gsPatchPreconditionersCreator.hpp

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#pragma once
 
#include <gsSolver/gsSumOp.h>
#include <gsSolver/gsProductOp.h>
#include <gsSolver/gsKroneckerOp.h>
#include <gsSolver/gsMatrixOp.h>
#include <gsAssembler/gsExprAssembler.h>
#include <gsNurbs/gsTensorBSplineBasis.h>
 
namespace gismo
{
 
namespace {
 
template<typename T>
gsBoundaryConditions<T> boundaryConditionsForDirection( const gsBoundaryConditions<T>& bc, index_t direction )
{
    gsBoundaryConditions<T> result;
 
    for ( index_t i = 1; i <= 2; ++i)
    {
        patchSide global(0,i+2*direction), local(0,i);
        const boundary_condition<T>* cond = bc.getConditionFromSide(global);
        if (cond)
            result.addCondition(local,cond->type(),cond->function());
    }
    return result;
}
 
template<typename T>
void eliminateDirichlet1D(const gsBoundaryConditions<T>& bc,
                          const gsOptionList& opt, gsSparseMatrix<T> & result)
{
    dirichlet::strategy ds = (dirichlet::strategy)opt.askInt("DirichletStrategy",dirichlet::elimination);
    if (ds == dirichlet::elimination)
    {
        patchSide west(0,boundary::west), east(0,boundary::east);
        index_t i = 0;
        if (bc.getConditionFromSide( west ) && bc.getConditionFromSide( west )->type() == condition_type::dirichlet ) i += 1;
        if (bc.getConditionFromSide( east ) && bc.getConditionFromSide( east )->type() == condition_type::dirichlet ) i += 2;
        if (i%2 + i/2 >= result.rows() || i%2 + i/2 >= result.cols())
            result.resize(0,0);
        else switch ( i )
             {
             case 0: break;
             case 1: result = result.block( 1, 1, result.rows()-1, result.cols()-1 ); break;
             case 2: result = result.block( 0, 0, result.rows()-1, result.cols()-1 ); break;
             case 3: result = result.block( 1, 1, result.rows()-2, result.cols()-2 ); break;
             }
    }
    else
        GISMO_ERROR("Unknown Dirichlet strategy.");
}
 
template<typename T>
gsSparseMatrix<T> assembleMass(const gsBasis<T>& basis)
{
    gsExprAssembler<T> mass(1,1);
    gsMultiBasis<T> mb(basis);
    mass.setIntegrationElements(mb);
    typename gsExprAssembler<T>::space u = mass.getSpace(mb);
    mass.initMatrix();
    mass.assemble( u * u.tr() );
    gsSparseMatrix<T> result;
    mass.matrix_into(result);
    return result;
}
 
template<typename T>
gsSparseMatrix<T> assembleStiffness(const gsBasis<T>& basis)
{
    gsExprAssembler<T> stiff(1,1);
    gsMultiBasis<T> mb(basis);
    stiff.setIntegrationElements(mb);
    typename gsExprAssembler<T>::space u = stiff.getSpace(mb);
    stiff.initMatrix();
    stiff.assemble( grad(u) * grad(u).tr() );
    gsSparseMatrix<T> result;
    stiff.matrix_into(result);
    return result;
}
 
 
template<typename T>
std::vector< gsSparseMatrix<T> > assembleTensorMass(
    const gsBasis<T>& basis,
    const gsBoundaryConditions<T>& bc,
    const gsOptionList& opt
    )
{
    const index_t d = basis.dim();
    std::vector< gsSparseMatrix<T> > result(d);
    for ( index_t i=0; i!=d; ++i )
    {
        result[i] = assembleMass(basis.component(d-1-i));
        eliminateDirichlet1D(boundaryConditionsForDirection(bc,d-1-i), opt, result[i]);
    }
    return result;
}
 
template<typename T>
std::vector< gsSparseMatrix<T> > assembleTensorStiffness(
    const gsBasis<T>& basis,
    const gsBoundaryConditions<T>& bc,
    const gsOptionList& opt
    )
{
    const index_t d = basis.dim();
    std::vector< gsSparseMatrix<T> > result(d);
    for ( index_t i=0; i!=d; ++i )
    {
        result[i] = assembleStiffness(basis.component(d-1-i));
        eliminateDirichlet1D(boundaryConditionsForDirection(bc,d-1-i), opt, result[i]);
    }
    return result;
}
 
