mixedlocalgfeadolcstiffness.hh

#ifndef DUNE_GFE_MIXEDLOCALGFEADOLCSTIFFNESS_HH
#define DUNE_GFE_MIXEDLOCALGFEADOLCSTIFFNESS_HH

#include <adolc/adouble.h>            // use of active doubles
#include <adolc/drivers/drivers.h>    // use of "Easy to Use" drivers
// gradient(.) and hessian(.)
#include <adolc/interfaces.h>    // use of "Easy to Use" drivers
#include <adolc/taping.h>             // use of taping

#include <dune/gfe/adolcnamespaceinjections.hh>

#include <dune/common/fmatrix.hh>
#include <dune/istl/matrix.hh>

#include <dune/gfe/mixedlocalgeodesicfestiffness.hh>

#define ADOLC_VECTOR_MODE

/** \brief Assembles energy gradient and Hessian with ADOL-C (automatic differentiation)
 */
template<class GridView,
         class LocalFiniteElement0, class TargetSpace0,
         class LocalFiniteElement1, class TargetSpace1>
class MixedLocalGFEADOLCStiffness
    : public MixedLocalGeodesicFEStiffness<GridView,
                                           LocalFiniteElement0,TargetSpace0,
                                           LocalFiniteElement1,TargetSpace1>
{
    // grid types
    typedef typename GridView::Grid::ctype DT;
    typedef typename TargetSpace0::ctype RT;
    typedef typename GridView::template Codim<0>::Entity Entity;

    // The 'active' target spaces, i.e., the number type is replaced by adouble
    typedef typename TargetSpace0::template rebind<adouble>::other ATargetSpace0;
    typedef typename TargetSpace1::template rebind<adouble>::other ATargetSpace1;

    // some other sizes
    enum {gridDim=GridView::dimension};

public:

    //! Dimension of a tangent space
    enum { blocksize0 = TargetSpace0::TangentVector::dimension };
    enum { blocksize1 = TargetSpace1::TangentVector::dimension };

    //! Dimension of the embedding space
    enum { embeddedBlocksize0 = TargetSpace0::EmbeddedTangentVector::dimension };
    enum { embeddedBlocksize1 = TargetSpace1::EmbeddedTangentVector::dimension };

    MixedLocalGFEADOLCStiffness(const MixedLocalGeodesicFEStiffness<GridView,
                                                               LocalFiniteElement0, ATargetSpace0,
                                                               LocalFiniteElement1, ATargetSpace1>* energy)
    : localEnergy_(energy)
    {}

    /** \brief Compute the energy at the current configuration */
    virtual RT energy (const Entity& e,
                       const LocalFiniteElement0& localFiniteElement0,
                       const std::vector<TargetSpace0>& localConfiguration0,
                       const LocalFiniteElement1& localFiniteElement1,
                       const std::vector<TargetSpace1>& localConfiguration1) const;
#if 0
    /** \brief Assemble the element gradient of the energy functional

    This uses the automatic differentiation toolbox ADOL_C.
    */
    virtual void assembleGradient(const Entity& element,
                          const LocalFiniteElement& localFiniteElement,
                          const std::vector<TargetSpace>& solution,
                          std::vector<typename TargetSpace::TangentVector>& gradient) const;
#endif
    /** \brief Assemble the local stiffness matrix at the current position

    This uses the automatic differentiation toolbox ADOL_C.
    */
    virtual void assembleGradientAndHessian(const Entity& e,
                                            const LocalFiniteElement0& localFiniteElement0,
                                            const std::vector<TargetSpace0>& localConfiguration0,
                                            const LocalFiniteElement1& localFiniteElement1,
                                            const std::vector<TargetSpace1>& localConfiguration1,
                                            std::vector<typename TargetSpace0::TangentVector>& localGradient0,
                                            std::vector<typename TargetSpace1::TangentVector>& localGradient1);

    const MixedLocalGeodesicFEStiffness<GridView, LocalFiniteElement0, ATargetSpace0, LocalFiniteElement1, ATargetSpace1>* localEnergy_;

