
            if (std::abs(val[k]) >= thresh) {
                col[nk] = col[k];
                val[nk] = val[k] * ((val[k] > T(0)) ? fpos : fneg);
                ++nk;
            }
        }
    }
    return nk;
}
#endif
 
}
 
namespace detail {

template <typename T, typename F>
typename SpMatrix<T>::local_csr_type
compact_rows (ParCsr<T const> const& p, Long nc, int const* AMREX_RESTRICT state,
              Long const* AMREX_RESTRICT cidx, int c, F const& colmap)
{
    Long const nrows = p.csr0.nrows;
 
    // Offsets by fine row, then gathered at the selected rows.
    Gpu::DeviceVector<Long> off(nrows+1);
    auto* AMREX_RESTRICT poff = off.data();
    Long const nnz = Scan::PrefixSum<Long>
        (nrows+1,
         [=] AMREX_GPU_DEVICE (Long i) -> Long {
             if (i >= nrows || state[i] != c) { return 0; }
             Long n = 0;
             auto const& c0 = p.csr0;
             for (Long idx = c0.row_offset[i]; idx < c0.row_offset[i+1]; ++idx) {
                 if (colmap(c0.col_index[idx], false) >= 0) { ++n; }
             }
             if (has_remote_row(p, i)) {
                 auto const& c1 = p.csr1;
                 Long const ii = p.row_map[i];
                 for (Long idx = c1.row_offset[ii]; idx < c1.row_offset[ii+1]; ++idx) {
                     if (colmap(c1.col_index[idx], true) >= 0) { ++n; }
                 }
             }
             return n;
         },
         [=] AMREX_GPU_DEVICE (Long i, Long const& x) { poff[i] = x; },
         Scan::Type::exclusive, Scan::retSum);
 
    typename SpMatrix<T>::local_csr_type csr;
    csr.resize(nc, nnz);
    auto* AMREX_RESTRICT prow = csr.row_offset.data();
    auto* AMREX_RESTRICT pcol = csr.col_index.data();
    auto* AMREX_RESTRICT pmat = csr.mat.data();
    ParallelForOMP(nrows+1, [=] AMREX_GPU_DEVICE (Long i) noexcept
    {
        if (i == nrows) { prow[nc] = nnz; return; }
        if (state[i] != c) { return; }
        Long q = poff[i];
        prow[cidx[i]] = q;
        auto const& c0 = p.csr0;
        for (Long idx = c0.row_offset[i]; idx < c0.row_offset[i+1]; ++idx) {
            int const g = colmap(c0.col_index[idx], false);
            if (g >= 0) { pcol[q] = g; pmat[q] = c0.mat[idx]; ++q; }
        }
        if (has_remote_row(p, i)) {
            auto const& c1 = p.csr1;
            Long const ii = p.row_map[i];
            for (Long idx = c1.row_offset[ii]; idx < c1.row_offset[ii+1]; ++idx) {
                int const g = colmap(c1.col_index[idx], true);
                if (g >= 0) { pcol[q] = g; pmat[q] = c1.mat[idx]; ++q; }
            }
        }
    });
    Gpu::streamSynchronize();
    return csr;
}
 
}
 
template <typename T>
SpMatrix<T> AlgMG<T>::interpolation (SpMatrix<T>& A, SpMatrix<TS>& S, SpMatrix<TS> const& ST,
                                   Gpu::DeviceVector<int> const& state,
                                   Gpu::DeviceVector<Long> const& cidx,
                                   AlgPartition const& cpart, T scale) const
{
    if (m_interp_type == InterpType::direct) {
        return build_interp(A, S, state, cidx, cpart);
    } else {
        return build_interp_mm(A, S, ST, state, cidx, cpart, scale,
                               m_interp_type == InterpType::mm_ext_i);
    }
}
 
template <typename T>
SpMatrix<typename AlgMG<T>::TS> AlgMG<T>::second_pass_strength (SpMatrix<TS>& S,
                                          Gpu::DeviceVector<int> const& state,
                                          Gpu::DeviceVector<Long> const& cidx,
                                          AlgPartition const& cpart)
{
    BL_PROFILE("AlgMG::second_pass_strength");
 
