targetspacetest.cc

#include <config.h>

#include <dune/gfe/spaces/hyperbolichalfspacepoint.hh>
#include <dune/gfe/spaces/productmanifold.hh>
#include <dune/gfe/spaces/unitvector.hh>
#include <dune/gfe/spaces/realtuple.hh>
#include <dune/gfe/spaces/rotation.hh>

#include "valuefactory.hh"

using namespace Dune;


/** \file
    \brief Unit tests for classes that implement value manifolds for geodesic FE functions
 */

/** \brief Computes the diameter of a set */
template <class TargetSpace>
double diameter(const std::vector<TargetSpace>& v)
{
  double d = 0;
  for (size_t i=0; i<v.size(); i++)
    for (size_t j=0; j<v.size(); j++)
      d = std::max(d, TargetSpace::distance(v[i],v[j]));
  return d;
}

const double eps = 1e-4;

template <class TargetSpace>
void testExpLog(const TargetSpace& a, const TargetSpace& b)
{
  // Check whether exp and log are mutually inverse
  typename TargetSpace::EmbeddedTangentVector logarithm = TargetSpace::log(a,b);
  TargetSpace exponential = TargetSpace::exp(a, logarithm);

  if (TargetSpace::distance(b, exponential) > eps)
  {
    std::cout << className(a) << ": Exp and log are not mutually inverse." << std::endl;
    std::cout << "exp(a,log(a,b)): " << exponential << std::endl;
    std::cout << "b              : " << b << std::endl;
    assert(false);
  }
}

template <class TargetSpace>
double distanceSquared(const TargetSpace& a, const TargetSpace& b)
{
  return Dune::power(TargetSpace::distance(a,b), 2);
}

// Squared distance between two points slightly off the manifold.
// This is required for finite difference approximations.
template <class TargetSpace, int dim>
double distanceSquared(const FieldVector<double,dim>& a, const FieldVector<double,dim>& b)
{
  return Dune::power(TargetSpace::distance(TargetSpace(a),TargetSpace(b)), 2);
}

/** \brief Compute the Riemannian Hessian of the squared distance function in global coordinates

    The formula for the Riemannian Hessian has been taken from Absil, Mahony, Sepulchre:
    'Optimization algorithms on matrix manifolds', page 107
 */
template <class TargetSpace, int worldDim>
FieldMatrix<double,worldDim,worldDim> getSecondDerivativeOfSecondArgumentFD(const TargetSpace& a, const TargetSpace& b)
{

  const size_t spaceDim = TargetSpace::dim;

  // finite-difference approximation
  FieldMatrix<double,spaceDim,spaceDim> d2d2_fd(0);

  FieldMatrix<double,spaceDim,worldDim> B = b.orthonormalFrame();

  for (size_t i=0; i<spaceDim; i++) {
    for (size_t j=0; j<spaceDim; j++) {

      FieldVector<double,worldDim> epsXi  = eps * B[i];
      FieldVector<double,worldDim> epsEta = eps * B[j];

      FieldVector<double,worldDim> minusEpsXi  = -1 * epsXi;
      FieldVector<double,worldDim> minusEpsEta = -1 * epsEta;

      double forwardValue  = distanceSquared(a,TargetSpace::exp(b,epsXi+epsEta)) - distanceSquared(a, TargetSpace::exp(b,epsXi)) - distanceSquared(a,TargetSpace::exp(b,epsEta));
      double centerValue   = distanceSquared(a,b) - distanceSquared(a,b) - distanceSquared(a,b);
      double backwardValue = distanceSquared(a,TargetSpace::exp(b,minusEpsXi + minusEpsEta)) - distanceSquared(a, TargetSpace::exp(b,minusEpsXi)) - distanceSquared(a,TargetSpace::exp(b,minusEpsEta));

      d2d2_fd[i][j] = 0.5 * (forwardValue - 2*centerValue + backwardValue) / (eps*eps);

    }
  }

  //B.invert();
  FieldMatrix<double,worldDim,spaceDim> BT;
  for (int i=0; i<worldDim; i++)
    for (size_t j=0; j<spaceDim; j++)
      BT[i][j] = B[j][i];


  FieldMatrix<double,spaceDim,worldDim> ret1;
  FMatrixHelp::multMatrix(d2d2_fd,B,ret1);