} // anonymous namespace
 
template<typename T>
gsSparseMatrix<T> gsPatchPreconditionersCreator<T>::massMatrix(
    const gsBasis<T>& basis,
    const gsBoundaryConditions<T>& bc,
    const gsOptionList& opt
    )
{
    std::vector< gsSparseMatrix<T> > local_mass = assembleTensorMass(basis, bc, opt);
    const index_t d = local_mass.size();
    gsSparseMatrix<T> result = local_mass[d-1];
    for (index_t i=d-2; i>-1; --i)
        result = local_mass[i].kron(result);
    return result;
}
 
template<typename T>
typename gsPatchPreconditionersCreator<T>::OpUPtr gsPatchPreconditionersCreator<T>::massMatrixOp(
    const gsBasis<T>& basis,
    const gsBoundaryConditions<T>& bc,
    const gsOptionList& opt
    )
{
    const index_t d = basis.dim();
 
    std::vector< gsSparseMatrix<T> > local_mass = assembleTensorMass(basis, bc, opt);
 
    std::vector<OpPtr> local_mass_op(d);
    for (index_t i=0; i<d; ++i)
        local_mass_op[i] = makeMatrixOp(local_mass[i].moveToPtr());
 
    return gsKroneckerOp<T>::make(local_mass_op);
}
 
template<typename T>
typename gsPatchPreconditionersCreator<T>::OpUPtr gsPatchPreconditionersCreator<T>::massMatrixInvOp(
    const gsBasis<T>& basis,
    const gsBoundaryConditions<T>& bc,
    const gsOptionList& opt
    )
{
    const index_t d = basis.dim();
 
    std::vector< gsSparseMatrix<T> > local_mass = assembleTensorMass(basis, bc, opt);
 
    std::vector<OpPtr> local_mass_op(d);
    for (index_t i=0; i<d; ++i)
        local_mass_op[i] = makeSparseCholeskySolver(local_mass[i]);
 
    return gsKroneckerOp<T>::make(local_mass_op);
}
 
template<typename T>
gsSparseMatrix<T> gsPatchPreconditionersCreator<T>::stiffnessMatrix(
    const gsBasis<T>& basis,
    const gsBoundaryConditions<T>& bc,
    const gsOptionList& opt,
    T alpha,
    T beta
    )
{
    const index_t d = basis.dim();
 
    std::vector< gsSparseMatrix<T> > local_stiff = assembleTensorStiffness(basis, bc, opt);
    std::vector< gsSparseMatrix<T> > local_mass  = assembleTensorMass(basis, bc, opt);
 
    if ( beta!=1 )
    {
        for (index_t i=0; i!=d; ++i)
            local_stiff[i] *= beta;
    }
 
    gsSparseMatrix<T> K = give(local_stiff[d-1]);
    gsSparseMatrix<T> M = give(local_mass [d-1]);
 
    for (index_t i=d-2; i>-1; --i)
    {
        K  = local_mass[i].kron(K);
        K += local_stiff[i].kron(M);
        if ( i != 0 || alpha != 0 )
            M = local_mass[i].kron(M);
    }
    if (alpha==1)
        K += M;
    else if (alpha!=0)
        K += alpha * M;
    return K;
}
 
template<typename T>
typename gsPatchPreconditionersCreator<T>::OpUPtr gsPatchPreconditionersCreator<T>::stiffnessMatrixOp(
    const gsBasis<T>& basis,
    const gsBoundaryConditions<T>& bc,
    const gsOptionList& opt,
    T alpha,
    T beta
    )
{
    const index_t d = basis.dim();
 
    std::vector< gsSparseMatrix<T> > local_stiff = assembleTensorStiffness(basis, bc, opt);
    std::vector< gsSparseMatrix<T> > local_mass  = assembleTensorMass(basis, bc, opt);
 
    std::vector<OpUPtr> local_stiff_op(d);
    std::vector<OpPtr > local_mass_op (d);
    for (index_t i=0; i<d; ++i)
    {
        if (beta!=1)
            local_stiff[i] *= beta;
        local_stiff_op[i] = makeMatrixOp(local_stiff[i].moveToPtr());
        local_mass_op [i] = makeMatrixOp(local_mass [i].moveToPtr());
    }
    OpUPtr K = give(local_stiff_op[0]);
    OpPtr  M = give(local_mass_op [0]);
 