    // Distance-two C-C connections: S_CF S_FC. The few direct C-C
    // connections PMIS leaves are ignored. Both factors are built in
    // compact C and F numberings, so the product never forms the full S^2.
    auto const& s = S.const_parcsr();
    Long const nrows = s.csr0.nrows;
    Long const nc = cpart.numLocalRows();
    int const C = PointType::coarse;
    int const F = PointType::fine;
 
    Gpu::DeviceVector<Long> fidx;
    auto const fpart = coarse_numbering(state, fidx, PointType::fine);
    Long const nf = fpart.numLocalRows();
    int const nlocal_c = int(nc);
    int const nlocal_f = int(nf);
 
    auto const* AMREX_RESTRICT ps = state.data();
    auto const* AMREX_RESTRICT pc = cidx.data();
    auto const* AMREX_RESTRICT pf = fidx.data();
    Long const cbegin = cpart.globalRowBegin();
    Long const fbegin = fpart.globalRowBegin();
    Gpu::DeviceVector<Long> gcidx(nrows);
    Gpu::DeviceVector<Long> gfidx(nrows);
    auto* AMREX_RESTRICT pgc = gcidx.data();
    auto* AMREX_RESTRICT pgf = gfidx.data();
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
    {
        pgc[i] = pc[i] + cbegin;
        pgf[i] = pf[i] + fbegin;
    });
    auto const state_r = S.gatherRemote(ps);
    auto const gcidx_r = S.gatherRemote(pgc);
    auto const gfidx_r = S.gatherRemote(pgf);
    auto const* AMREX_RESTRICT ps_r = state_r.data();
 
    // S_CF: C rows, F columns in F numbering (remote: nlocal_f + remote index).
    auto csr_cf = detail::compact_rows<TS>
        (s, nc, ps, pc, C,
         [=] AMREX_GPU_DEVICE (int j, bool remote) -> int {
             if (remote) {
                 return (ps_r[j] == F) ? nlocal_f + j : -1;
             } else {
                 return (ps[j] == F) ? int(pf[j]) : -1;
             }
         });
    SpMatrix<TS> SCF;
    SCF.define_split(cpart, fpart, std::move(csr_cf), nlocal_f, gfidx_r.data(), 0);
 
    // S_FC: F rows, C columns in C numbering.
    auto csr_fc = detail::compact_rows<TS>
        (s, nf, ps, pf, F,
         [=] AMREX_GPU_DEVICE (int j, bool remote) -> int {
             if (remote) {
                 return (ps_r[j] == C) ? nlocal_c + j : -1;
             } else {
                 return (ps[j] == C) ? int(pc[j]) : -1;
             }
         });
    SpMatrix<TS> SFC;
    SFC.define_split(fpart, cpart, std::move(csr_fc), nlocal_c, gcidx_r.data(), 0);
 
    auto S2 = amrex::SpGEMM(SCF, SFC, cpart);
 
    // Drop the diagonal: a C-point is not its own distance-two neighbor.
    auto const& s2 = S2.const_parcsr();
    typename SpMatrix<TS>::local_csr_type out;
    out.row_offset.resize(nc+1);
    auto* AMREX_RESTRICT orow = out.row_offset.data();
    Long const nnz = Scan::PrefixSum<Long>
        (nc+1,
         [=] AMREX_GPU_DEVICE (Long r) -> Long {
             if (r >= nc) { return 0; }
             Long n = 0;
             auto const& c0 = s2.csr0;
             for (Long idx = c0.row_offset[r]; idx < c0.row_offset[r+1]; ++idx) {
                 n += (c0.col_index[idx] != int(r));
             }
             if (detail::has_remote_row(s2, r)) {
                 Long const rr = s2.row_map[r];
                 n += s2.csr1.row_offset[rr+1] - s2.csr1.row_offset[rr];
             }
             return n;
         },
         [=] AMREX_GPU_DEVICE (Long r, Long const& x) { orow[r] = x; },
         Scan::Type::exclusive, Scan::retSum);
    out.resize(nc, nnz);
    auto* AMREX_RESTRICT ocol = out.col_index.data();
    auto* AMREX_RESTRICT omat = out.mat.data();
    ParallelForOMP(nc, [=] AMREX_GPU_DEVICE (Long r) noexcept
    {
        Long q = orow[r];
        auto const& c0 = s2.csr0;
        for (Long idx = c0.row_offset[r]; idx < c0.row_offset[r+1]; ++idx) {
            if (c0.col_index[idx] != int(r)) {
                ocol[q] = c0.col_index[idx];
                omat[q] = c0.mat[idx];
                ++q;
            }
        }
        if (detail::has_remote_row(s2, r)) {
            auto const& c1 = s2.csr1;
            Long const rr = s2.row_map[r];
            for (Long idx = c1.row_offset[rr]; idx < c1.row_offset[rr+1]; ++idx) {
                ocol[q] = nlocal_c + c1.col_index[idx];
                omat[q] = c1.mat[idx];
                ++q;
            }
        }
    });
    Gpu::streamSynchronize();
    SpMatrix<TS> S2d;
    S2d.define_split(cpart, cpart, std::move(out), nlocal_c, s2.col_map, 0);
    return S2d;
}
 
