LSProblemImplem.H

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#ifdef CH_LANG_CC
/*
 *      _______              __
 *     / ___/ /  ___  __ _  / /  ___
 *    / /__/ _ \/ _ \/  V \/ _ \/ _ \
 *    \___/_//_/\___/_/_/_/_.__/\___/
 *    Please refer to Copyright.txt, in Chombo's root directory.
 */
#endif

#ifndef _LSPROBLEMIMPLEM_H_
#define _LSPROBLEMIMPLEM_H_

#if defined(CH_Darwin) && defined(__GNUC__) && ( __GNUC__ == 3 )
// deal with the broken isnan()/isinf() in GCC on MacOS
#include <unistd.h>
#define _GLIBCPP_USE_C99 1
#endif

#include <cmath>
#include <cstdio>
#include <cstdlib>
#include <fstream>
#include <iostream>
#include <string>
using std::endl;

#include "ConstrainedLS.H"
#include "LSquares.H"
#include "MultiIndex.H"
#include "IFData.H"
#include "CutCellMoments.H"

#include "NamespaceHeader.H"

template<int dim> int LSProblem<dim>::nChooseR(int a_n,
                                               int a_r)
{
  if (a_r == 0) return 1;
  int num = 1;
  int den = 1;
  for (int i = 1; i <= a_r; ++i)
  {
    num *= (a_n+1-i);
    den *= i;
  }
  return num/den;
}

template<int dim> void LSProblem<dim>::monoMaxMin(Real                   & a_maxVal,
                                                  Real                   & a_minVal,
                                                  const IndexTM<int,dim> & a_mono,
                                                  const IFData<dim>      & a_IFData)
{
  a_maxVal = 0.0;
  a_minVal = 0.0;

  for (typename IFData<dim>::CornerSigns::const_iterator cornerIt = a_IFData.m_cornerSigns.begin();
       cornerIt != a_IFData.m_cornerSigns.end();
       ++cornerIt)
    {
      const typename IFData<dim>::Vertex & vertex = cornerIt->first;

      // represent the vertex as an RvDim in cell centered coordinates
      typename IFData<dim>::RvDim corner;
      for (int idir = 0; idir < dim; ++idir)
        {
          corner[idir] = vertex[idir] - 0.5;
          corner[idir] *= a_IFData.m_cellCenterCoord.m_dx[idir];
        }

      // compute coordinates of corner in local coordinates
      typename IFData<dim>::RvDim cornerCoord = a_IFData.m_localCoord.convert(corner,a_IFData.m_cellCenterCoord);

      // monomial value at the corner
      Real monoCorner = 1.0;
      for (int idir = 0; idir < dim ; ++idir)
        {
          monoCorner *= pow(cornerCoord[idir],a_mono[idir]);
        }

      if (monoCorner > a_maxVal)
        {
          a_maxVal = monoCorner;
        }

      if (monoCorner < a_minVal)
        {
          a_minVal = monoCorner;
        }
    }
}

template<int dim> void LSProblem<dim>::computeBounds(const IndexTM<Real,dim>   & a_dx,
                                                     const CutCellMoments<dim> & a_ccm)
{
  m_lowerBound.assign(-HUGE);
  m_upperBound.assign(HUGE);

  int nEBBounds = 0;
  if (m_degreeP == 0 && dim == 2) // || m_degreeP == 1)
    {
      nEBBounds = m_numP;       // A lower bound for each EB. Haven't figured out a useful u.b. yet
    }