    for (index_t i=1; i<d; ++i)
    {
        K = gsSumOp<T>::make(
            gsKroneckerOp<T>::make( give(K),      local_mass_op [i]  ),
            gsKroneckerOp<T>::make( M,       give(local_stiff_op[i]) )
            );
        if ( i < d-1 || alpha != 0 )
            M = gsKroneckerOp<T>::make(M, local_mass_op[i]);
    }
    if (alpha==1)
        K = gsSumOp<T>::make( give(K), M );
    else if (alpha!=0)
        K = gsSumOp<T>::make( give(K), gsScaledOp<T>::make( M, alpha ) );
 
    return K;
}
 
template<typename T>
typename gsPatchPreconditionersCreator<T>::OpUPtr gsPatchPreconditionersCreator<T>::fastDiagonalizationOp(
    const gsBasis<T>& basis,
    const gsBoundaryConditions<T>& bc,
    const gsOptionList& opt,
    T alpha,
    T beta,
    T gamma
    )
{
    GISMO_ASSERT ( beta != 0, "gsPatchPreconditionersCreator<T>::fastDiagonalizationOp() does not work for beta==0." );
 
    const index_t d = basis.dim();
 
    // Assemble univariate
    std::vector< gsSparseMatrix<T> > local_stiff = assembleTensorStiffness(basis, bc, opt);
    std::vector< gsSparseMatrix<T> > local_mass  = assembleTensorMass(basis, bc, opt);
 
    if (beta!=0)
    {
        for (index_t i=0; i!=d; ++i)
            local_stiff[i] *= beta;
    }
 
    // Determine overall size
    index_t sz = 1;
    for ( index_t i=0; i<d; ++i )
        sz *= local_stiff[i].rows();
 
    // Initialize the diagonal with 1
    gsMatrix<T> diag;
    diag.setConstant(sz,1,alpha); // This is the pure-mass part!
 
    index_t glob = sz; // Indexing value for setting up the Kronecker product
 
    typedef typename gsMatrix<T>::GenSelfAdjEigenSolver EVSolver;
    typedef typename EVSolver::EigenvectorsType evMatrix;
    typedef typename EVSolver::RealVectorType evVector;
    EVSolver ges;
 
    std::vector<OpPtr> Qop(d);
    std::vector<OpPtr> QTop(d);
    gsMatrix<T> ev;
 
    T avg_term = gamma;
 
    // Now, setup the Q's and update the D's
    for ( index_t i=0; i<d; ++i )
    {
        // Solve generalized eigenvalue problem
        ges.compute(local_stiff[i], local_mass[i], gsEigen::ComputeEigenvectors);
        // Q^T M Q = I, or M = Q^{-T} Q^{-1}
        // Q^T K Q = D, or K = Q^{-T} D Q^{-1}
 
        if (gamma != (T)(0))
        {
            gsMatrix<T> etrans = ges.eigenvectors().transpose()*local_mass[i]*gsMatrix<T>::Ones(local_mass[i].rows(),1);
            GISMO_ASSERT((etrans.block(1, 0, etrans.rows()-1, 1).array() < (T)(1)/100000000).all(),
                "gsPatchPreconditionerCreator::fastDiagonalizationOp: gamma!=0 only allowed for pure Neumann.");
            avg_term *= etrans(0,0) * etrans(0,0);
        }
 
        // From the eigenvalues, we setup the matrix D already in an Kroneckerized way.
        const evVector & D = ges.eigenvalues();
 
        const index_t loc = D.rows();
        glob /= loc;
        const index_t glob2 = sz / loc / glob;
 
        for ( index_t l=0; l<loc; ++l )
            for ( index_t m=0; m<glob; ++m )
                for ( index_t n=0; n<glob2; ++n )
                    diag( m + l*glob + n*loc*glob, 0 ) += D(l,0);
 
        // Finally, we store the eigenvectors
        ev.swap(const_cast<evMatrix&>(ges.eigenvectors()));
 