template <typename T>
SpMatrix<T> AlgMG<T>::select_coarse_rows (SpMatrix<T> const& P,
                                        Gpu::DeviceVector<int> const& state,
                                        Gpu::DeviceVector<Long> const& cidx,
                                        AlgPartition const& cpart)
{
    auto const& p = P.const_parcsr();
    int const nlocal = int(P.columnPartition().numLocalRows());
    auto csr = detail::compact_rows<T>
        (p, cpart.numLocalRows(), state.data(), cidx.data(), int(PointType::coarse),
         [=] AMREX_GPU_DEVICE (int j, bool remote) -> int {
             return remote ? nlocal + j : j;
         });
    SpMatrix<T> Q;
    Q.define_split(cpart, P.columnPartition(), std::move(csr), nlocal, p.col_map, 0);
    return Q;
}
 
template <typename T>
SpMatrix<T> AlgMG<T>::truncate_interp (SpMatrix<T>& P, AlgPartition const& cpart,
                                     int max_elmts, T trunc_factor)
{
    BL_PROFILE("AlgMG::truncate_interp");
 
    constexpr int KMAX = max_p_elmts; // rows of P are short
    AMREX_ALWAYS_ASSERT(max_elmts <= KMAX);
 
    P.setColumnPartition(cpart);
    auto const& p = P.const_parcsr();
    Long const nrows = p.csr0.nrows;
    int const nlocal = int(cpart.numLocalRows());
    detail::TruncRow<T,KMAX> const tr{p, max_elmts, trunc_factor, nlocal};
 
    // Drop threshold and signed sums of each row, computed once.
    Gpu::DeviceVector<T> rthr(nrows), rpos(nrows), rneg(nrows);
    auto* AMREX_RESTRICT pthr = rthr.data();
    auto* AMREX_RESTRICT ppos = rpos.data();
    auto* AMREX_RESTRICT pneg = rneg.data();
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
    {
        pthr[i] = tr.threshold(i, ppos[i], pneg[i]);
    });
 
    using local_csr_type = typename SpMatrix<T>::local_csr_type;
    local_csr_type csr;
    csr.row_offset.resize(nrows+1);
    auto* AMREX_RESTRICT prow = csr.row_offset.data();
    Long const nnz = Scan::PrefixSum<Long>
        (nrows+1,
         [=] AMREX_GPU_DEVICE (Long i) -> Long {
             if (i >= nrows) { return 0; }
             int const n = tr.count(i, pthr[i]);
             return (max_elmts > 0) ? std::min(n, max_elmts) : n;
         },
         [=] AMREX_GPU_DEVICE (Long i, Long const& x) { prow[i] = x; },
         Scan::Type::exclusive, Scan::retSum);
 
