#ifndef AMGCL_PRECONDITIONER_SCHUR_PRESSURE_CORRECTION_HPP
#define AMGCL_PRECONDITIONER_SCHUR_PRESSURE_CORRECTION_HPP

/*
The MIT License

Copyright (c) 2012-2022 Denis Demidov <dennis.demidov@gmail.com>
Copyright (c) 2016, Riccardo Rossi, CIMNE (International Center for Numerical Methods in Engineering)

Permission is hereby granted, free of charge, to any person obtaining a copy
of this software and associated documentation files (the "Software"), to deal
in the Software without restriction, including without limitation the rights
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all copies or substantial portions of the Software.

THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT.  IN NO EVENT SHALL THE
AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN
THE SOFTWARE.
*/

/**
 * \file   amgcl/preconditioner/schur_pressure_correction.hpp
 * \author Denis Demidov <dennis.demidov@gmail.com>
 * \brief  Schur-complement pressure correction preconditioning scheme.
 *
 * [1] Elman, Howard, et al. "A taxonomy and comparison of parallel block
 *     multi-level preconditioners for the incompressible Navier–Stokes
 *     equations." Journal of Computational Physics 227.3 (2008): 1790-1808.
 * [2] Gmeiner, Björn, et al. "A quantitative performance analysis for Stokes
 *     solvers at the extreme scale." arXiv preprint arXiv:1511.02134 (2015).
 * [3] Vincent, C., and R. Boyer. "A preconditioned conjugate gradient
 *     Uzawa‐type method for the solution of the Stokes problem by mixed Q1–P0
 *     stabilized finite elements." International journal for numerical methods
 *     in fluids 14.3 (1992): 289-298.
 */

#include <vector>

#include <memory>

#include <amgcl/backend/builtin.hpp>
#include <amgcl/backend/detail/mixing.hpp>
#include <amgcl/util.hpp>
#include <amgcl/io/mm.hpp>

namespace amgcl {
namespace preconditioner {

/// Schur-complement pressure correction preconditioner
template <class USolver, class PSolver>
class schur_pressure_correction {
    static_assert(
            backend::backends_compatible<
                typename USolver::backend_type,
                typename PSolver::backend_type
                >::value,
            "Backends for pressure and flow preconditioners should coincide!"
            );
    public:
        typedef
            typename backend::detail::common_scalar_backend<
                typename USolver::backend_type,
                typename PSolver::backend_type
                >::type
            backend_type;

        typedef typename backend_type::value_type value_type;
        typedef typename backend_type::col_type col_type;
        typedef typename backend_type::ptr_type ptr_type;
        typedef typename math::scalar_of<value_type>::type scalar_type;
        typedef typename backend_type::matrix     matrix;
        typedef typename backend_type::vector     vector;
        typedef typename backend_type::params     backend_params;

        typedef typename backend::builtin<value_type, col_type, ptr_type>::matrix build_matrix;

        struct params {
            typedef typename USolver::params usolver_params;
            typedef typename PSolver::params psolver_params;

            usolver_params usolver;
            psolver_params psolver;

            std::vector<char> pmask;

            // Variant of block preconditioner to use in apply()
            // 1: schur pressure correction:
            //      S p = fp - Kpu Kuu^-1 fu
            //      Kuu u = fu - Kup p
            // 2: Block triangular:
            //      S p = fp
            //      Kuu u = fu - Kup p
            int type;

            // Approximate Kuu^-1 with inverted diagonal of Kuu during
            // construction of matrix-less Schur complement.
            // When false, USolver is used instead.
            bool approx_schur;

            // Adjust preconditioner matrix for the Schur complement system.
            // That is, use
            //   Kpp                                 when adjust_p == 0,
            //   Kpp - dia(Kpu * dia(Kuu)^-1 * Kup)  when adjust_p == 1,
            //   Kpp - Kpu * dia(Kuu)^-1 * Kup       when adjust_p == 2
            int adjust_p;