        // These are the operators representing the eigenvectors
        typename gsMatrixOp< gsMatrix<T> >::Ptr matrOp = makeMatrixOp( ev.moveToPtr() );
        Qop [i] = matrOp;
        // Here we are safe as long as we do not want to apply QTop after Qop got destroyed.
        QTop[i] = makeMatrixOp( matrOp->matrix().transpose() );
    }
 
    GISMO_ASSERT( glob == 1,
        "gsPatchPreconditionerCreator::fastDiagonalizationOp: Internal error." );
 
    diag(0,0) += avg_term;
 
    for ( index_t l=0; l<sz; ++l )
        diag( l, 0 ) = (T)(1)/diag( l, 0 );
 
    memory::unique_ptr< gsEigen::DiagonalMatrix<T,Dynamic> > diag_mat( new gsEigen::DiagonalMatrix<T,Dynamic>( give(diag) ) );
 
    return gsProductOp<T>::make(
        gsKroneckerOp<T>::make(QTop),
        makeMatrixOp(give(diag_mat)),
        gsKroneckerOp<T>::make(Qop)
        );
}
 
namespace {
 
// Get the tilde basis
template<typename T>
void tildeSpaceBasis_oneside(const gsTensorBSplineBasis<1,T>& basis, bool isLeftHandSide, gsMatrix<T>& tildeBasis, gsMatrix<T>& complBasis, bool bc = 0, const bool odd = true)
{
    // bc == false: Neumann (or any other not-eliminating bc), bc == true: Dirichlet
 
    const index_t b = (index_t)bc;
    const index_t p = basis.degree();
    const T h = basis.knots().maxIntervalLength();
 
    if (p-b<=0)
    {
        tildeBasis.resize(0,0); complBasis.resize(0,0);
        return;
    }
 
    const T u = (isLeftHandSide ? basis.knots().first() : basis.knots().last());
    gsMatrix<T> U(1,1);
    U(0,0) = u;
 
    std::vector< gsMatrix<T> > allDerivs;
    basis.evalAllDers_into(U, p-1, allDerivs);
 
    // Collect all derivatives in matrix (rows: derivatives, columns: basis functions)
    // normalize with h^(deriv)
    // use only odd derivatives if odd = true and only even derivatives if odd = false
    // skip last (on left) or first (on right) basis function since it's always in S-tilde
    gsMatrix<T> derivs;
    // If we have Dirichlet bc, we reduce the number of rows and columns by 1. We basically
    // eliminate the first row and the first column (first basis function, 0th derivative)
    // for the right-side, we have to remove the last basis function.
    derivs.setZero(p-b, p-b);
 
    const index_t offset = (isLeftHandSide ? b : 1);
 
    index_t i_start;
    if (odd) i_start = 1;
    else i_start = 2*b;
 
    for (index_t i = i_start; i < p; i += 2)
        for (index_t j = 0; j < p-b; ++j)
            derivs(i-b, j) = math::pow(h, i) * allDerivs[i](j+offset);
 
    typename gsMatrix<T>::JacobiSVD svd = derivs.jacobiSvd(gsEigen::ComputeFullV);
 
    index_t n_tilde;
    if (odd) n_tilde = (p + 1) / 2 - b;
    else n_tilde = p / 2 - b;
 
    tildeBasis = svd.matrixV().rightCols(n_tilde);
    complBasis = svd.matrixV().leftCols(p - b - n_tilde);
}
 
// Compute a basis for S-tilde and one for its orthogonal complement
template<typename T>
void tildeSpaceBasis(const gsBasis<T>& basis, gsSparseMatrix<T>& B_tilde, gsSparseMatrix<T>& B_compl, const gsBoundaryConditions<T>& bc, const bool odd = true)
{
    GISMO_ASSERT( nullptr != (dynamic_cast<const gsTensorBSplineBasis<1,T>*>(&basis)),
                    "gsPatchPreconditionersCreator<T>::getTildeSpaceBasisTransformation and "
                    "gsPatchPreconditionersCreator<T>::subspaceCorrectedMassSmootherOp only work with tensor-B-spline bases." );
    const gsTensorBSplineBasis<1,T>& bbasis = static_cast<const gsTensorBSplineBasis<1,T>&>(basis);
 
    patchSide west(0,boundary::west), east(0,boundary::east);
    bool bwest = ( bc.getConditionFromSide( west ) && bc.getConditionFromSide( west )->type() == condition_type::dirichlet );
    bool beast = ( bc.getConditionFromSide( east ) && bc.getConditionFromSide( east )->type() == condition_type::dirichlet );
 
    gsMatrix<T> b_L, b_compl_L;
    gsMatrix<T> b_R, b_compl_R;
 