    csr.resize(nrows, nnz);
    auto* AMREX_RESTRICT pmat = csr.mat.data();
    auto* AMREX_RESTRICT pcol = csr.col_index.data();
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
    {
        T const thresh = pthr[i];
        T const sum_pos = ppos[i];
        T const sum_neg = pneg[i];
        Long q = prow[i];
        if (max_elmts > 0) {
            int cols[KMAX];
            T vals[KMAX];
            int const n = tr.select(i, thresh, cols, vals);
            T kept_pos = T(0), kept_neg = T(0);
            for (int k = 0; k < n; ++k) {
                if (vals[k] > T(0)) { kept_pos += vals[k]; } else { kept_neg += vals[k]; }
            }
            T const fpos = (kept_pos != T(0)) ? sum_pos/kept_pos : T(1);
            T const fneg = (kept_neg != T(0)) ? sum_neg/kept_neg : T(1);
            for (int k = 0; k < n; ++k) {
                pcol[q] = cols[k];
                pmat[q] = vals[k] * ((vals[k] > T(0)) ? fpos : fneg);
                ++q;
            }
        } else { // threshold only, no limit on the count
            T kept_pos = T(0), kept_neg = T(0);
            tr.walk(i, [&] (int, T v) {
                if (std::abs(v) >= thresh) {
                    if (v > T(0)) { kept_pos += v; } else { kept_neg += v; }
                }
            });
            T const fpos = (kept_pos != T(0)) ? sum_pos/kept_pos : T(1);
            T const fneg = (kept_neg != T(0)) ? sum_neg/kept_neg : T(1);
            tr.walk(i, [&] (int col, T v) {
                if (std::abs(v) >= thresh) {
                    pcol[q] = col;
                    pmat[q] = v * ((v > T(0)) ? fpos : fneg);
                    ++q;
                }
            });
        }
    });
    Gpu::streamSynchronize();
 
    // Compact columns sort like the global ones within each block.
    csr.sort();
    SpMatrix<T> Pt;
    Pt.define_split(P.partition(), cpart, std::move(csr), nlocal, p.col_map, 0);
    return Pt;
}
 
template <typename T>
SpMatrix<T> AlgMG<T>::build_interp_mm (SpMatrix<T>& A, SpMatrix<TS>& S,
                                     SpMatrix<TS> const& ST,
                                     Gpu::DeviceVector<int> const& state,
                                     Gpu::DeviceVector<Long> const& cidx,
                                     AlgPartition const& cpart, T sscale,
                                     bool plus_i, int const* row_state,
                                     int max_elmts, T trunc_factor, Long* nnz_untruncated)
{
    BL_PROFILE("AlgMG::build_interp_mm");
 
    // W = -M_F B_F of Li, Sjögreen & Yang (2021), Eqs. (3.4)-(3.12). Both
    // factors keep the natural row numbering with unit C rows, so P is one
    // product SpGEMM(M, B).
 
    auto const& s = S.const_parcsr();
    Long const nrows = s.csr0.nrows;
    auto const* AMREX_RESTRICT ps = state.data();
    auto const* AMREX_RESTRICT pc = cidx.data();
    Long const cbegin = cpart.globalRowBegin();
    int const nlocal_f = int(A.numLocalRows());
    int const nlocal_c = int(cpart.numLocalRows());
    int const C = PointType::coarse;
    int const F = PointType::fine;
 
    // alpha_i = a_ii + weak off-diagonal row sum = rowsum(A) - rowsum(S),
    // in the units of S. The weights do not depend on the scaling.
    auto const rsA = A.rowSum();
    auto const rsS = S.rowSum();
    auto const* AMREX_RESTRICT prsA = rsA.data();
    auto const* AMREX_RESTRICT prsS = rsS.data();
 
    Gpu::DeviceVector<Long> gcidx(nrows);
    auto* AMREX_RESTRICT pgc = gcidx.data();
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept { pgc[i] = pc[i] + cbegin; });
    auto const state_r = S.gatherRemote(ps);
    auto const gcidx_r = S.gatherRemote(pgc);
    auto const* AMREX_RESTRICT pstate_r = state_r.data();
 
    // D_beta: row sum of strong connections to C-points.
    Gpu::DeviceVector<T> dbeta(nrows);
    auto* AMREX_RESTRICT pdb = dbeta.data();
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
    {
        T d = T(0);
        auto const& sc = s.csr0;
        for (Long idx = sc.row_offset[i]; idx < sc.row_offset[i+1]; ++idx) {
            if (ps[sc.col_index[idx]] == C) { d += T(sc.mat[idx]); }
        }
        if (detail::has_remote_row(s, i)) {
            auto const& sc1 = s.csr1;
            Long const ii = s.row_map[i];
            for (Long idx = sc1.row_offset[ii]; idx < sc1.row_offset[ii+1]; ++idx) {
                if (pstate_r[sc1.col_index[idx]] == C) { d += T(sc1.mat[idx]); }
            }
        }
        pdb[i] = d;
    });
    Gpu::streamSynchronize();
    auto const dbeta_r = S.gatherRemote(pdb);
 
    detail::MMInterpRow<T,TS> const w{s, ST.const_parcsr(), ps, pdb, pc,
                                   pstate_r, dbeta_r.data(), nlocal_f, nlocal_c};
 