};


template <class GridView, class LocalFiniteElement0, class TargetSpace0, class LocalFiniteElement1, class TargetSpace1>
typename MixedLocalGFEADOLCStiffness<GridView, LocalFiniteElement0, TargetSpace0, LocalFiniteElement1, TargetSpace1>::RT
MixedLocalGFEADOLCStiffness<GridView, LocalFiniteElement0, TargetSpace0, LocalFiniteElement1, TargetSpace1>::
energy(const Entity& element,
       const LocalFiniteElement0& localFiniteElement0,
       const std::vector<TargetSpace0>& localConfiguration0,
       const LocalFiniteElement1& localFiniteElement1,
       const std::vector<TargetSpace1>& localConfiguration1) const
{
    double pureEnergy;

    std::vector<ATargetSpace0> localAConfiguration0(localConfiguration0.size());
    std::vector<ATargetSpace1> localAConfiguration1(localConfiguration1.size());

    trace_on(1);

    adouble energy = 0;

    // The following loop is not quite intuitive: we copy the localSolution into an
    // array of FieldVector<double>, go from there to FieldVector<adouble> and
    // only then to ATargetSpace.
    // Rationale: The constructor/assignment-from-vector of TargetSpace frequently
    // contains a projection onto the manifold from the surrounding Euclidean space.
    // ADOL-C needs a function on the whole Euclidean space, hence that projection
    // is part of the function and needs to be taped.

    // The following variable cannot be declared inside of the loop, or ADOL-C will report wrong results
    // (Presumably because several independent variables use the same memory location.)
    std::vector<typename ATargetSpace0::CoordinateType> aRaw0(localConfiguration0.size());
    for (size_t i=0; i<localConfiguration0.size(); i++) {
      typename TargetSpace0::CoordinateType raw = localConfiguration0[i].globalCoordinates();
      for (size_t j=0; j<raw.size(); j++)
        aRaw0[i][j] <<= raw[j];
      localAConfiguration0[i] = aRaw0[i];  // may contain a projection onto M -- needs to be done in adouble
    }

    std::vector<typename ATargetSpace1::CoordinateType> aRaw1(localConfiguration1.size());
    for (size_t i=0; i<localConfiguration1.size(); i++) {
      typename TargetSpace1::CoordinateType raw = localConfiguration1[i].globalCoordinates();
      for (size_t j=0; j<raw.size(); j++)
        aRaw1[i][j] <<= raw[j];
      localAConfiguration1[i] = aRaw1[i];  // may contain a projection onto M -- needs to be done in adouble
    }

    energy = localEnergy_->energy(element,
                                  localFiniteElement0,localAConfiguration0,
                                  localFiniteElement1,localAConfiguration1);

    energy >>= pureEnergy;

    trace_off();
#if 0
    size_t tape_stats[STAT_SIZE];
    tapestats(1,tape_stats);             // reading of tape statistics
    cout<<"maxlive "<<tape_stats[NUM_MAX_LIVES]<<"\n";
    cout<<"tay_stack_size "<<tape_stats[TAY_STACK_SIZE]<<"\n";
    cout<<"total number of operations "<<tape_stats[NUM_OPERATIONS]<<"\n";
    // ..... print other tape stats
#endif
    return pureEnergy;
}

#if 0
template <class GridView, class LocalFiniteElement, class TargetSpace>
void LocalGeodesicFEADOLCStiffness<GridView, LocalFiniteElement, TargetSpace>::
assembleGradient(const Entity& element,
                 const LocalFiniteElement& localFiniteElement,
                 const std::vector<TargetSpace>& localSolution,
                 std::vector<typename TargetSpace::TangentVector>& localGradient) const
{
    // Tape energy computation.  We may not have to do this every time, but it's comparatively cheap.
    energy(element, localFiniteElement, localSolution);

    // Compute the actual gradient
    size_t nDofs = localSolution.size();
    size_t nDoubles = nDofs*embeddedBlocksize;
    std::vector<double> xp(nDoubles);
    int idx=0;
    for (size_t i=0; i<nDofs; i++)
        for (size_t j=0; j<embeddedBlocksize; j++)
            xp[idx++] = localSolution[i].globalCoordinates()[j];