    // Contruct space with vanishing odd derivatives
    tildeSpaceBasis_oneside(bbasis, true,  b_L, b_compl_L, bwest, odd);
    tildeSpaceBasis_oneside(bbasis, false, b_R, b_compl_R, beast, odd);
 
    const index_t n = bbasis.size() - (index_t)bwest - (index_t)beast;
    const index_t n_L = b_L.cols();
    const index_t m_L = b_L.rows();
    const index_t n_R = b_R.cols();
    const index_t m_R = b_R.rows();
    const index_t n_c_L = b_compl_L.cols();
    const index_t m_c_L = b_compl_L.rows();
    const index_t n_c_R = b_compl_R.cols();
    const index_t m_c_R = b_compl_R.rows();
    const index_t n_I = n - n_L - n_R - n_c_L - n_c_R;
 
    //GISMO_ENSURE ( n_I >= 0, "tildeSpaceBasis: Too few knots for that spline degree." );
    if ( n_I <= 0 )
    {
        static bool warned = false;
        if (!warned)
        {
            gsWarn << "tildeSpaceBasis was called with too few knots for that spline degree.\n"
                   << "So, we assume that S_tilde is empty.\n";
            warned = true;
        }
 
        B_tilde.resize(n,0);
 
        gsSparseEntries<T> E_compl;
        E_compl.reserve(n);
        for (index_t i = 0; i < n; ++i)
            E_compl.add(i,i,1.0);
        B_compl.resize(n,n);
        B_compl.setFrom(E_compl);
        return;
    }
 
    gsSparseEntries<T> E_tilde, E_compl;
 
    // put b_L into upper left block of S-tilde basis
    for (index_t j = 0; j < n_L; ++j)
    {
        for (index_t i = 0; i < m_L; ++i)
            E_tilde.add(i, j, b_L(i, j));
    }
    // fill identity matrix into interior part of S-tilde basis
    for (index_t j = 0; j < n_I; ++j)
    {
        E_tilde.add(m_L + j, n_L + j, 1.0);
    }
    // put b_R into lower right block of S-tilde basis
    for (index_t j = 0; j < n_R; ++j)
    {
        for (index_t i = 0; i < m_R; ++i)
            E_tilde.add(m_L + n_I + i, n_L + n_I + j, b_R(i, j));
    }
    B_tilde.resize(n, n_L + n_I + n_R);
    B_tilde.setFrom(E_tilde);
 
    // put b_compl_L into upper left block of complement basis
    for (index_t j = 0; j < n_c_L; ++j)
    {
        for (index_t i = 0; i < m_c_L; ++i)
            E_compl.add(i, j, b_compl_L(i, j));
    }
    // put b_compl_R into lower right block of complement basis
    for (index_t j = 0; j < n_c_R; ++j)
    {
        for (index_t i = 0; i < m_c_R; ++i)
            E_compl.add(m_c_L + n_I + i, n_c_L + j, b_compl_R(i, j));
    }
    B_compl.resize(n, n_c_L + n_c_R);
    B_compl.setFrom(E_compl);
 
}
 
template<typename T>
void constructTildeSpaceBasis(
    const gsBasis<T>& basis,
    const gsBoundaryConditions<T>& bc,
    const gsOptionList& opt,
    std::vector< gsSparseMatrix<T> >& B_tilde,
    std::vector< gsSparseMatrix<T> >& B_l2compl,
    const bool odd = true
    )
{
    const index_t d = basis.dim();
    B_tilde.resize(d);
    B_l2compl.resize(d);
    dirichlet::strategy ds = (dirichlet::strategy)opt.askInt("DirichletStrategy",dirichlet::elimination);
    if (ds == dirichlet::elimination)
    {
        for ( index_t i=0; i<d; ++i )
            tildeSpaceBasis(basis.component(d-1-i), B_tilde[i], B_l2compl[i], boundaryConditionsForDirection(bc,d-1-i), odd);
    }
    else
        GISMO_ERROR("Unknown Dirichlet strategy.");
 