  // int nVar=m_numP + m_numPLess1; // total variables in problem

  // This constraint is a lower bound on the zero monomial of the EB.
  // It is based on inverting integral(mono)<Max(mono)*integral(1)
  if (m_degreeP == 0 && dim == 2)
    {
      Real lobnd = 0.0; // minimum, but we can do better
      IvDim whichMono; // for tracking
      for (typename CutCellMoments<dim>::PthMoment::const_iterator it = a_ccm.m_EBmoments.begin();
          it != a_ccm.m_EBmoments.end();++it)
        {
          IvDim mono = it->first;
          Real maxMono = 0.0;
          Real minMono = 0.0;
          monoMaxMin(maxMono,minMono,mono,a_ccm.m_IFData);
          Real val = it->second;
          if (minMono < 0.0)
            {
              Real newLoBnd = val/minMono;
              if (newLoBnd > lobnd)
                {
                  lobnd = newLoBnd;
                  whichMono = mono;
                }
            }
          if (maxMono > 0.0)
            {
              Real newLoBnd = val/maxMono;
              if (newLoBnd > lobnd)
                {
                  lobnd = newLoBnd;
                  whichMono = mono;
                }
            }
        }

      // The constraints matrix fits CiT x + ci >=0
      // so the rows are variables and the columns are constraints (may be a FORTRAN holdover)
      IvDim mono = IvDim::Zero;

      int ndx = 0; // todo: guarantee there should be only one for (0,0)
      int variableNdx = ndx;
      m_lowerBound[variableNdx]=lobnd;
      m_upperBound[variableNdx]=HUGE;

    }

  // Now create constraints for the volume integrals.
  for (typename PthMomentLoc::const_iterator it = m_monoLocPLess1.begin();
      it != m_monoLocPLess1.end();++it)
    {
      IvDim mono = it->first;
      // The constraints matrix fits CiT x + ci >=0
      // so the rows are variables and the columns are constraints (may be a FORTRAN holdover)
      int ndx = it->second;
      int variableNdx = ndx + m_numP;
      Real lobnd, hibnd;

      // Compute the lower and upper bounds
      // based on quadrature of all the regions that contribute
      // negatively or positively.
      momentBounds(lobnd,hibnd,mono, a_ccm.m_IFData);



      bool isZeroDegree = (mono.sum() == 0);

      if (isZeroDegree )
        {
          Real zeroDegreeLoBnd = 0.0;

          // Refine the lower bound of the zero degree monomial
          // based by inverting the identity
          // integral(mono)/integral(zeromono) < Max(mono)
          // Note that we already have integral(mono) from previous steps
          IvDim whichMono;
          // Real valAtWhichMono=LARGEREALVAL;
          for (typename CutCellMoments<dim>::PthMoment::const_iterator it = a_ccm.m_moments.begin();
              it != a_ccm.m_moments.end();++it)
            {
              IvDim mono = it->first;
              Real val = it->second;
              Real maxMono = 0.0;
              Real minMono = 0.0;
              monoMaxMin(maxMono,minMono,mono,a_ccm.m_IFData);
              if (minMono < 0.0)
                {
                  Real newLoBnd = val/minMono;
                  if (newLoBnd > zeroDegreeLoBnd)
                    {
                      zeroDegreeLoBnd = newLoBnd;
                      whichMono = mono;
                    }
                }
              if (maxMono > 0.0)
                {
                  Real newLoBnd = val/maxMono;
                  if (newLoBnd > zeroDegreeLoBnd)
                    {
                      zeroDegreeLoBnd = newLoBnd;
                      whichMono = mono;
                    }
                }
            }
          lobnd = Max(lobnd, zeroDegreeLoBnd);
#ifdef CH_REALM
          // This represents the assumption that kappa3d < kappa2d
          IndexTM<Real,dim> largestVol=IndexTM<Real,dim>::Zero;