  // Compute gradient
    std::vector<double> g(nDoubles);
    gradient(1,nDoubles,xp.data(),g.data());                  // gradient evaluation

    // Copy into Dune type
    std::vector<typename TargetSpace::EmbeddedTangentVector> localEmbeddedGradient(localSolution.size());

    idx=0;
    for (size_t i=0; i<nDofs; i++)
        for (size_t j=0; j<embeddedBlocksize; j++)
            localEmbeddedGradient[i][j] = g[idx++];

//     std::cout << "localEmbeddedGradient:\n";
//     for (size_t i=0; i<nDofs; i++)
//       std::cout << localEmbeddedGradient[i] << std::endl;

    // Express gradient in local coordinate system
    for (size_t i=0; i<nDofs; i++) {
        Dune::FieldMatrix<RT,blocksize,embeddedBlocksize> orthonormalFrame = localSolution[i].orthonormalFrame();
        orthonormalFrame.mv(localEmbeddedGradient[i],localGradient[i]);
    }
}
#endif

// ///////////////////////////////////////////////////////////
//   Compute gradient and Hessian together
//   To compute the Hessian we need to compute the gradient anyway, so we may
//   as well return it.  This saves assembly time.
// ///////////////////////////////////////////////////////////
template <class GridType, class LocalFiniteElement0, class TargetSpace0, class LocalFiniteElement1, class TargetSpace1>
void MixedLocalGFEADOLCStiffness<GridType, LocalFiniteElement0, TargetSpace0, LocalFiniteElement1, TargetSpace1>::
assembleGradientAndHessian(const Entity& element,
                           const LocalFiniteElement0& localFiniteElement0,
                           const std::vector<TargetSpace0>& localConfiguration0,
                           const LocalFiniteElement1& localFiniteElement1,
                           const std::vector<TargetSpace1>& localConfiguration1,
                           std::vector<typename TargetSpace0::TangentVector>& localGradient0,
                           std::vector<typename TargetSpace1::TangentVector>& localGradient1)
{
    // Tape energy computation.  We may not have to do this every time, but it's comparatively cheap.
    energy(element, localFiniteElement0, localConfiguration0, localFiniteElement1, localConfiguration1);

    /////////////////////////////////////////////////////////////////
    // Compute the gradient.  It is needed to transform the Hessian
    // into the correct coordinates.
    /////////////////////////////////////////////////////////////////

    // Compute the actual gradient
    size_t nDofs0 = localConfiguration0.size();
    size_t nDofs1 = localConfiguration1.size();
    size_t nDoubles = nDofs0*embeddedBlocksize0 + nDofs1*embeddedBlocksize1;

    std::vector<double> xp(nDoubles);
    int idx=0;
    for (size_t i=0; i<localConfiguration0.size(); i++)
        for (size_t j=0; j<embeddedBlocksize0; j++)
            xp[idx++] = localConfiguration0[i].globalCoordinates()[j];

    for (size_t i=0; i<localConfiguration1.size(); i++)
        for (size_t j=0; j<embeddedBlocksize1; j++)
            xp[idx++] = localConfiguration1[i].globalCoordinates()[j];

  // Compute gradient
    std::vector<double> g(nDoubles);
    gradient(1,nDoubles,xp.data(),g.data());                  // gradient evaluation

    // Copy into Dune type
    std::vector<typename TargetSpace0::EmbeddedTangentVector> localEmbeddedGradient0(localConfiguration0.size());
    std::vector<typename TargetSpace1::EmbeddedTangentVector> localEmbeddedGradient1(localConfiguration1.size());

    idx=0;
    for (size_t i=0; i<localConfiguration0.size(); i++) {
        for (size_t j=0; j<embeddedBlocksize0; j++)
            localEmbeddedGradient0[i][j] = g[idx++];

        // Express gradient in local coordinate system
        localConfiguration0[i].orthonormalFrame().mv(localEmbeddedGradient0[i],localGradient0[i]);
    }

    for (size_t i=0; i<localConfiguration1.size(); i++) {
        for (size_t j=0; j<embeddedBlocksize1; j++)
            localEmbeddedGradient1[i][j] = g[idx++];