            // Use 1/sum_j(abs(Kuu_{i,j})) instead of dia(Kuu)^-1
            // as approximation for the Kuu^-1 (as in SIMPLEC algorithm)
            bool simplec_dia;

            int verbose;

            params() : type(1), approx_schur(false), adjust_p(1), simplec_dia(true), verbose(0) {}

#ifndef AMGCL_NO_BOOST
            params(const boost::property_tree::ptree &p)
                : AMGCL_PARAMS_IMPORT_CHILD(p, usolver),
                  AMGCL_PARAMS_IMPORT_CHILD(p, psolver),
                  AMGCL_PARAMS_IMPORT_VALUE(p, type),
                  AMGCL_PARAMS_IMPORT_VALUE(p, approx_schur),
                  AMGCL_PARAMS_IMPORT_VALUE(p, adjust_p),
                  AMGCL_PARAMS_IMPORT_VALUE(p, simplec_dia),
                  AMGCL_PARAMS_IMPORT_VALUE(p, verbose)
            {
                size_t n = 0;

                n = p.get("pmask_size", n);

                precondition(n > 0,
                        "Error in schur_complement parameters: "
                        "pmask_size is not set");

                if (p.count("pmask_pattern")) {
                    pmask.resize(n, 0);

                    std::string pattern = p.get("pmask_pattern", std::string());
                    switch (pattern[0]) {
                        case '%':
                            {
                                int start  = std::atoi(pattern.substr(1).c_str());
                                int stride = std::atoi(pattern.substr(3).c_str());
                                for(size_t i = start; i < n; i += stride) pmask[i] = 1;
                            }
                            break;
                        case '<':
                            {
                                size_t m = std::atoi(pattern.c_str()+1);
                                for(size_t i = 0; i < std::min(m, n); ++i) pmask[i] = 1;
                            }
                            break;
                        case '>':
                            {
                                size_t m = std::atoi(pattern.c_str()+1);
                                for(size_t i = m; i < n; ++i) pmask[i] = 1;
                            }
                            break;
                        default:
                            precondition(false, "Unknown pattern in pmask_pattern");
                    }
                } else if (p.count("pmask")) {
                    void *pm = 0;
                    pm = p.get("pmask", pm);
                    pmask.assign(static_cast<char*>(pm), static_cast<char*>(pm) + n);
                } else {
                    precondition(false,
                            "Error in schur_complement parameters: "
                            "neither pmask_pattern, nor pmask is set"
                            );
                }

                check_params(p, {"usolver", "psolver", "type", "approx_schur", "adjust_p", "simplec_dia", "pmask_size", "verbose"},
                        {"pmask", "pmask_pattern"});
            }

            void get(boost::property_tree::ptree &p, const std::string &path = "") const
            {
                AMGCL_PARAMS_EXPORT_CHILD(p, path, usolver);
                AMGCL_PARAMS_EXPORT_CHILD(p, path, psolver);
                AMGCL_PARAMS_EXPORT_VALUE(p, path, type);
                AMGCL_PARAMS_EXPORT_VALUE(p, path, approx_schur);
                AMGCL_PARAMS_EXPORT_VALUE(p, path, adjust_p);
                AMGCL_PARAMS_EXPORT_VALUE(p, path, simplec_dia);
                AMGCL_PARAMS_EXPORT_VALUE(p, path, verbose);
            }
#endif
        } prm;

        template <class Matrix>
        schur_pressure_correction(
                const Matrix &K,
                const params &prm = params(),
                const backend_params &bprm = backend_params()
                )
            : prm(prm), n(backend::rows(K)), np(0), nu(0)
        {
            init(std::make_shared<build_matrix>(K), bprm);
        }

        schur_pressure_correction(
                std::shared_ptr<build_matrix> K,
                const params &prm = params(),
                const backend_params &bprm = backend_params()
                )
            : prm(prm), n(backend::rows(*K)), np(0), nu(0)
        {
            init(K, bprm);
        }

        template <class Vec1, class Vec2>
        void apply(const Vec1 &rhs, Vec2 &&x) const {
            const auto one = math::identity<scalar_type>();
            const auto zero = math::zero<scalar_type>();

            backend::spmv(one, *x2u, rhs, zero, *rhs_u);
            backend::spmv(one, *x2p, rhs, zero, *rhs_p);

            if (prm.type == 1) {
                // Kuu u = rhs_u
                backend::clear(*u);
                report("U1", (*U)(*rhs_u, *u));

