NumLib::LocalLinearLeastSquaresExtrapolator Class Reference

Detailed Description

Extrapolator doing a linear least squares extrapolation locally in each element.

The results of the element-wise least squares are averaged at each node.

Note

The described procedure is not least-squares optimal on the whole mesh! But the residuals are computed from the actual interpolation result that is being returned by this class.

The number of integration points in each element must be greater than or equal to the number of nodes of that element. This restriction is due to the use of the least squares which requires an exact or overdetermined equation system.

Definition at line 31 of file LocalLinearLeastSquaresExtrapolator.h.

#include <LocalLinearLeastSquaresExtrapolator.h>

Inheritance diagram for NumLib::LocalLinearLeastSquaresExtrapolator:

Collaboration diagram for NumLib::LocalLinearLeastSquaresExtrapolator:

Classes
struct	CachedData
	Stores a matrix and its Moore-Penrose pseudo-inverse. More...

Public Member Functions
	LocalLinearLeastSquaresExtrapolator (NumLib::LocalToGlobalIndexMap const &dof_table)
void	extrapolate (const unsigned num_components, ExtrapolatableElementCollection const &extrapolatables, const double t, std::vector< GlobalVector * > const &x, std::vector< NumLib::LocalToGlobalIndexMap const * > const &dof_table) override
	Extrapolates the given property from the given local assemblers.
void	calculateResiduals (const unsigned num_components, ExtrapolatableElementCollection const &extrapolatables, const double t, std::vector< GlobalVector * > const &x, std::vector< NumLib::LocalToGlobalIndexMap const * > const &dof_table) override
GlobalVector const &	getNodalValues () const override
GlobalVector const &	getElementResiduals () const override
Public Member Functions inherited from NumLib::Extrapolator
virtual	~Extrapolator ()=default

Private Member Functions
void	extrapolateElement (std::size_t const element_index, const unsigned num_components, ExtrapolatableElementCollection const &extrapolatables, const double t, std::vector< GlobalVector * > const &x, std::vector< NumLib::LocalToGlobalIndexMap const * > const &dof_table, GlobalVector &counts)
	Extrapolate one element.
void	calculateResidualElement (std::size_t const element_index, const unsigned num_components, ExtrapolatableElementCollection const &extrapolatables, const double t, std::vector< GlobalVector * > const &x, std::vector< NumLib::LocalToGlobalIndexMap const * > const &dof_table)
	Compute the residuals for one element.

Private Attributes
std::unique_ptr< GlobalVector >	_nodal_values
	extrapolated nodal values
std::unique_ptr< GlobalVector >	_residuals
	extrapolation residuals
NumLib::LocalToGlobalIndexMap const &	_dof_table_single_component
	DOF table used for writing to global vectors.
std::vector< double >	_integration_point_values_cache
	Avoids frequent reallocations.
std::map< std::pair< unsigned, unsigned >, CachedData >	_qr_decomposition_cache

Constructor & Destructor Documentation

LocalLinearLeastSquaresExtrapolator()

NumLib::LocalLinearLeastSquaresExtrapolator::LocalLinearLeastSquaresExtrapolator ( NumLib::LocalToGlobalIndexMap const & dof_table )

explicit

Constructs a new instance.

Note

The dof_table must point to a d.o.f. table for one single-component variable.

Definition at line 18 of file LocalLinearLeastSquaresExtrapolator.cpp.

    : _dof_table_single_component(dof_table)
{
    /* Note in case the following assertion fails:
     * If you copied the extrapolation code, for your processes from
     * somewhere, note that the code from the groundwater flow process might
     * not suit your needs: It is a special case and is therefore most
     * likely too simplistic. You better adapt the extrapolation code from
     * some more advanced process, like the TES process.
     */
    if (dof_table.getNumberOfGlobalComponents() != 1)
    {
        OGS_FATAL(
            "The d.o.f. table passed must be for one variable that has only "
            "one component!");
    }
}

References _dof_table_single_component, and NumLib::LocalToGlobalIndexMap::getNumberOfGlobalComponents().