        // Express gradient in local coordinate system
        localConfiguration1[i].orthonormalFrame().mv(localEmbeddedGradient1[i],localGradient1[i]);
    }

    /////////////////////////////////////////////////////////////////
    // Compute Hessian
    /////////////////////////////////////////////////////////////////

    // We compute the Hessian of the energy functional using the ADOL-C system.
    // Since ADOL-C does not know about nonlinear spaces, what we get is actually
    // the Hessian of a prolongation of the energy functional into the surrounding
    // Euclidean space.  To obtain the Riemannian Hessian from this we apply the
    // formula described in Absil, Mahoney, Trumpf, "An extrinsic look at the Riemannian Hessian".
    // This formula consists of two steps:
    // 1) Remove all entries of the Hessian pertaining to the normal space of the
    //    manifold.  In the aforementioned paper this is done by projection onto the
    //    tangent space.  Since we want a matrix that is really smaller (but full rank again),
    //    we can achieve the same effect by multiplying the embedded Hessian from the left
    //    and from the right by the matrix of orthonormal frames.
    // 2) Add a correction involving the Weingarten map.
    //
    // This works, and is easy to implement using the ADOL-C "hessian" driver.
    // However, here we implement a small shortcut.  Computing the embedded Hessian and
    // multiplying one side by the orthonormal frame is the same as evaluating the Hessian
    // (seen as an operator from R^n to R^n) in the directions of the vectors of the
    // orthonormal frame.  By luck, ADOL-C can compute the evaluations of the Hessian in
    // a given direction directly (in fact, this is also how the "hessian" driver works).
    // Since there are less frame vectors than the dimension of the embedding space,
    // this reinterpretation allows to reduce the number of calls to ADOL-C.
    // In my Cosserat shell tests this reduced assembly time by about 10%.

    std::vector<Dune::FieldMatrix<RT,blocksize0,embeddedBlocksize0> > orthonormalFrame0(nDofs0);

    for (size_t i=0; i<nDofs0; i++)
        orthonormalFrame0[i] = localConfiguration0[i].orthonormalFrame();

    std::vector<Dune::FieldMatrix<RT,blocksize1,embeddedBlocksize1> > orthonormalFrame1(nDofs1);

    for (size_t i=0; i<nDofs1; i++)
        orthonormalFrame1[i] = localConfiguration1[i].orthonormalFrame();

    Dune::Matrix<Dune::FieldMatrix<double,blocksize0, embeddedBlocksize0> > embeddedHessian00(nDofs0,nDofs0);
    Dune::Matrix<Dune::FieldMatrix<double,blocksize0, embeddedBlocksize1> > embeddedHessian01(nDofs0,nDofs1);
    Dune::Matrix<Dune::FieldMatrix<double,blocksize1, embeddedBlocksize0> > embeddedHessian10(nDofs1,nDofs0);
    Dune::Matrix<Dune::FieldMatrix<double,blocksize1, embeddedBlocksize1> > embeddedHessian11(nDofs1,nDofs1);

#ifndef ADOLC_VECTOR_MODE
#error ADOL-C scalar mode not implemented
#if 0
    std::vector<double> v(nDoubles);
    std::vector<double> w(nDoubles);

    std::fill(v.begin(), v.end(), 0.0);

    for (int i=0; i<nDofs; i++)
      for (int ii=0; ii<blocksize; ii++)
      {
        // Evaluate Hessian in the direction of each vector of the orthonormal frame
        for (size_t k=0; k<embeddedBlocksize; k++)
          v[i*embeddedBlocksize + k] = orthonormalFrame[i][ii][k];

        int rc= 3;
        MINDEC(rc, hess_vec(1, nDoubles, xp.data(), v.data(), w.data()));
        if (rc < 0)
          DUNE_THROW(Dune::Exception, "ADOL-C has returned with error code " << rc << "!");

        for (int j=0; j<nDoubles; j++)
          embeddedHessian[i][j/embeddedBlocksize][ii][j%embeddedBlocksize] = w[j];

