DTK_LocalMLSProblem_impl.hpp

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//---------------------------------------------------------------------------//
/*
  Copyright (c) 2012, Stuart R. Slattery
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  *: Redistributions in binary form must reproduce the above copyright
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  documentation and/or other materials provided with the distribution.

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  names of its contributors may be used to endorse or promote products
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*/
//---------------------------------------------------------------------------//
//---------------------------------------------------------------------------//

#ifndef DTK_LOCALMLSPROBLEM_IMPL_HPP
#define DTK_LOCALMLSPROBLEM_IMPL_HPP

#include <algorithm>
#include <functional>
#include <limits>

#include "DTK_DBC.hpp"
#include "DTK_EuclideanDistance.hpp"
#include "DTK_LocalMLSProblem.hpp"
#include "DTK_RadialBasisPolicy.hpp"

#include <Teuchos_LAPACK.hpp>
#include <Teuchos_SerialDenseSolver.hpp>
#include <Teuchos_SerialDenseVector.hpp>

namespace DataTransferKit
{
//---------------------------------------------------------------------------//
template <class Basis, int DIM>
LocalMLSProblem<Basis, DIM>::LocalMLSProblem(
    const Teuchos::ArrayView<const double> &target_center,
    const Teuchos::ArrayView<const unsigned> &source_lids,
    const Teuchos::ArrayView<const double> &source_centers, const Basis &basis,
    const double radius )
    : d_shape_function( source_lids.size() )
{
    DTK_REQUIRE( 0 == source_centers.size() % DIM );
    DTK_REQUIRE( 0 == target_center.size() % DIM );

    // Number of source centers supporting this target center.
    int num_sources = source_lids.size();
    int poly_size = 0;
    Teuchos::SerialDenseMatrix<int, double> P;
    Teuchos::SerialDenseVector<int, double> target_poly;

    // Make Phi.
    Teuchos::SerialDenseMatrix<int, double> phi( num_sources, num_sources );
    Teuchos::ArrayView<const double> source_center_view;
    double dist = 0.0;
    for ( int i = 0; i < num_sources; ++i )
    {
        source_center_view = source_centers( DIM * source_lids[i], DIM );
        dist = EuclideanDistance<DIM>::distance(
            target_center.getRawPtr(), source_center_view.getRawPtr() );
        phi( i, i ) = BP::evaluateValue( basis, radius, dist );
    }

    // Make P.
    Teuchos::Array<int> poly_ids( 1, 0 );
    int poly_id = 0;
    P.reshape( num_sources, poly_id + 1 );
    for ( int i = 0; i < num_sources; ++i )
    {
        source_center_view = source_centers( DIM * source_lids[i], DIM );
        P( i, poly_id ) = polynomialCoefficient( 0, source_center_view );
    }
    ++poly_id;

    // Add polynomial columns until we are full rank.
    bool full_rank = false;
    int num_poly = 10;
    int total_poly = std::min( num_poly, num_sources );
    for ( int j = 1; j < total_poly; ++j )
    {
        // Add the next column.
        P.reshape( num_sources, poly_id + 1 );
        for ( int i = 0; i < num_sources; ++i )
        {
            source_center_view = source_centers( DIM * source_lids[i], DIM );
            P( i, poly_id ) = polynomialCoefficient( j, source_center_view );
        }

        // Check for rank deficiency.
        full_rank = isFullRank( P );

        // If we are full rank, add this coefficient.
        if ( full_rank )
        {
            poly_ids.push_back( j );
            ++poly_id;
        }

        // If we are rank deficient, remove the last column.
        else
        {
            P.reshape( num_sources, poly_id );
        }
    }

    // Make p.
    poly_size = poly_ids.size();
    target_poly.resize( poly_size );
    for ( int i = 0; i < poly_size; ++i )
    {
        target_poly( i ) = polynomialCoefficient( poly_ids[i], target_center );
    }

    // Construct b.
    Teuchos::SerialDenseMatrix<int, double> b( poly_size, num_sources );
    b.multiply( Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, P, phi, 0.0 );

    // Construct the A matrix.
    Teuchos::SerialDenseMatrix<int, double> A( poly_size, poly_size );
    {
        // Build A.
        Teuchos::SerialDenseMatrix<int, double> work( num_sources, poly_size );
        work.multiply( Teuchos::NO_TRANS, Teuchos::NO_TRANS, 1.0, phi, P, 0.0 );
        A.multiply( Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, P, work, 0.0 );
    }

    // Apply the inverse of the A matrix to b.
    Teuchos::LAPACK<int, double> lapack;
    double A_rcond = std::numeric_limits<double>::epsilon();
    Teuchos::Array<double> work( 4 * A.numCols() );
    Teuchos::SerialDenseVector<int, double> s( poly_size );
    int rank = 0;
    int info = 0;