    // Counts: M gets a diagonal plus the usable strong F neighbors; B gets
    // the strong C neighbors. C rows get one unit entry each.
    using local_csr_type = typename SpMatrix<T>::local_csr_type;
    local_csr_type mcsr, bcsr;
    mcsr.row_offset.resize(nrows+1);
    bcsr.row_offset.resize(nrows+1);
    auto* AMREX_RESTRICT pmrow = mcsr.row_offset.data();
    auto* AMREX_RESTRICT pbrow = bcsr.row_offset.data();
 
    auto usable_f = [=] AMREX_GPU_DEVICE (T aji, int st_j, T db_j) -> bool {
        return st_j == F && (db_j + (plus_i ? aji : T(0))) != T(0);
    };
 
    // Rows of P: all, or the C-points of `row_state` (the others stay
    // empty in M and so in P, but B needs every row).
    auto const* AMREX_RESTRICT prow_state = row_state;
    auto has_row = [=] AMREX_GPU_DEVICE (Long i) -> bool {
        return prow_state == nullptr || prow_state[i] == C;
    };
 
    Long const mnnz = Scan::PrefixSum<Long>
        (nrows+1,
         [=] AMREX_GPU_DEVICE (Long i) -> Long {
             if (i >= nrows || !has_row(i)) { return 0; }
             Long n = 1;
             if (ps[i] == F) {
                 w.walk(i, [&] (int, int, T, T aji, int st_j, T db_j) {
                     if (usable_f(aji, st_j, db_j)) { ++n; }
                 });
             }
             return n;
         },
         [=] AMREX_GPU_DEVICE (Long i, Long const& x) { pmrow[i] = x; },
         Scan::Type::exclusive, Scan::retSum);
 
    Long const bnnz = Scan::PrefixSum<Long>
        (nrows+1,
         [=] AMREX_GPU_DEVICE (Long i) -> Long {
             if (i >= nrows) { return 0; }
             if (ps[i] == C) { return 1; }
             Long n = 0;
             w.walk(i, [&] (int, int, T, T, int st_j, T) {
                 if (st_j == C) { ++n; }
             });
             return n;
         },
         [=] AMREX_GPU_DEVICE (Long i, Long const& x) { pbrow[i] = x; },
         Scan::Type::exclusive, Scan::retSum);
 
    mcsr.resize(nrows, mnnz);
    bcsr.resize(nrows, bnnz);
    auto* AMREX_RESTRICT pmmat = mcsr.mat.data();
    auto* AMREX_RESTRICT pmcol = mcsr.col_index.data();
    auto* AMREX_RESTRICT pbmat = bcsr.mat.data();
    auto* AMREX_RESTRICT pbcol = bcsr.col_index.data();
 
    ParallelForOMP(nrows, [=] AMREX_GPU_DEVICE (Long i) noexcept
    {
        Long mp = pmrow[i];
        Long bp = pbrow[i];
        bool const in_p = has_row(i);
        if (ps[i] != F) {
            if (in_p) {
                pmcol[mp] = int(i);
                pmmat[mp] = T(1);
            }
            pbcol[bp] = int(pc[i]);
            pbmat[bp] = T(1);
            return;
        }
        T const db_i = pdb[i];
        T const bscale = plus_i ? T(1) : ((db_i != T(0)) ? T(1)/db_i : T(0));
        if (!in_p) {
            w.walk(i, [&] (int, int jc, T aij, T, int st_j, T) {
                if (st_j == C) {
                    pbcol[bp] = jc;
                    pbmat[bp] = bscale*aij;
                    ++bp;
                }
            });
            return;
        }
 