          //Refine lower bound based on assumption that kappa3d < kappa2d*dz
          for (typename CutCellMoments<dim>::BdCutCellMoments::const_iterator it = a_ccm.m_bdCutCellMoments.begin();
               it != a_ccm.m_bdCutCellMoments.end();
               ++it)
            {
              Iv2 bdId = it->first;
              int idir = bdId[BDID_DIR];
              EBorVol doVol = VolMoment;
              Real volSide = it->second.getVol(doVol);
              if (volSide > largestVol[idir])
                {
                  largestVol[idir] = volSide;
                }
            }
          Real noUndercutBnd = LARGEREALVAL;
          if (a_ccm.m_IFData.m_globalCoord.m_dx[0] != a_ccm.m_IFData.m_globalCoord.m_dx[dim-1])
            {
              Real vol= largestVol[dim-1];
              Real normalDx = a_ccm.m_IFData.m_globalCoord.m_dx[dim-1];
              noUndercutBnd = Min(noUndercutBnd, vol*normalDx);
            }
          hibnd = Min(hibnd, noUndercutBnd);
#endif
        }


      m_lowerBound[variableNdx] = lobnd;
      m_upperBound[variableNdx] = hibnd;

    }


}

// Calculate upper and lower bounds for a moment
template<int dim> void LSProblem<dim>::momentBounds(Real              & a_lobnd,
                                                    Real              & a_hibnd,
                                                    const IvDim       & a_mono,
                                                    const IFData<dim> & a_IFData)
{
  a_lobnd = 0.0;
  a_hibnd = 0.0;
  // Enumerate though every vertex, and integrate the contributions
  // from the (local) cell center to the vertex in local coordinates
  typename IFData<dim>::CornerSigns::const_iterator cornerIt;
  // pout() << a_mono << endl;
  for (cornerIt =  a_IFData.m_cornerSigns.begin();
       cornerIt != a_IFData.m_cornerSigns.end();
       ++cornerIt)
    {
      const typename IFData<dim>::Vertex & vertex = cornerIt->first;
     // represent the vertex as an RvDim in cell centered coordinates
      typename IFData<dim>::RvDim corner;
      for (int idir = 0; idir < dim; ++idir)
      {
        corner[idir] = vertex[idir] - 0.5;
        corner[idir] *= a_IFData.m_cellCenterCoord.m_dx[idir];
      }

      // compute coordinates of corner in local coordinates
      typename IFData<dim>::RvDim cornerCoord
        = a_IFData.m_localCoord.convert(corner,a_IFData.m_cellCenterCoord);

      Real partialCellMag = 1.0;
      for (int idir = 0; idir < dim ; ++idir)
        {
          Real pPlus1 =(Real)( a_mono[idir]+1);
          partialCellMag *= pow(cornerCoord[idir],pPlus1)/pPlus1;
          if (cornerCoord[idir] < 0.0) partialCellMag *=-1.0;
        }
      if (partialCellMag > 0.0) a_hibnd += partialCellMag;
      if (partialCellMag < 0.0) a_lobnd += partialCellMag;
    }
}

template<int dim> LSProblem<dim>::~LSProblem()
{
  if (m_matrix != NULL)
    {
      // free m_matrix
      int numRows = dim*m_numP;
      int numCols = m_numP + m_numPLess1;
      freeArray(numRows,numCols,m_matrix);

    }
}

// this constructor is used when just a list of monomials is wanted
template<int dim> LSProblem<dim>::LSProblem(const int  & a_degreeP,
                                            const bool & a_useConstraints)
  :m_degreeP(a_degreeP),
   m_numActiveBounds(0),
   m_useConstraints(a_useConstraints)
{
   fillMap(m_monoLocP,m_locMonoP,m_degreeP);
   fillMap(m_monoLocPLess1,m_locMonoPLess1,m_degreeP-1);
   m_numP      = numMonomials(m_degreeP);
   m_numPLess1 = numMonomials(m_degreeP-1);
   m_matrix = NULL;
}