  FieldMatrix<double,worldDim,worldDim> ret2;
  FMatrixHelp::multMatrix(BT,ret1,ret2);
  return ret2;
}

template <class TargetSpace>
void testOrthonormalFrame(const TargetSpace& a)
{
  const size_t spaceDim = TargetSpace::dim;
  const size_t embeddedDim = TargetSpace::embeddedDim;

  FieldMatrix<double,spaceDim,embeddedDim> B = a.orthonormalFrame();

  for (size_t i=0; i<spaceDim; i++)
    for (size_t j=0; j<spaceDim; j++)
      assert( std::fabs(a.metric(B[i],B[j]) - (i==j)) < 1e-10 );
}

template <class TargetSpace>
void testDerivativeOfDistanceSquared(const TargetSpace& a, const TargetSpace& b)
{
  static const size_t embeddedDim = TargetSpace::embeddedDim;

  ///////////////////////////////////////////////////////////////////
  //  Test derivative with respect to second argument
  ///////////////////////////////////////////////////////////////////
  typename TargetSpace::EmbeddedTangentVector d2 =  TargetSpace::derivativeOfDistanceSquaredWRTSecondArgument(a, b);

  // finite-difference approximation
  Dune::FieldMatrix<double,TargetSpace::TangentVector::dimension,embeddedDim> B = b.orthonormalFrame();

  typename TargetSpace::TangentVector d2_fd;
  for (size_t i=0; i<TargetSpace::TangentVector::dimension; i++) {

    typename TargetSpace::EmbeddedTangentVector fwVariation =  eps * B[i];
    typename TargetSpace::EmbeddedTangentVector bwVariation = -eps * B[i];
    TargetSpace bPlus  = TargetSpace::exp(b,fwVariation);
    TargetSpace bMinus = TargetSpace::exp(b,bwVariation);

    d2_fd[i] = (distanceSquared(a,bPlus) - distanceSquared(a,bMinus)) / (2*eps);
  }

  // transform into embedded coordinates
  typename TargetSpace::EmbeddedTangentVector d2_fd_embedded;
  B.mtv(d2_fd,d2_fd_embedded);

  if ( (d2 - d2_fd_embedded).infinity_norm() > 200*eps ) {
    std::cout << className(a) << ": Analytical gradient does not match fd approximation." << std::endl;
    std::cout << "d2 Analytical: " << d2 << std::endl;
    std::cout << "d2 FD        : " << d2_fd << std::endl;
    assert(false);
  }

}

template <class TargetSpace>
void testHessianOfDistanceSquared(const TargetSpace& a, const TargetSpace& b)
{
  static const int embeddedDim = TargetSpace::embeddedDim;

  ///////////////////////////////////////////////////////////////////
  //  Test second derivative with respect to second argument
  ///////////////////////////////////////////////////////////////////
  FieldMatrix<double,embeddedDim,embeddedDim> d2d2 = (TargetSpace::secondDerivativeOfDistanceSquaredWRTSecondArgument(a, b)).matrix();

  // finite-difference approximation
  FieldMatrix<double,embeddedDim,embeddedDim> d2d2_fd = getSecondDerivativeOfSecondArgumentFD<TargetSpace,embeddedDim>(a,b);

  if ( (d2d2 - d2d2_fd).infinity_norm() > 200*eps) {
    std::cout << className(a) << ": Analytical second derivative does not match fd approximation." << std::endl;
    std::cout << "d2d2 Analytical:" << std::endl << d2d2 << std::endl;
    std::cout << "d2d2 FD        :" << std::endl << d2d2_fd << std::endl;
    assert(false);
  }

}

template <class TargetSpace>
void testMixedDerivativesOfDistanceSquared(const TargetSpace& a, const TargetSpace& b)
{
  static const size_t embeddedDim = TargetSpace::embeddedDim;

  //////////////////////////////////////////////////////////////////////////////
  //  Test mixed second derivative with respect to first and second argument
  //////////////////////////////////////////////////////////////////////////////

  FieldMatrix<double,embeddedDim,embeddedDim> d1d2 = TargetSpace::secondDerivativeOfDistanceSquaredWRTFirstAndSecondArgument(a, b);

  // finite-difference approximation
  FieldMatrix<double,embeddedDim,embeddedDim> d1d2_fd;

  for (size_t i=0; i<embeddedDim; i++) {
    for (size_t j=0; j<embeddedDim; j++) {

      FieldVector<double,embeddedDim> aPlus  = a.globalCoordinates();
      FieldVector<double,embeddedDim> aMinus = a.globalCoordinates();
      aPlus[i]  += eps;
      aMinus[i] -= eps;