                // rhs_p -= Kpu u
                backend::spmv(-one, *Kpu, *u, one, *rhs_p);

                // S p = rhs_p
                backend::clear(*p);
                report("P1", (*P)(*this, *rhs_p, *p));

                // rhs_u -= Kup p
                backend::spmv(-one, *Kup, *p, one, *rhs_u);

                // Kuu u = rhs_u
                backend::clear(*u);
                report("U2", (*U)(*rhs_u, *u));
            } else if (prm.type == 2) {
                // S p = fp
                backend::clear(*p);
                report("P", (*P)(*this, *rhs_p, *p));

                // Kuu u = fu - Kup p
                backend::spmv(-one, *Kup, *p, one, *rhs_u);
                backend::clear(*u);
                report("U", (*U)(*rhs_u, *u));
            }

            backend::spmv(one, *u2x, *u, zero, x);
            backend::spmv(one, *p2x, *p, one, x);
        }

        template <class Alpha, class Vec1, class Beta, class Vec2>
        void spmv(Alpha alpha, const Vec1 &x, Beta beta, Vec2 &y) const {
            const auto one = math::identity<scalar_type>();
            const auto zero = math::zero<scalar_type>();

            // y = beta y + alpha S x, where S = Kpp - Kpu Kuu^-1 Kup
            if (prm.adjust_p == 1) {
                backend::spmv( alpha, P->system_matrix(), x, beta, y);
                backend::vmul( alpha, *Ld, x, one, y);
            } else if (prm.adjust_p == 2) {
                backend::spmv( alpha, *Lm, x, beta, y);
            } else {
                backend::spmv( alpha, P->system_matrix(), x, beta, y);
            }

            backend::spmv(one, *Kup, x, zero, *tmp);

            if (prm.approx_schur) {
                backend::vmul(one, *M, *tmp, zero, *u);
            } else {
                backend::clear(*u);
                (*U)(*tmp, *u);
            }

            backend::spmv(-alpha, *Kpu, *u, one, y);
        }

        std::shared_ptr<matrix> system_matrix_ptr() const {
            return K;
        }

        const matrix& system_matrix() const {
            return *K;
        }

        size_t bytes() const {
            size_t b = 0;

            b += backend::bytes(*K);
            b += backend::bytes(*Kup);
            b += backend::bytes(*Kpu);
            b += backend::bytes(*x2u);
            b += backend::bytes(*x2p);
            b += backend::bytes(*u2x);
            b += backend::bytes(*p2x);
            b += backend::bytes(*rhs_u);
            b += backend::bytes(*rhs_p);
            b += backend::bytes(*u);
            b += backend::bytes(*p);
            b += backend::bytes(*tmp);
            b += backend::bytes(*U);
            b += backend::bytes(*P);

            if (M) b += backend::bytes(*M);
            if (Ld) b += backend::bytes(*Ld);
            if (Lm) b += backend::bytes(*Lm);

            return b;
        }

    private:
        size_t n, np, nu;

        std::shared_ptr<matrix> K, Lm, Kup, Kpu, x2u, x2p, u2x, p2x;
        std::shared_ptr<vector> rhs_u, rhs_p, u, p, tmp;
        std::shared_ptr<typename backend_type::matrix_diagonal> M;
        std::shared_ptr<typename backend_type::matrix_diagonal> Ld;

        std::shared_ptr<USolver> U;
        std::shared_ptr<PSolver> P;

        void init(const std::shared_ptr<build_matrix> &K, const backend_params &bprm)
        {
            this->K = backend_type::copy_matrix(K, bprm);