Member Function Documentation

calculateResidualElement()

void NumLib::LocalLinearLeastSquaresExtrapolator::calculateResidualElement ( std::size_t const element_index, const unsigned num_components, ExtrapolatableElementCollection const & extrapolatables, const double t, std::vector< GlobalVector * > const & x, std::vector< NumLib::LocalToGlobalIndexMap const * > const & dof_table )

private

Compute the residuals for one element.

Definition at line 274 of file LocalLinearLeastSquaresExtrapolator.cpp.

{
    auto const& int_pt_vals = extrapolatables.getIntegrationPointValues(
        element_index, t, x, dof_table, _integration_point_values_cache);
 
    auto const num_values = static_cast<unsigned>(int_pt_vals.size());
    if (num_values % num_components != 0)
    {
        OGS_FATAL(
            "The number of computed integration point values is not divisible "
            "by the number of num_components. Maybe the computed property is "
            "not a {:d}-component vector for each integration point.",
            num_components);
    }
 
    // number of integration points in the element
    const auto num_int_pts = num_values / num_components;
 
    const auto& global_indices =
        _dof_table_single_component(element_index, 0).rows;
    const auto num_nodes = static_cast<unsigned>(global_indices.size());
 
    auto const& interpolation_matrix =
        _qr_decomposition_cache.find({num_nodes, num_int_pts})->second.A;
 
    Eigen::VectorXd nodal_vals_element(num_nodes);
    auto const int_pt_vals_mat =
        MathLib::toMatrix(int_pt_vals, num_components, num_int_pts);
 
    MathLib::LinAlg::setLocalAccessibleVector(
        *_nodal_values);  // For access in the for-loop.
    for (unsigned comp = 0; comp < num_components; ++comp)
    {
        // filter nodal values of the current element
        for (unsigned i = 0; i < num_nodes; ++i)
        {
            // TODO PETSc negative indices?
            auto const idx = num_components * global_indices[i] + comp;
            nodal_vals_element[i] = _nodal_values->get(idx);
        }
 
        double const residual = (interpolation_matrix * nodal_vals_element -
                                 int_pt_vals_mat.row(comp).transpose())
                                    .squaredNorm();
 
        auto const eidx =
            static_cast<GlobalIndexType>(num_components * element_index + comp);
        // The residual is set to the root mean square value.
        auto const root_mean_square = std::sqrt(residual / num_int_pts);
        _residuals->set(eidx, root_mean_square);
    }
}

References _dof_table_single_component, _integration_point_values_cache, _nodal_values, _qr_decomposition_cache, _residuals, NumLib::ExtrapolatableElementCollection::getIntegrationPointValues(), OGS_FATAL, MathLib::LinAlg::setLocalAccessibleVector(), and MathLib::toMatrix().

Referenced by calculateResiduals().

calculateResiduals()

void NumLib::LocalLinearLeastSquaresExtrapolator::calculateResiduals ( const unsigned num_components, ExtrapolatableElementCollection const & extrapolatables, const double t, std::vector< GlobalVector * > const & x, std::vector< NumLib::LocalToGlobalIndexMap const * > const & dof_table )

overridevirtual

Computes residuals from the extrapolation of the given property.

The residuals are computed as element values.

Precondition

extrapolate() must have been called before with the same arguments.

The computed residuals are root-mean-square of the difference between the integration point values obtained from the local assemblers and the extrapolation results when interpolated back to the integration points again.

Implements NumLib::Extrapolator.

Definition at line 100 of file LocalLinearLeastSquaresExtrapolator.cpp.

{
    auto const num_element_dof_result = static_cast<GlobalIndexType>(
        _dof_table_single_component.size() * num_components);
 
    if (!_residuals || _residuals->size() != num_element_dof_result)
    {
#ifndef USE_PETSC
        _residuals.reset(new GlobalVector{num_element_dof_result});
#else
        _residuals.reset(new GlobalVector{num_element_dof_result, false});
#endif
    }
 
    if (static_cast<std::size_t>(num_element_dof_result) !=
        extrapolatables.size() * num_components)
    {
        OGS_FATAL("mismatch in number of D.o.F.");
    }
 
    auto const size = extrapolatables.size();
    for (std::size_t i = 0; i < size; ++i)
    {
        calculateResidualElement(i, num_components, extrapolatables, t, x,
                                 dof_table);
    }
    MathLib::LinAlg::finalizeAssembly(*_residuals);
}

References _dof_table_single_component, _residuals, calculateResidualElement(), MathLib::LinAlg::finalizeAssembly(), OGS_FATAL, and NumLib::ExtrapolatableElementCollection::size().

extrapolate()

void NumLib::LocalLinearLeastSquaresExtrapolator::extrapolate ( const unsigned num_components, ExtrapolatableElementCollection const & extrapolatables, const double t, std::vector< GlobalVector * > const & x, std::vector< NumLib::LocalToGlobalIndexMap const * > const & dof_table )

overridevirtual

Extrapolates the given property from the given local assemblers.