        // Make v the null vector again
        std::fill(&v[i*embeddedBlocksize], &v[(i+1)*embeddedBlocksize], 0.0);
      }
#endif
#else
    int n = nDoubles;
    int nDirections = nDofs0 * blocksize0 + nDofs1 * blocksize1;
    double* tangent[nDoubles];
    for(size_t i=0; i<nDoubles; i++)
        tangent[i] = (double*)malloc(nDirections*sizeof(double));

    double* rawHessian[nDoubles];
    for(size_t i=0; i<nDoubles; i++)
        rawHessian[i] = (double*)malloc(nDirections*sizeof(double));

    // Initialize directions field with zeros
    for (int j=0; j<nDirections; j++)
      for (int i=0; i<n; i++)
        tangent[i][j] = 0.0;

    for (int j=0; j<nDofs0*blocksize0; j++)
      for (int i=0; i<embeddedBlocksize0; i++)
        tangent[(j/blocksize0)*embeddedBlocksize0+i][j] = orthonormalFrame0[j/blocksize0][j%blocksize0][i];

    for (int j=0; j<nDofs1*blocksize1; j++)
      for (int i=0; i<embeddedBlocksize1; i++)
        tangent[nDofs0*embeddedBlocksize0 + (j/blocksize1)*embeddedBlocksize1+i][nDofs0*blocksize0 + j] = orthonormalFrame1[j/blocksize1][j%blocksize1][i];

    hess_mat(1,nDoubles,nDirections,xp.data(),tangent,rawHessian);

    // Copy Hessian into Dune data type
    size_t offset0 = nDofs0*embeddedBlocksize0;
    size_t offset1 = nDofs0*blocksize0;

    // upper left block
    for(size_t i=0; i<nDofs0*embeddedBlocksize0; i++)
      for (size_t j=0; j<nDofs0*blocksize0; j++)
        embeddedHessian00[j/blocksize0][i/embeddedBlocksize0][j%blocksize0][i%embeddedBlocksize0] = rawHessian[i][j];

    // upper right block
    for(size_t i=0; i<nDofs1*embeddedBlocksize1; i++)
      for (size_t j=0; j<nDofs0*blocksize0; j++)
        embeddedHessian01[j/blocksize0][i/embeddedBlocksize1][j%blocksize0][i%embeddedBlocksize1] = rawHessian[offset0+i][j];

    // lower left block
    for(size_t i=0; i<nDofs0*embeddedBlocksize0; i++)
      for (size_t j=0; j<nDofs1*blocksize1; j++)
        embeddedHessian10[j/blocksize1][i/embeddedBlocksize0][j%blocksize1][i%embeddedBlocksize0] = rawHessian[i][offset1+j];

    // lower right block
    for(size_t i=0; i<nDofs1*embeddedBlocksize1; i++)
      for (size_t j=0; j<nDofs1*blocksize1; j++)
        embeddedHessian11[j/blocksize1][i/embeddedBlocksize1][j%blocksize1][i%embeddedBlocksize1] = rawHessian[offset0+i][offset1+j];

    for(size_t i=0; i<nDoubles; i++) {
        free(rawHessian[i]);
        free(tangent[i]);
    }
#endif

    // From this, compute the Hessian with respect to the manifold (which we assume here is embedded
    // isometrically in a Euclidean space.
    // For the detailed explanation of the following see: Absil, Mahoney, Trumpf, "An extrinsic look
    // at the Riemannian Hessian".

    this->A00_.setSize(nDofs0,nDofs0);

    for (size_t col=0; col<nDofs0; col++) {

        for (size_t subCol=0; subCol<blocksize0; subCol++) {

            typename TargetSpace0::EmbeddedTangentVector z = orthonormalFrame0[col][subCol];

            // P_x \partial^2 f z
            for (size_t row=0; row<nDofs0; row++) {
                typename TargetSpace0::TangentVector semiEmbeddedProduct;
                embeddedHessian00[row][col].mv(z,semiEmbeddedProduct);

                for (int subRow=0; subRow<blocksize0; subRow++)
                    this->A00_[row][col][subRow][subCol] = semiEmbeddedProduct[subRow];
            }

        }

    }

    this->A01_.setSize(nDofs0,nDofs1);

    for (size_t col=0; col<nDofs1; col++) {

        for (size_t subCol=0; subCol<blocksize1; subCol++) {

            typename TargetSpace1::EmbeddedTangentVector z = orthonormalFrame1[col][subCol];