            // Extract matrix subblocks.
            auto Kuu = std::make_shared<build_matrix>();
            auto Kpu = std::make_shared<build_matrix>();
            auto Kup = std::make_shared<build_matrix>();
            auto Kpp = std::make_shared<build_matrix>();

            std::vector<ptrdiff_t> idx(n);

            for(size_t i = 0; i < n; ++i)
                idx[i] = (prm.pmask[i] ? np++ : nu++);

            Kuu->set_size(nu, nu, true);
            Kup->set_size(nu, np, true);
            Kpu->set_size(np, nu, true);
            Kpp->set_size(np, np, true);

#pragma omp parallel for
            for(ptrdiff_t i = 0; i < static_cast<ptrdiff_t>(n); ++i) {
                ptrdiff_t ci = idx[i];
                char      pi = prm.pmask[i];
                for(auto k = backend::row_begin(*K, i); k; ++k) {
                    char pj = prm.pmask[k.col()];

                    if (pi) {
                        if (pj) {
                            ++Kpp->ptr[ci+1];
                        } else {
                            ++Kpu->ptr[ci+1];
                        }
                    } else {
                        if (pj) {
                            ++Kup->ptr[ci+1];
                        } else {
                            ++Kuu->ptr[ci+1];
                        }
                    }
                }
            }

            Kuu->set_nonzeros(Kuu->scan_row_sizes());
            Kup->set_nonzeros(Kup->scan_row_sizes());
            Kpu->set_nonzeros(Kpu->scan_row_sizes());
            Kpp->set_nonzeros(Kpp->scan_row_sizes());

#pragma omp parallel for
            for(ptrdiff_t i = 0; i < static_cast<ptrdiff_t>(n); ++i) {
                ptrdiff_t ci = idx[i];
                char      pi = prm.pmask[i];

                ptrdiff_t uu_head = 0, up_head = 0, pu_head = 0, pp_head = 0;

                if(pi) {
                    pu_head = Kpu->ptr[ci];
                    pp_head = Kpp->ptr[ci];
                } else {
                    uu_head = Kuu->ptr[ci];
                    up_head = Kup->ptr[ci];
                }

                for(auto k = backend::row_begin(*K, i); k; ++k) {
                    ptrdiff_t  j = k.col();
                    value_type v = k.value();
                    ptrdiff_t cj = idx[j];
                    char      pj = prm.pmask[j];

                    if (pi) {
                        if (pj) {
                            Kpp->col[pp_head] = cj;
                            Kpp->val[pp_head] = v;
                            ++pp_head;
                        } else {
                            Kpu->col[pu_head] = cj;
                            Kpu->val[pu_head] = v;
                            ++pu_head;
                        }
                    } else {
                        if (pj) {
                            Kup->col[up_head] = cj;
                            Kup->val[up_head] = v;
                            ++up_head;
                        } else {
                            Kuu->col[uu_head] = cj;
                            Kuu->val[uu_head] = v;
                            ++uu_head;
                        }
                    }
                }
            }

            if (prm.verbose >= 2) {
                io::mm_write("Kuu.mtx", *Kuu);
                io::mm_write("Kpp.mtx", *Kpp);
            }

            std::shared_ptr<backend::numa_vector<value_type>> Kuu_dia;

            if (prm.simplec_dia) {
                Kuu_dia = std::make_shared<backend::numa_vector<value_type>>(nu);
#pragma omp parallel for
                for(ptrdiff_t i = 0; i < static_cast<ptrdiff_t>(nu); ++i) {
                    value_type s = math::zero<value_type>();
                    for(ptrdiff_t j = Kuu->ptr[i], e = Kuu->ptr[i+1]; j < e; ++j) {
                        s += math::norm(Kuu->val[j]);
                    }
                    (*Kuu_dia)[i] = math::inverse(s);
                }
            } else {
                Kuu_dia = diagonal(*Kuu, /*invert = */true);
            }

            if (prm.adjust_p == 1) {
                // Use (Kpp - dia(Kpu * dia(Kuu)^-1 * Kup))
                // to setup the P preconditioner.
                auto L = std::make_shared<backend::numa_vector<value_type>>(np, false);