Implements NumLib::Extrapolator.

Definition at line 37 of file LocalLinearLeastSquaresExtrapolator.cpp.

{
    auto const num_nodal_dof_result =
        _dof_table_single_component.dofSizeWithoutGhosts() * num_components;
 
    std::vector<GlobalIndexType> ghost_indices;
    {  // Create num_components times version of ghost_indices arranged by
       // location. For example for 3 components and ghost_indices {5,6,10} we
       // compute {15, 16, 17,  18, 19, 20,  30, 31, 32}.
        auto const& single_component_ghost_indices =
            _dof_table_single_component.getGhostIndices();
        auto const single_component_ghost_indices_size =
            single_component_ghost_indices.size();
        ghost_indices.reserve(single_component_ghost_indices_size *
                              num_components);
        for (unsigned i = 0; i < single_component_ghost_indices_size; ++i)
        {
            for (unsigned c = 0; c < num_components; ++c)
            {
                ghost_indices.push_back(
                    single_component_ghost_indices[i] * num_components + c);
            }
        }
    }
 
    if (!_nodal_values ||
#ifdef USE_PETSC
        _nodal_values->getLocalSize() + _nodal_values->getGhostSize()
#else
        _nodal_values->size()
#endif
            != static_cast<GlobalIndexType>(num_nodal_dof_result))
    {
        _nodal_values = MathLib::MatrixVectorTraits<GlobalVector>::newInstance(
            {num_nodal_dof_result, num_nodal_dof_result, &ghost_indices,
             nullptr});
    }
    _nodal_values->setZero();
 
    // counts the writes to each nodal value, i.e., the summands in order to
    // compute the average afterwards
    auto counts =
        MathLib::MatrixVectorTraits<GlobalVector>::newInstance(*_nodal_values);
    counts->setZero();
 
    auto const size = extrapolatables.size();
    for (std::size_t i = 0; i < size; ++i)
    {
        extrapolateElement(i, num_components, extrapolatables, t, x, dof_table,
                           *counts);
    }
    MathLib::LinAlg::finalizeAssembly(*_nodal_values);
    MathLib::LinAlg::finalizeAssembly(*counts);
 
    MathLib::LinAlg::componentwiseDivide(*_nodal_values, *_nodal_values,
                                         *counts);
}

References _dof_table_single_component, _nodal_values, MathLib::LinAlg::componentwiseDivide(), extrapolateElement(), MathLib::LinAlg::finalizeAssembly(), and NumLib::ExtrapolatableElementCollection::size().

extrapolateElement()

void NumLib::LocalLinearLeastSquaresExtrapolator::extrapolateElement ( std::size_t const element_index, const unsigned num_components, ExtrapolatableElementCollection const & extrapolatables, const double t, std::vector< GlobalVector * > const & x, std::vector< NumLib::LocalToGlobalIndexMap const * > const & dof_table, GlobalVector & counts )

private

Extrapolate one element.

Definition at line 134 of file LocalLinearLeastSquaresExtrapolator.cpp.

{
    auto const& integration_point_values =
        extrapolatables.getIntegrationPointValues(
            element_index, t, x, dof_table, _integration_point_values_cache);
 
    // Empty vector means to ignore the values and not to change the counts.
    if (integration_point_values.empty())
    {
        return;
    }
 
    auto const& N_0 = extrapolatables.getShapeMatrix(element_index, 0);
    auto const num_nodes = static_cast<unsigned>(N_0.cols());
    auto const num_values =
        static_cast<unsigned>(integration_point_values.size());
 
    if (num_values % num_components != 0)
    {
        OGS_FATAL(
            "The number of computed integration point values is not divisible "
            "by the number of num_components. Maybe the computed property is "
            "not a {:d}-component vector for each integration point.",
            num_components);
    }
 