        // Strong F neighbors without a usable denominator are lumped into
        // the diagonal as weak connections (remark after Eq. (3.8)).
        T alpha = prsA[i]/sscale - T(prsS[i]);
        T theta = T(0);
        w.walk(i, [&] (int, int, T aij, T aji, int st_j, T db_j) {
            if (st_j == F) {
                T const den = db_j + (plus_i ? aji : T(0));
                if (den != T(0)) {
                    if (plus_i) { theta += aij * aji / den; }
                } else {
                    alpha += aij;
                }
            }
        });
        T const denom = alpha + theta;
        T const scale = (denom != T(0)) ? T(-1)/denom : T(0);
 
        pmcol[mp] = int(i);
        pmmat[mp] = plus_i ? scale : scale*db_i;
        ++mp;
        w.walk(i, [&] (int jf, int jc, T aij, T aji, int st_j, T db_j) {
            if (st_j == C) {
                pbcol[bp] = jc;
                pbmat[bp] = bscale*aij;
                ++bp;
            } else if (usable_f(aji, st_j, db_j)) {
                pmcol[mp] = jf;
                pmmat[mp] = plus_i ? scale*aij/(db_j + aji) : scale*aij;
                ++mp;
            }
        });
    });
    Gpu::streamSynchronize();
 
    // The diagonal of M is emitted first and must be sorted into place.
    mcsr.sort();
    SpMatrix<T> M, B;
    M.define_split(A.partition(), A.partition(), std::move(mcsr), nlocal_f, s.col_map, 0);
    B.define_split(A.partition(), cpart, std::move(bcsr), nlocal_c, gcidx_r.data(), 0);
#ifndef AMREX_USE_GPU
    if (max_elmts > 0 || trunc_factor > T(0)) {
        // Each row is truncated as the product makes it.
        constexpr int KMAX = max_p_elmts;
        AMREX_ALWAYS_ASSERT(max_elmts <= KMAX);
        Vector<Long> n0(nrows, 0);
        auto P = amrex::SpGEMM(M, B, cpart, [&] (Long i, int* col, T* val, Long n) {
            n0[i] = n;
            return detail::truncate_row_host<T,KMAX>(col, val, n, max_elmts, trunc_factor,
                                                     nlocal_c);
        });
        if (nnz_untruncated) {
            *nnz_untruncated = std::accumulate(n0.begin(), n0.end(), Long(0));
        }
        return P;
    }
#else
    amrex::ignore_unused(max_elmts, trunc_factor, nnz_untruncated);
#endif
    return amrex::SpGEMM(M, B, cpart);
}
 
#ifndef AMREX_USE_GPU
template <typename T>
SpMatrix<T> AlgMG<T>::build_interp_mm_host (SpMatrix<T>& A, SpMatrix<TS>& S,
                                          SpMatrix<TS> const& ST,
                                          Gpu::DeviceVector<int> const& state,
                                          Gpu::DeviceVector<Long> const& cidx,
                                          AlgPartition const& cpart, T sscale, bool plus_i,
                                          int max_elmts, T trunc_factor,
                                          Long& nnz_untruncated)
{
    BL_PROFILE("AlgMG::build_interp_mm_host");
 
    // Row i of P = M B (see build_interp_mm) is summed from the rows of B
    // in the order of the SpGEMM, then truncated as in truncate_interp.
    constexpr int KMAX = max_p_elmts;
    AMREX_ALWAYS_ASSERT(max_elmts <= KMAX);
 
    auto const& s = S.const_parcsr();
    Long const nrows = s.csr0.nrows;
    auto const* AMREX_RESTRICT ps = state.data();
    auto const* AMREX_RESTRICT pc = cidx.data();
    int const nc = int(cpart.numLocalRows());
    int const C = PointType::coarse;
    int const F = PointType::fine;
 
    auto const rsA = A.rowSum();
    auto const rsS = S.rowSum();
    auto const* AMREX_RESTRICT prsA = rsA.data();
    auto const* AMREX_RESTRICT prsS = rsS.data();
 
    // D_beta and the rows of B at the F-points: the strong C neighbors.
    Gpu::DeviceVector<T> dbeta(nrows);
    auto* AMREX_RESTRICT pdb = dbeta.data();
    Vector<Long> boff(nrows+1, 0);
    auto const& sc = s.csr0;
    ParallelForOMP(nrows, [&] (Long i) noexcept
    {
        T d = T(0);
        Long n = 0;
        for (Long idx = sc.row_offset[i]; idx < sc.row_offset[i+1]; ++idx) {
            if (ps[sc.col_index[idx]] == C) {
                d += T(sc.mat[idx]);
                ++n;
            }
        }
        pdb[i] = d;
        boff[i+1] = (ps[i] == F) ? n : 0;
    });
    std::partial_sum(boff.begin(), boff.end(), boff.begin());
    Vector<int> bcol(boff[nrows]);
    Vector<T> bval(boff[nrows]);
    ParallelForOMP(nrows, [&] (Long i) noexcept
    {
        if (ps[i] != F) { return; }
        T const bscale = plus_i ? T(1) : ((pdb[i] != T(0)) ? T(1)/pdb[i] : T(0));
        Long q = boff[i];
        for (Long idx = sc.row_offset[i]; idx < sc.row_offset[i+1]; ++idx) {
            int const j = sc.col_index[idx];
            if (ps[j] == C) {
                bcol[q] = int(pc[j]);
                bval[q] = bscale*T(sc.mat[idx]);
                ++q;
            }
        }
    });
 