    // Estimate the reciprocal of the condition number.
    Teuchos::Array<int> ipiv( std::min( A.numRows(), A.numCols() ) );
    Teuchos::SerialDenseMatrix<int, double> LU_A( A );
    lapack.GETRF( LU_A.numRows(), LU_A.numCols(), LU_A.values(), LU_A.numRows(),
                  ipiv.getRawPtr(), &info );
    DTK_CHECK( 0 == info );

    Teuchos::Array<int> iwork( A.numCols() );
    lapack.GECON( '1', LU_A.numCols(), LU_A.values(), LU_A.numRows(),
                  A.normOne(), &A_rcond, work.getRawPtr(), iwork.getRawPtr(),
                  &info );
    DTK_CHECK( 0 == info );

    // Get the optimal work size.
    lapack.GELSS( A.numRows(), A.numCols(), b.numCols(), A.values(),
                  A.numRows(), b.values(), b.numRows(), s.values(), A_rcond,
                  &rank, work.getRawPtr(), -1, &info );
    DTK_CHECK( 0 == info );

    // Apply the inverse of A to b.
    work.resize( work[0] );
    lapack.GELSS( A.numRows(), A.numCols(), b.numCols(), A.values(),
                  A.numRows(), b.values(), b.numRows(), s.values(), A_rcond,
                  &rank, work.getRawPtr(), work.size(), &info );
    DTK_CHECK( 0 == info );

    // Construct the basis.
    Teuchos::SerialDenseMatrix<int, double> shape_matrix(
        Teuchos::View, d_shape_function.getRawPtr(), 1, 1,
        d_shape_function.size() );
    shape_matrix.multiply( Teuchos::TRANS, Teuchos::NO_TRANS, 1.0, target_poly,
                           b, 0.0 );
}

//---------------------------------------------------------------------------//
// Get a polynomial coefficient.
template <class Basis, int DIM>
double LocalMLSProblem<Basis, DIM>::polynomialCoefficient(
    const int coeff, const Teuchos::ArrayView<const double> &center ) const
{
    switch ( coeff )
    {
    // Linear.
    case ( 0 ):
        return 1.0;
        break;
    case ( 1 ):
        return center[0];
        break;
    case ( 2 ):
        return center[1];
        break;
    case ( 3 ):
        return center[2];
        break;

    // Quadratic
    case ( 4 ):
        return center[0] * center[1];
        break;
    case ( 5 ):
        return center[0] * center[2];
        break;
    case ( 6 ):
        return center[1] * center[2];
        break;
    case ( 7 ):
        return center[0] * center[0];
        break;
    case ( 8 ):
        return center[1] * center[1];
        break;
    case ( 9 ):
        return center[2] * center[2];
        break;
    }
    return 0.0;
}

//---------------------------------------------------------------------------//
// Check if a matrix is full rank.
template <class Basis, int DIM>
bool LocalMLSProblem<Basis, DIM>::isFullRank(
    const Teuchos::SerialDenseMatrix<int, double> &matrix ) const
{
    // Copy the matrix.
    Teuchos::SerialDenseMatrix<int, double> A = matrix;

    // Determine the full rank.
    int full_rank = std::min( A.numRows(), A.numCols() );

    // Compute the singular value decomposition.
    Teuchos::LAPACK<int, double> lapack;
    Teuchos::Array<double> S( full_rank );
    Teuchos::SerialDenseMatrix<int, double> U( A.numRows(), A.numRows() );
    Teuchos::SerialDenseMatrix<int, double> VT( A.numCols(), A.numCols() );
    Teuchos::Array<double> work( full_rank );
    Teuchos::Array<double> rwork( full_rank );
    int info = 0;
    lapack.GESVD( 'A', 'A', A.numRows(), A.numCols(), A.values(), A.numRows(),
                  S.getRawPtr(), U.values(), U.numRows(), VT.values(),
                  VT.numRows(), work.getRawPtr(), -1, rwork.getRawPtr(),
                  &info );
    DTK_CHECK( 0 == info );

    work.resize( work[0] );
    rwork.resize( work.size() );
    lapack.GESVD( 'A', 'A', A.numRows(), A.numCols(), A.values(), A.numRows(),
                  S.getRawPtr(), U.values(), U.numRows(), VT.values(),
                  VT.numRows(), work.getRawPtr(), work.size(),
                  rwork.getRawPtr(), &info );
    DTK_CHECK( 0 == info );

    // Check the singular values. If they are greater than delta they count.
    double epsilon = std::numeric_limits<double>::epsilon();
    double delta = S[0] * epsilon;
    int rank = std::count_if( S.begin(), S.end(),
                              [=]( double s ) { return ( s > delta ); } );

    return ( rank == full_rank );
}

//---------------------------------------------------------------------------//

} // end namespace DataTransferKit

//---------------------------------------------------------------------------//

#endif // end DTK_LOCALMLSPROBLEM_IMPL_HPP

//---------------------------------------------------------------------------//
// end DTK_LocalMLSProblem_impl.hpp
//---------------------------------------------------------------------------//