}
 
// Constructs a matrix for swapping a tensor product
// from A (x) B (x) C (x) D (x) E  to   A (x) D (x) C (x) B (x) E,
// where only the dimensions of those matrices have to be given.
// Note that also those A, B, etc. could be tensor products (here as dimension just provide the products)
// Note that also those A, B, etc. could vanish. Then just provide a 1 as dimension, i.e., a scalar.
// So, literally every thinkable swap is possible.
template<typename T>
gsSparseMatrix<T> kroneckerSwap( index_t e, index_t d, index_t c, index_t b, index_t a )
{
    const index_t sz = a*b*c*d*e;
    gsSparseMatrix<T> result(sz,sz);
    gsSparseEntries<T> entries;
    entries.reserve(sz);
    for ( index_t i=0; i<a; ++i )
        for ( index_t j=0; j<b; ++j )
            for ( index_t k=0; k<c; ++k )
                for ( index_t l=0; l<d; ++l )
                    for ( index_t m=0; m<e; ++m )
                        entries.add( i+a*(j+b*(k+c*(l+d*m))), i+a*(l+d*(k+c*(j+b*m))), 1. );
 
    result.setFrom(entries);
    result.makeCompressed();
 
    return result;
}
 
} // anonymous namespace
 
template<typename T>
std::pair< std::vector< gsSparseMatrix<T> >, std::vector< gsSparseMatrix<T> > > gsPatchPreconditionersCreator<T>::getTildeSpaceBasisTransformation(
    const gsBasis<T>& basis,
    const gsBoundaryConditions<T>& bc,
    const gsOptionList& opt
    )
{
    std::vector< gsSparseMatrix<T> > B_tilde, B_l2compl;
    constructTildeSpaceBasis( basis, bc, opt, B_tilde, B_l2compl, !opt.askSwitch( "UseVanishingEvenDerivatives", false ) );
    return std::pair< std::vector< gsSparseMatrix<T> >, std::vector< gsSparseMatrix<T> > >( B_tilde, B_l2compl );
}
 
template<typename T>
typename gsPatchPreconditionersCreator<T>::OpUPtr gsPatchPreconditionersCreator<T>::subspaceCorrectedMassSmootherOp(
    const gsBasis<T>& basis,
    const gsBoundaryConditions<T>& bc,
    const gsOptionList& opt,
    T sigma,
    T alpha,
    T beta
    )
{
    GISMO_ASSERT ( beta != 0, "gsPatchPreconditionersCreator<T>::subspaceCorrectedMassSmootherOp() does not work for beta==0." );
 
    // Get some properties
    const index_t d = basis.dim();
    const T h = basis.getMinCellLength();
 
    // Assemble univariate
    std::vector< gsSparseMatrix<T> > local_stiff = assembleTensorStiffness(basis, bc, opt);
    std::vector< gsSparseMatrix<T> > local_mass  = assembleTensorMass(basis, bc, opt);
 
    if (beta != 1)
    {
        for (index_t i=0; i!=d; ++i)
            local_stiff[i] *= beta;
    }
 
    // Setup of basis
    std::vector< gsSparseMatrix<T> > B_tilde(d), B_l2compl(d);
    constructTildeSpaceBasis(basis, bc, opt, B_tilde, B_l2compl);
 
    std::vector< gsSparseMatrix<T> > M_compl(d), K_compl(d), B_compl(d);
    std::vector< typename gsLinearOperator<T>::Ptr > M_tilde_inv(d);
    for ( index_t i=0; i<d; ++i )
    {
        // Transform the complement
        typename gsLinearOperator<T>::Ptr M_inv = makeSparseCholeskySolver(local_mass[i]);
        gsMatrix<T> B_compl_dense;
        M_inv->apply( B_l2compl[i], B_compl_dense );
        B_compl[i] = B_compl_dense.sparseView();
 
        // Setup of matrices and corresponding solvers
        gsSparseMatrix<T> M_tilde = B_tilde[i].transpose() * local_mass[i] * B_tilde[i];
        M_tilde_inv[i] = makeSparseCholeskySolver( M_tilde );
        M_compl[i] = B_compl[i].transpose() * local_mass[i] * B_compl[i];
        K_compl[i] = B_compl[i].transpose() * local_stiff[i] * B_compl[i];
    }
 