            // P_x \partial^2 f z
            for (size_t row=0; row<nDofs0; row++) {
                typename TargetSpace0::TangentVector semiEmbeddedProduct;
                embeddedHessian01[row][col].mv(z,semiEmbeddedProduct);

                for (int subRow=0; subRow<blocksize0; subRow++)
                    this->A01_[row][col][subRow][subCol] = semiEmbeddedProduct[subRow];
            }

        }

    }

    this->A10_.setSize(nDofs1,nDofs0);

    for (size_t col=0; col<nDofs0; col++) {

        for (size_t subCol=0; subCol<blocksize0; subCol++) {

            typename TargetSpace0::EmbeddedTangentVector z = orthonormalFrame0[col][subCol];

            // P_x \partial^2 f z
            for (size_t row=0; row<nDofs1; row++) {
                typename TargetSpace1::TangentVector semiEmbeddedProduct;
                embeddedHessian10[row][col].mv(z,semiEmbeddedProduct);

                for (int subRow=0; subRow<blocksize1; subRow++)
                    this->A10_[row][col][subRow][subCol] = semiEmbeddedProduct[subRow];
            }

        }

    }

    this->A11_.setSize(nDofs1,nDofs1);

    for (size_t col=0; col<nDofs1; col++) {

        for (size_t subCol=0; subCol<blocksize1; subCol++) {

            typename TargetSpace1::EmbeddedTangentVector z = orthonormalFrame1[col][subCol];

            // P_x \partial^2 f z
            for (size_t row=0; row<nDofs1; row++) {
                typename TargetSpace1::TangentVector semiEmbeddedProduct;
                embeddedHessian11[row][col].mv(z,semiEmbeddedProduct);

                for (int subRow=0; subRow<blocksize1; subRow++)
                    this->A11_[row][col][subRow][subCol] = semiEmbeddedProduct[subRow];
            }

        }

    }

    //////////////////////////////////////////////////////////////////////////////////////
    //  Further correction due to non-planar configuration space
    //  + \mathfrak{A}_x(z,P^\orth_x \partial f)
    //////////////////////////////////////////////////////////////////////////////////////

    // Project embedded gradient onto normal space
    std::vector<typename TargetSpace0::EmbeddedTangentVector> projectedGradient0(nDofs0);
    for (size_t i=0; i<nDofs0; i++)
      projectedGradient0[i] = localConfiguration0[i].projectOntoNormalSpace(localEmbeddedGradient0[i]);

    std::vector<typename TargetSpace1::EmbeddedTangentVector> projectedGradient1(nDofs1);
    for (size_t i=0; i<nDofs1; i++)
      projectedGradient1[i] = localConfiguration1[i].projectOntoNormalSpace(localEmbeddedGradient1[i]);

    // The Weingarten map has only diagonal entries
    for (size_t row=0; row<nDofs0; row++) {

      for (size_t subRow=0; subRow<blocksize0; subRow++) {

        typename TargetSpace0::EmbeddedTangentVector z = orthonormalFrame0[row][subRow];
        typename TargetSpace0::EmbeddedTangentVector tmp1 = localConfiguration0[row].weingarten(z,projectedGradient0[row]);

        typename TargetSpace0::TangentVector tmp2;
        orthonormalFrame0[row].mv(tmp1,tmp2);

        this->A00_[row][row][subRow] += tmp2;
      }

    }

    for (size_t row=0; row<nDofs1; row++) {

      for (size_t subRow=0; subRow<blocksize1; subRow++) {

        typename TargetSpace1::EmbeddedTangentVector z = orthonormalFrame1[row][subRow];
        typename TargetSpace1::EmbeddedTangentVector tmp1 = localConfiguration1[row].weingarten(z,projectedGradient1[row]);

        typename TargetSpace1::TangentVector tmp2;
        orthonormalFrame1[row].mv(tmp1,tmp2);

        this->A11_[row][row][subRow] += tmp2;
      }

    }

//     std::cout << "ADOL-C stiffness:\n";
//     printmatrix(std::cout, this->A_, "foo", "--");
}

#endif