    detail::MMInterpRow<T,TS> const w{s, ST.const_parcsr(), ps, pdb, pc,
                                   nullptr, nullptr, int(nrows), nc};
    bool const trunc = (max_elmts > 0 || trunc_factor > T(0));
    struct Nbr { int j; T aij; T den; };
 
    Vector<Long> nnz0(OpenMP::get_max_threads(), 0);
    auto csr = detail::csr_from_rows_cpu<T,SpMatrix<T>::template container_type,int>
        (nrows, [&] (Long i) { return sc.row_offset[i+1] - sc.row_offset[i] + 1; },
         [&] (int t) {
        return [&, t, marker = Vector<int>(nc, -1), col = Vector<int>(1), val = Vector<T>(1),
                tmp = Vector<std::pair<int,T>>(), nbr = Vector<Nbr>()]
            (Long i, int const*& rc, T const*& rv) mutable -> Long
        {
            rc = col.data();
            rv = val.data();
            if (ps[i] != F) {
                col[0] = int(pc[i]);
                val[0] = T(1);
                ++nnz0[t];
                return 1;
            }
 
            // Scale of row i, as in build_interp_mm, and the usable strong
            // F neighbors (column order) for M.
            T alpha = prsA[i]/sscale - T(prsS[i]);
            T theta = T(0);
            Long maxlen = boff[i+1] - boff[i];
            nbr.clear();
            w.walk(i, [&] (int jf, int, T aij, T aji, int st_j, T db_j) {
                if (st_j == F) {
                    T const den = db_j + (plus_i ? aji : T(0));
                    if (den != T(0)) {
                        if (plus_i) { theta += aij * aji / den; }
                        maxlen += boff[jf+1] - boff[jf];
                        nbr.push_back({jf, aij, den});
                    } else {
                        alpha += aij;
                    }
                }
            });
            T const denom = alpha + theta;
            T const scale = (denom != T(0)) ? T(-1)/denom : T(0);
            maxlen = std::min(maxlen, Long(nc));
            if (Long(col.size()) < maxlen) {
                col.resize(maxlen);
                val.resize(maxlen);
                rc = col.data();
                rv = val.data();
            }
 
            // M_ik times row k of B, k in increasing order (M's row sorted).
            Long n = 0;
            auto add = [&] (Long k, T m) {
                for (Long bp = boff[k]; bp < boff[k+1]; ++bp) {
                    int const j = bcol[bp];
                    Long const pos = marker[j];
                    if (pos >= 0 && pos < n && col[pos] == j) {
                        val[pos] += m * bval[bp];
                    } else {
                        marker[j] = int(n);
                        col[n] = j;
                        val[n] = m * bval[bp];
                        ++n;
                    }
                }
            };
            T const mdiag = plus_i ? scale : scale*pdb[i];
            bool diag_done = false;
            for (auto const& nb : nbr) {
                if (!diag_done && nb.j > i) {
                    add(i, mdiag);
                    diag_done = true;
                }
                add(nb.j, plus_i ? scale*nb.aij/nb.den : scale*nb.aij);
            }
            if (!diag_done) { add(i, mdiag); }
            detail::sort_row_cpu(col.data(), val.data(), n, tmp);
            nnz0[t] += n;
            if (!trunc) { return n; }
 
            return detail::truncate_row_host<T,KMAX>(col.data(), val.data(), n, max_elmts,
                                                     trunc_factor, nc);
        };
    });
 
    nnz_untruncated = std::accumulate(nnz0.begin(), nnz0.end(), Long(0));
    SpMatrix<T> P;
    P.define_split(A.partition(), cpart, std::move(csr), nc, nullptr, 0);
    return P;
}
#endif
 
}
 
#endif