#pragma omp parallel for
                for(ptrdiff_t i = 0; i < static_cast<ptrdiff_t>(np); ++i) {
                    value_type s = math::zero<value_type>();
                    for(ptrdiff_t j = Kpu->ptr[i], e = Kpu->ptr[i+1]; j < e; ++j) {
                        ptrdiff_t  k = Kpu->col[j];
                        value_type v = Kpu->val[j];
                        for(ptrdiff_t jj = Kup->ptr[k], ee = Kup->ptr[k+1]; jj < ee; ++jj) {
                            if (Kup->col[jj] == i) {
                                s += v * (*Kuu_dia)[k] * Kup->val[jj];
                                break;
                            }
                        }
                    }

                    (*L)[i] = s;
                    for(ptrdiff_t j = Kpp->ptr[i], e = Kpp->ptr[i+1]; j < e; ++j) {
                        if (Kpp->col[j] == i) {
                            Kpp->val[j] -= s;
                            break;
                        }
                    }
                }
                Ld = backend_type::copy_vector(L, bprm);
            } else if (prm.adjust_p == 2) {
                Lm = backend_type::copy_matrix(Kpp, bprm);

                // Use (Kpp - Kpu * dia(Kuu)^-1 * Kup)
                // to setup the P preconditioner.
                backend::numa_vector<value_type> val(Kup->nnz);

#pragma omp parallel for
                for(ptrdiff_t i = 0; i < static_cast<ptrdiff_t>(nu); ++i) {
                    value_type d = (*Kuu_dia)[i];
                    for(ptrdiff_t j = Kup->ptr[i], e = Kup->ptr[i+1]; j < e; ++j) {
                        val[j] = d * Kup->val[j];
                    }
                }

                build_matrix Kup_hat;

                Kup_hat.own_data = false;
                Kup_hat.nrows    = nu;
                Kup_hat.ncols    = np;
                Kup_hat.nnz      = Kup->nnz;
                Kup_hat.ptr      = Kup->ptr;
                Kup_hat.col      = Kup->col;
                Kup_hat.val      = val.data();

                Kpp = backend::sum(
                        math::identity<value_type>(), *Kpp,
                       -math::identity<value_type>(), *backend::product(*Kpu, Kup_hat));
            }

            U = std::make_shared<USolver>(*Kuu, prm.usolver, bprm);
            P = std::make_shared<PSolver>(*Kpp, prm.psolver, bprm);

            this->Kup = backend_type::copy_matrix(Kup, bprm);
            this->Kpu = backend_type::copy_matrix(Kpu, bprm);

            rhs_u = backend_type::create_vector(nu, bprm);
            rhs_p = backend_type::create_vector(np, bprm);

            u = backend_type::create_vector(nu, bprm);
            p = backend_type::create_vector(np, bprm);

            tmp = backend_type::create_vector(nu, bprm);

            if (prm.approx_schur)
                M = backend_type::copy_vector(Kuu_dia, bprm);

            // Scatter/Gather matrices
            auto x2u = std::make_shared<build_matrix>();
            auto x2p = std::make_shared<build_matrix>();
            auto u2x = std::make_shared<build_matrix>();
            auto p2x = std::make_shared<build_matrix>();

            x2u->set_size(nu, n, true);
            x2p->set_size(np, n, true);
            u2x->set_size(n, nu, true);
            p2x->set_size(n, np, true);