// constructor for solving a LS problem
template<int dim> LSProblem<dim>::LSProblem(const int               & a_order,
                                            const int               & a_degreeP,
                                            const bool              & a_useConstraints,
                                            const IndexTM<Real,dim> & a_normal)
  :m_order(a_order),
   m_degreeP(a_degreeP),
   m_numActiveBounds(0),
   m_useConstraints(a_useConstraints),
   m_normal(a_normal)
{
  fillMap(m_monoLocP,m_locMonoP,m_degreeP);
  fillMap(m_monoLocPLess1,m_locMonoPLess1,m_degreeP-1);
  m_numP      = numMonomials(m_degreeP);
  m_numPLess1 = numMonomials(m_degreeP-1);
  setMatrix();

  m_unknowns.resize(m_numP + m_numPLess1);
  m_rhs.resize(m_numP*dim);

  if (a_useConstraints)
  {
    m_lowerBound.resize(m_numP+m_numPLess1);
    m_upperBound.resize(m_numP+m_numPLess1);
  }
}

//
template<int dim> int LSProblem<dim>::invertNormalEq(const Vector<Real> & a_rhs,
                                                     Vector<Real>       & a_residual)
{
  m_rhs = a_rhs;
  int retCode = -1;
  if (!m_useConstraints)
  {
    LSquares lsSolver;
    lsSolver.LeastSquares(m_matrix,m_unknowns,m_rhs);
    m_numActiveBounds = 0;
    retCode = 0;
  }
  else
  {
    ConstrainedLS cls;
    ConstrainedLS::LSResult result = cls.solveBoundConstrained(m_unknowns,
                                                               m_matrix,
                                                               m_rhs,
                                                               m_lowerBound,
                                                               m_upperBound);
    m_numActiveBounds = cls.numberActiveConstraints();
    retCode = (result == ConstrainedLS::SUCCESS) ? 0 : -1;
    switch(result)
      {
      case(ConstrainedLS::SUCCESS):
        break;
      case(ConstrainedLS::SINGULAR):
        pout() << "Singular LS problem. Linearly dependent columns (zero normal?)" << endl;
        break;
      case(ConstrainedLS::INCONSISTENT_BOUNDS):
        pout() << "Conflicting upper/lower bounds in LS problem. Zero or bad normal?" << endl;
        break;
      case(ConstrainedLS::UNDERDETERMINED):
        MayDay::Error("Malformed LS problem. Underdetermined");
      case(ConstrainedLS::UNCONVERGED):
        pout() << "BVLS failed to converge properly" << endl;
        break;
      }

  }

  a_residual.resize(3);
  a_residual[0] = 0.0;
  a_residual[1] = 0.0;
  a_residual[2] = 0.0;
  Real maxRi = 0.0;
  for (int i = 0 ; i < m_numP*dim ; i++)
    {
      Real AtimeX = 0.0;
      for (int j = 0 ; j < m_numP + m_numPLess1 ; j++)
        {
          AtimeX += m_matrix[i][j] * m_unknowns[j];
        }
      Real ri = Abs(AtimeX - m_rhs[i]);
      if (ri > maxRi)
        {
          a_residual[0] = ri;
          maxRi = ri;
        }
      a_residual[1] += ri;
      a_residual[2] += ri * ri;
    }
  a_residual[2] = sqrt(a_residual[2]);
  // pout() << "Residual : " << a_residual[1] << ":" << a_residual[2] << endl;
  return retCode;
}

template<int dim> int LSProblem<dim>::factorial(const int & a_n,
                                                const int & a_m)
{
  int retval = 1;
  if (a_n < 0 || a_m < 0)
    {
      MayDay::Abort("Attempting n! for n < 0");
    }
  for (int i = a_m+1; i <= a_n; ++i)
    {
      retval *= i;
    }
  return retval;
}

template<int dim> int LSProblem<dim>::numMonomials(const int & a_monoDegree)
{
  int retval = LARGEINTVAL;
  if (a_monoDegree == -1)
    {
      retval =  0;
    }
  else
    {
      int bigger;
      int smaller;
      if (dim- 1 > a_monoDegree)
        {
          bigger = dim-1;
          smaller = a_monoDegree;
    }
      else
        {
          smaller = dim-1;
          bigger = a_monoDegree;
        }
      int numerator = factorial(dim - 1 + a_monoDegree,bigger);
      int denominator = factorial(smaller);
      // pout() << "numerator = "   << numerator   << endl;
      // pout() << "denominator = " << denominator << endl;
      retval =  numerator/denominator;
    }
  return retval;
}