      FieldVector<double,embeddedDim> bPlus  = b.globalCoordinates();
      FieldVector<double,embeddedDim> bMinus = b.globalCoordinates();
      bPlus[j]  += eps;
      bMinus[j] -= eps;

      d1d2_fd[i][j] = (distanceSquared<TargetSpace>(aPlus,bPlus) + distanceSquared<TargetSpace>(aMinus,bMinus)
                       - distanceSquared<TargetSpace>(aPlus,bMinus) - distanceSquared<TargetSpace>(aMinus,bPlus)) / (4*eps*eps);

    }
  }

  if ( (d1d2 - d1d2_fd).infinity_norm() > 200*eps ) {
    std::cout << className(a) << ": Analytical mixed second derivative does not match fd approximation." << std::endl;
    std::cout << "d1d2 Analytical:" << std::endl << d1d2 << std::endl;
    std::cout << "d1d2 FD        :" << std::endl << d1d2_fd << std::endl;
    assert(false);
  }

}


template <class TargetSpace>
void testDerivativeOfHessianOfDistanceSquared(const TargetSpace& a, const TargetSpace& b)
{
  static const size_t embeddedDim = TargetSpace::embeddedDim;

  /////////////////////////////////////////////////////////////////////////////////////////////
  //  Test mixed third derivative with respect to first (once) and second (twice) argument
  /////////////////////////////////////////////////////////////////////////////////////////////

  Tensor3<double,embeddedDim,embeddedDim,embeddedDim> d2d2d2 = TargetSpace::thirdDerivativeOfDistanceSquaredWRTSecondArgument(a, b);

  Tensor3<double,embeddedDim,embeddedDim,embeddedDim> d2d2d2_fd;

  for (size_t i=0; i<embeddedDim; i++) {

    FieldVector<double,embeddedDim> bPlus  = b.globalCoordinates();
    FieldVector<double,embeddedDim> bMinus = b.globalCoordinates();
    bPlus[i]  += eps;
    bMinus[i] -= eps;

    FieldMatrix<double,embeddedDim,embeddedDim> hPlus  = getSecondDerivativeOfSecondArgumentFD<TargetSpace,embeddedDim>(a,TargetSpace(bPlus));
    FieldMatrix<double,embeddedDim,embeddedDim> hMinus = getSecondDerivativeOfSecondArgumentFD<TargetSpace,embeddedDim>(a,TargetSpace(bMinus));

    d2d2d2_fd[i] = hPlus;
    d2d2d2_fd[i] -= hMinus;
    d2d2d2_fd[i] /= 2*eps;

  }

  if ( (d2d2d2 - d2d2d2_fd).infinity_norm() > 200*eps) {
    std::cout << className(a) << ": Analytical third derivative does not match fd approximation." << std::endl;
    std::cout << "d2d2d2 Analytical:" << std::endl << d2d2d2 << std::endl;
    std::cout << "d2d2d2 FD        :" << std::endl << d2d2d2_fd << std::endl;
    assert(false);
  }

}


template <class TargetSpace>
void testMixedDerivativeOfHessianOfDistanceSquared(const TargetSpace& a, const TargetSpace& b)
{
  static const size_t embeddedDim = TargetSpace::embeddedDim;

  /////////////////////////////////////////////////////////////////////////////////////////////
  //  Test mixed third derivative with respect to first (once) and second (twice) argument
  /////////////////////////////////////////////////////////////////////////////////////////////

  Tensor3<double,embeddedDim,embeddedDim,embeddedDim> d1d2d2 = TargetSpace::thirdDerivativeOfDistanceSquaredWRTFirst1AndSecond2Argument(a, b);

  Tensor3<double,embeddedDim,embeddedDim,embeddedDim> d1d2d2_fd;

  for (size_t i=0; i<embeddedDim; i++) {

    FieldVector<double,embeddedDim> aPlus  = a.globalCoordinates();
    FieldVector<double,embeddedDim> aMinus = a.globalCoordinates();
    aPlus[i]  += eps;
    aMinus[i] -= eps;

    FieldMatrix<double,embeddedDim,embeddedDim> hPlus  = getSecondDerivativeOfSecondArgumentFD<TargetSpace,embeddedDim>(TargetSpace(aPlus),b);
    FieldMatrix<double,embeddedDim,embeddedDim> hMinus = getSecondDerivativeOfSecondArgumentFD<TargetSpace,embeddedDim>(TargetSpace(aMinus),b);

    d1d2d2_fd[i] = hPlus;
    d1d2d2_fd[i] -= hMinus;
    d1d2d2_fd[i] /= 2*eps;