            {
                ptrdiff_t x2u_head = 0, x2u_idx = 0;
                ptrdiff_t x2p_head = 0, x2p_idx = 0;
                ptrdiff_t u2x_head = 0, u2x_idx = 0;
                ptrdiff_t p2x_head = 0, p2x_idx = 0;

                for(size_t i = 0; i < n; ++i) {
                    if (prm.pmask[i]) {
                        x2p->ptr[++x2p_idx] = ++x2p_head;
                        ++p2x_head;
                    } else {
                        x2u->ptr[++x2u_idx] = ++x2u_head;
                        ++u2x_head;
                    }

                    p2x->ptr[++p2x_idx] = p2x_head;
                    u2x->ptr[++u2x_idx] = u2x_head;
                }
            }

            x2u->set_nonzeros();
            x2p->set_nonzeros();
            u2x->set_nonzeros();
            p2x->set_nonzeros();

            {
                ptrdiff_t x2u_head = 0;
                ptrdiff_t x2p_head = 0;
                ptrdiff_t u2x_head = 0;
                ptrdiff_t p2x_head = 0;

                for(size_t i = 0; i < n; ++i) {
                    ptrdiff_t j = idx[i];

                    if (prm.pmask[i]) {
                        x2p->col[x2p_head] = i;
                        x2p->val[x2p_head] = math::identity<value_type>();
                        ++x2p_head;

                        p2x->col[p2x_head] = j;
                        p2x->val[p2x_head] = math::identity<value_type>();
                        ++p2x_head;
                    } else {
                        x2u->col[x2u_head] = i;
                        x2u->val[x2u_head] = math::identity<value_type>();
                        ++x2u_head;

                        u2x->col[u2x_head] = j;
                        u2x->val[u2x_head] = math::identity<value_type>();
                        ++u2x_head;
                    }
                }
            }

            this->x2u = backend_type::copy_matrix(x2u, bprm);
            this->x2p = backend_type::copy_matrix(x2p, bprm);
            this->u2x = backend_type::copy_matrix(u2x, bprm);
            this->p2x = backend_type::copy_matrix(p2x, bprm);
        }

        friend std::ostream& operator<<(std::ostream &os, const schur_pressure_correction &p) {
            os << "Schur complement (two-stage preconditioner)" << std::endl;
            os << "  Unknowns: " << p.n << "(" << p.np << ")" << std::endl;
            os << "  Nonzeros: " << backend::nonzeros(p.system_matrix()) << std::endl;
            os << "  Memory:  " << human_readable_memory(p.bytes()) << std::endl;
            os << std::endl;
            os << "[ U ]\n" << *p.U << std::endl;
            os << "[ P ]\n" << *p.P << std::endl;

            return os;
        }

        template <typename I, typename E>
        void report(const std::string &name, const std::tuple<I, E> &c) const {
            if (prm.verbose >= 1) {
                std::cout << name << " (" << std::get<0>(c) << ", " << std::get<1>(c) << ")\n";
            }
        }
};

} // namespace preconditioner

namespace backend {

template <class US, class PS, class Alpha, class Beta, class Vec1, class Vec2>
struct spmv_impl< Alpha, preconditioner::schur_pressure_correction<US, PS>, Vec1, Beta, Vec2>
{
    static void apply(Alpha alpha, const preconditioner::schur_pressure_correction<US, PS> &A, const Vec1 &x, Beta beta, Vec2 &y)
    {
        A.spmv(alpha, x, beta, y);
    }
};

template <class US, class PS, class Vec1, class Vec2, class Vec3>
struct residual_impl< preconditioner::schur_pressure_correction<US, PS>, Vec1, Vec2, Vec3>
{
    static void apply(const Vec1 &rhs, const preconditioner::schur_pressure_correction<US, PS> &A, const Vec2 &x, Vec3 &r)
    {
        backend::copy(rhs, r);
        A.spmv(-1, x, 1, r);
    }
};

} // namespace backend
} // namespace amgcl

#endif