// uses dimension & degree create matrix for overdetermined system
template<int dim> void LSProblem<dim>::setMatrix()
{
  // solving m_matrix[x] = b

  // initializes m_matrix to zeros

  int numRows = dim*m_numP;
  int numCols = m_numP + m_numPLess1;
  allocArray(numRows,numCols,m_matrix);

  // iterate through the list of mono of Degree P
  // pout() << " Number of mono of DegreeP = " << m_monoLocP.size() << endl;
  for (typename PthMomentLoc::const_iterator it = m_monoLocP.begin();
      it != m_monoLocP.end();++it)
    {
      for (int idir = 0; idir< dim; ++idir)
        {
          // this entry corresponds to the integral over the boundary
          int row = dim*(it->second) + idir;
          int pcol = it->second;
          m_matrix[row][pcol] = -m_normal[idir];

          // differentiate the mono and
          IvDim Dmono;
          IvDim mono = it->first;
          int coeff = LARGEINTVAL;
          // diff(it->first) = coeff*Dmono
          differentiate(coeff,Dmono,idir,mono);
          int pLess1Col= LARGEINTVAL;
          if (coeff == LARGEINTVAL)
            {
              MayDay::Abort("problem wth differentiate");
            }
          if (coeff > 0)
            {
              // find which mono this is in the list
              if (m_monoLocPLess1.find(Dmono) != m_monoLocPLess1.end())
                {
                  pLess1Col = m_monoLocPLess1[Dmono] + m_numP;
                  m_matrix[row][pLess1Col] = coeff;
                }
              else
                {
                  MayDay::Abort("can't find derived mono");
                }
            }
        }
    }
}

// differentiate a_mono w.r.t. x_idir. Answer = a_coeff*a_Dmono
template<int dim> void LSProblem<dim>::differentiate(int         & a_coeff,
                                                     IvDim       & a_Dmono,
                                                     int         & a_idir,
                                                     const IvDim & a_mono)
{
  if (a_mono[a_idir] > 0)
    {
      a_coeff          = a_mono[a_idir];
      a_Dmono          = a_mono;
      a_Dmono[a_idir] -= 1;
    }
  else if (a_mono[a_idir] == 0)
    {
      a_coeff = 0;
      for (int idir = 0; idir < dim; ++idir)
        {
          a_Dmono[idir] = LARGEINTVAL;
        }
    }
  else
    {
      MayDay::Abort("Monomial has negative power");
    }
}

template<int dim> void LSProblem<dim>::setRhs(const Vector<Real> & a_rhs)
{
}

template<int dim> void LSProblem<dim>::fillMap(PthMomentLoc & a_monoLoc,
                                               LocPthMoment & a_locMono,
                                               const int    & a_degree)
{
  if (a_degree >= 0)
    {
      Vector<IvDim> monomials;

      generateMultiIndices(monomials,a_degree);

      for (int i = 0; i < monomials.size(); i++)
      {
        const IvDim & monomial = monomials[i];

        a_monoLoc[monomial] = i;
        a_locMono[i] = monomial;
      }
    }
}

template<int dim> void LSProblem<dim>::allocArray(const int & a_rows,
                                                  const int & a_cols,
                                                  Real**    & a_A)
{
  a_A = new Real* [a_rows];

  for (int i = 0; i < a_rows; i++)
    {
      a_A[i] = new Real [a_cols];