    // Setup of final operator
    typename gsSumOp<T>::uPtr result = gsSumOp<T>::make();
 
    for ( index_t type = 0; type < (1<<d); ++ type )
    {
        std::vector< typename gsLinearOperator<T>::Ptr > correction(0);
        gsSparseMatrix<T> transfer;
 
        std::vector< gsSparseMatrix<T>* > transfers(d);
 
        index_t numberInteriors = 0;
 
        // Setup of transfer
        for ( index_t j = 0; j<d; ++ j )
        {
            if ( type & ( 1 << j ) )
                transfers[j] = &(B_compl[d-1-j]);
            else
            {
                transfers[j] = &(B_tilde[d-1-j]);
                ++numberInteriors;
            }
 
            if ( j == 0 )
                transfer = *(transfers[j]);
            else
                transfer = transfers[j]->kron(transfer);
 
        }
 
        // If the subspace is not present, ignore it.
        if ( transfer.cols() == 0 )
            continue;
 
        // Setup of swap, where the boundary part is shifted to the begin
 
        index_t left = 1, current = transfers[d-1]->cols(), right = 1;
        for ( index_t j = 0; j < d-1; ++j )
            left *= transfers[j]->cols();
 
        for ( index_t j = d-1; j >= 0; --j )
        {
            if ( type & ( 1 << j ) )
            {
                transfer = transfer * kroneckerSwap<T>( right, current, left, 1, 1 );
                if ( j > 0 )
                {
                    left *= current;
                    current = transfers[j-1]->cols();
                    left /= current;
                }
            }
            else
            {
                if ( j > 0 )
                {
                    right *= current;
                    current = transfers[j-1]->cols();
                    left /= current;
                }
            }
 
        }
 
        // Setup of interior correction
        for ( index_t j = d-1; j>=0; --j )
        {
            if ( ! ( type & ( 1 << j ) ) )
                correction.push_back( M_tilde_inv[d-1-j] );
        }
 
        // If we are in the interior, we have to do the scaling here as there is no boundary correction.
        if ( numberInteriors == d )
            correction[0] = gsScaledOp<T>::make( correction[0], (T)(1)/( alpha + beta*(T)(numberInteriors)/(sigma*h*h) ) );
 
        // Setup of bondary correction
        if ( numberInteriors < d )
        {
            gsSparseMatrix<T> bc_matrix;
 
            {
                gsSparseMatrix<T> s(0,0);
                for ( index_t k = d-1; k>=0; --k )
                {
                    if ( type & ( 1 << k ) )
                    {
                        if ( s.rows() == 0 )
                            s = M_compl[d-1-k];
                        else
                            s = M_compl[d-1-k].kron(s);
                    }
                }
                bc_matrix = ( alpha + beta*(T)(numberInteriors)/(sigma*h*h) ) * s;
            }
 
            for ( index_t j = d-1; j>=0; --j )
            {
                if ( type & ( 1 << j ) )
                {
                    gsSparseMatrix<T> s(0,0);
                    for ( index_t k = d-1; k>=0; --k )
                    {
                        if ( type & ( 1 << k ) )
                        {
                            gsSparseMatrix<T>* chosenMatrix;
                            if ( j == k )
                                chosenMatrix = &(K_compl[d-1-k]);
                            else
                                chosenMatrix = &(M_compl[d-1-k]);
 
                            if ( s.rows() == 0 )
                                s = *chosenMatrix;
                            else
                                s = s.kron(*chosenMatrix);
                        }
                    }
                    bc_matrix += s;
                }
            }
 
            correction.push_back(makeSparseCholeskySolver(bc_matrix));
        }
 
        typename gsMatrixOp< gsSparseMatrix<T> >::Ptr transOp = makeMatrixOp(transfer.moveToPtr());
        // Setup of whole operator
        // The correction is the Kronecker-product of the operators in the vector correction.
        result->addOperator(
            gsProductOp<T>::make(
                makeMatrixOp( transOp->matrix().transpose() ),
                gsKroneckerOp<T>::make( correction ),
                transOp
                )
            );
    }
 
    return give(result);
 
}
 
} // namespace gismo