  }

  if ( (d1d2d2 - d1d2d2_fd).infinity_norm() > 200*eps ) {
    std::cout << className(a) << ": Analytical mixed third derivative does not match fd approximation." << std::endl;
    std::cout << "d1d2d2 Analytical:" << std::endl << d1d2d2 << std::endl;
    std::cout << "d1d2d2 FD        :" << std::endl << d1d2d2_fd << std::endl;
    assert(false);
  }

}


template <class TargetSpace>
void testDerivativesOfDistanceSquared(const TargetSpace& a, const TargetSpace& b)
{

  ///////////////////////////////////////////////////////////////////
  //  Test derivative with respect to second argument
  ///////////////////////////////////////////////////////////////////

  testDerivativeOfDistanceSquared<TargetSpace>(a,b);

  ///////////////////////////////////////////////////////////////////
  //  Test second derivative with respect to second argument
  ///////////////////////////////////////////////////////////////////

  testHessianOfDistanceSquared<TargetSpace>(a,b);

  //////////////////////////////////////////////////////////////////////////////
  //  Test mixed second derivative with respect to first and second argument
  //////////////////////////////////////////////////////////////////////////////

  testMixedDerivativesOfDistanceSquared<TargetSpace>(a,b);

  /////////////////////////////////////////////////////////////////////////////////////////////
  //  Test third derivative with respect to second argument
  /////////////////////////////////////////////////////////////////////////////////////////////

  testDerivativeOfHessianOfDistanceSquared<TargetSpace>(a,b);

  /////////////////////////////////////////////////////////////////////////////////////////////
  //  Test mixed third derivative with respect to first (once) and second (twice) argument
  /////////////////////////////////////////////////////////////////////////////////////////////

  testMixedDerivativeOfHessianOfDistanceSquared<TargetSpace>(a,b);

}


// The class ProductManifold had a bug that lead to a build failure
// when using std::cout in the namespace Dune::GFE.
//
// See https://gitlab.mn.tu-dresden.de/osander/dune-gfe/-/merge_requests/114
//
// The following method is here to make sure that this bug does not
// come back.
namespace Dune::GFE
{
  void testUsingIOStream()
  {
    std::cout << "dummy text" << std::endl;
  }
}


template <class TargetSpace>
void test()
{
  std::cout << "Testing class " << className<TargetSpace>() << std::endl;

  std::vector<TargetSpace> testPoints;
  ValueFactory<TargetSpace>::get(testPoints);

  int nTestPoints = testPoints.size();

  // Test each element in the list
  for (int i=0; i<nTestPoints; i++) {

    //testOrthonormalFrame<TargetSpace>(testPoints[i]);

    for (int j=0; j<nTestPoints; j++) {

      std::vector<TargetSpace> testPointPair(2);
      testPointPair[0] = testPoints[i];
      testPointPair[1] = testPoints[j];
      if (diameter(testPointPair) > TargetSpace::convexityRadius)
        continue;

      // Test the exponential map and the logarithm
      testExpLog(testPoints[i], testPoints[j]);

      // Test the various derivatives of the squared distance
      testDerivativesOfDistanceSquared<TargetSpace>(testPoints[i], testPoints[j]);

    }

  }

  // Test whether we can rebind to another number type
  using FTargetSpace = typename TargetSpace::template rebind<float>::other;

  // Can we construct an object of that rebound type?
  FTargetSpace fTargetSpace;
}


int main() try
{
  // Test the RealTuple class
  test<RealTuple<double,1> >();
  test<RealTuple<double,3> >();

  // Test the UnitVector class
  test<UnitVector<double,2> >();
  test<UnitVector<double,3> >();
  test<UnitVector<double,4> >();

  // Test the rotation class
  test<Rotation<double,3> >();

  // Test the ProductManifold class
  test<Dune::GFE::ProductManifold<RealTuple<double,1>,Rotation<double,3>,UnitVector<double,2> > >();
  test<Dune::GFE::ProductManifold<Rotation<double,3>,UnitVector<double,5> > >();

  //
  //     test<HyperbolicHalfspacePoint<double,2> >();

}
catch (Exception& e) {

  std::cout << e.what() << std::endl;

}