      Real* scanA = a_A[i];
      for (int j = 0; j < a_cols; j++)
        {
          *(scanA++) = 0.0;
        }
    }
}

template<int dim> void LSProblem<dim>::freeArray(const int & a_rows,
                                                 const int & a_cols,
                                                 Real**    & a_A)
{
  for (int i = 0; i < a_rows; i++)
    {
      delete[] a_A[i];
    }

  delete[] a_A;
}

template<int dim > void LSProblem<dim>::print(ostream & a_out) const
{
  a_out << "Dim = " << dim << ", degree = " << m_degreeP << '\n';
  a_out << "m_monoLocP has " << m_monoLocP.size() << " elements" << '\n';

   for (typename PthMomentLoc::const_iterator it = m_monoLocP.begin();
       it != m_monoLocP.end();++it)
  {
    a_out << "Monomial =  " << it->first << ", Loc = " << it->second << '\n';
  }

    a_out << "Dim = " << dim << '\n';
  a_out << "m_locMonoP has " << m_locMonoP.size() << " elements" << '\n';

   for (typename LocPthMoment::const_iterator it = m_locMonoP.begin();
       it != m_locMonoP.end();++it)
  {
    a_out << "Loc =  " << it->first << ", Monomial = " << it->second << '\n';
  }
   // one degree lower
   a_out << "m_locMonoPLess1 has " << m_locMonoPLess1.size() << " elements" << '\n';
   for (typename PthMomentLoc::const_iterator it = m_monoLocPLess1.begin();
       it != m_monoLocPLess1.end();++it)
     {
    a_out << "Monomial =  " << it->first << ", Loc = " << it->second << '\n';
  }

  a_out << "m_locMonoPLess1 has " << m_locMonoPLess1.size() << " elements" << '\n';

   for (typename LocPthMoment::const_iterator it = m_locMonoPLess1.begin();
       it != m_locMonoPLess1.end();++it)
  {
    a_out << "Loc =  " << it->first << ", Monomial = " << it->second << '\n';
  }

   a_out << "Matrix and rhs for least squares problem of dim = " << dim << endl;
   outputMatrix();
   outputRhs();
   outputUnknowns();
   outputBounds();
}

template<int dim> ostream& operator<<(ostream        & a_out,
                                      LSProblem<dim> & a_lsProblem)
{
  a_lsProblem.print(a_out);
  return a_out;
}

template<int dim> void LSProblem<dim>::outputMatrix() const
{
  int rows = m_locMonoP.size()*dim;
  int cols = m_locMonoPLess1.size() + m_locMonoP.size();
  pout() << "numRows = " << rows << ", numCols = " << cols << endl;
  // pout() << "outputting " << name << endl;
  for (int i = 0; i < rows; i++)
    {
      for (int j = 0; j < cols; j++)
       {
         pout() << m_matrix[i][j] << " ";
       }
      pout() << endl;
    }
}

template<int dim> void LSProblem<dim>::outputRhs() const
{
  pout() << "Outputting Rhs" << endl;
  for (int i = 0; i < m_rhs.size(); i++)
    {
      pout() << "m_rhs[" << i << "] = " << m_rhs[i] << endl;
    }
}

template<int dim> void LSProblem<dim>::outputUnknowns()const
{
  pout() << "Outputting Unknowns" << endl;
  for (int i = 0; i < m_unknowns.size(); i++)
    {
      pout() << "m_unknowns[" << i << "] = " << m_unknowns[i] << endl;
    }
}

template<int dim> void LSProblem<dim>::outputBounds()const
{
  if (m_useConstraints)
    {
      pout() << "Outputting Lower/Upper bounds" << endl;
      for (int i = 0; i < m_unknowns.size(); i++)
        {
          pout() << "m_lowerBound[" << setw(2) << i << "] = " << setw(14) << m_lowerBound[i]
                 << " m_upperBound[" << setw(2) << i << "] = " << setw(14) << m_upperBound[i] << endl;
        }
    }
  else
    {
      pout() << "Problem is unconstrained" << endl;
    }
}

#include "NamespaceFooter.H"

#endif