    // number of integration points in the element
    const auto num_int_pts = num_values / num_components;
 
    if (num_int_pts < num_nodes)
    {
        OGS_FATAL(
            "Least squares is not possible if there are more nodes than "
            "integration points.");
    }
 
    auto const pair_it_inserted = _qr_decomposition_cache.emplace(
        std::make_pair(num_nodes, num_int_pts), CachedData{});
 
    auto& cached_data = pair_it_inserted.first->second;
    if (pair_it_inserted.second)
    {
        DBUG("Computing new singular value decomposition");
 
        // interpolation_matrix * nodal_values = integration_point_values
        // We are going to pseudo-invert this relation now using singular value
        // decomposition.
        auto& interpolation_matrix = cached_data.A;
        interpolation_matrix.resize(num_int_pts, num_nodes);
 
        interpolation_matrix.row(0) = N_0;
        for (unsigned int_pt = 1; int_pt < num_int_pts; ++int_pt)
        {
            auto const& shp_mat =
                extrapolatables.getShapeMatrix(element_index, int_pt);
            assert(shp_mat.cols() == num_nodes);
 
            // copy shape matrix to extrapolation matrix row-wise
            interpolation_matrix.row(int_pt) = shp_mat;
        }
 
        // JacobiSVD is extremely reliable, but fast only for small matrices.
        // But we usually have small matrices and we don't compute very often.
        // Cf.
        // http://eigen.tuxfamily.org/dox/group__TopicLinearAlgebraDecompositions.html
        //
        // Decomposes interpolation_matrix = U S V^T.
        Eigen::JacobiSVD<Eigen::MatrixXd> svd(
            interpolation_matrix, Eigen::ComputeThinU | Eigen::ComputeThinV);
 
        auto const& S = svd.singularValues();
        auto const& U = svd.matrixU();
        auto const& V = svd.matrixV();
 
        // Compute and save the pseudo inverse V * S^{-1} * U^T.
        auto const rank = svd.rank();
        assert(rank == num_nodes);
 
        // cf. http://eigen.tuxfamily.org/dox/JacobiSVD_8h_source.html
        cached_data.A_pinv.noalias() = V.leftCols(rank) *
                                       S.head(rank).asDiagonal().inverse() *
                                       U.leftCols(rank).transpose();
    }
    else if (cached_data.A.row(0) != N_0)
    {
        OGS_FATAL("The cached and the passed shapematrices differ.");
    }
 
    auto const& global_indices =
        _dof_table_single_component(element_index, 0).rows;
 
    if (num_components == 1)
    {
        auto const integration_point_values_vec =
            MathLib::toVector(integration_point_values);
 
        // Apply the pre-computed pseudo-inverse.
        Eigen::VectorXd const nodal_values =
            cached_data.A_pinv * integration_point_values_vec;
 
        // TODO does that give rise to PETSc problems? E.g., writing to ghost
        // nodes? Furthermore: Is ghost nodes communication necessary for PETSc?
        _nodal_values->add(global_indices, nodal_values);
        counts.add(global_indices,
                   std::vector<double>(global_indices.size(), 1.0));
    }
    else
    {
        auto const integration_point_values_mat = MathLib::toMatrix(
            integration_point_values, num_components, num_int_pts);
 
        // Apply the pre-computed pseudo-inverse.
        Eigen::MatrixXd const nodal_values =
            cached_data.A_pinv * integration_point_values_mat.transpose();
 
        std::vector<GlobalIndexType> indices;
        indices.reserve(num_components * global_indices.size());
 
        // _nodal_values is ordered location-wise
        for (unsigned comp = 0; comp < num_components; ++comp)
        {
            transform(cbegin(global_indices), cend(global_indices),
                      back_inserter(indices),
                      [&](auto const i) { return num_components * i + comp; });
        }
 
        // Nodal_values are passed as a raw pointer, because PETScVector and
        // EigenVector implementations differ slightly.
        _nodal_values->add(indices, nodal_values.data());
        counts.add(indices, std::vector<double>(indices.size(), 1.0));
    }
}

References _dof_table_single_component, _integration_point_values_cache, _nodal_values, _qr_decomposition_cache, MathLib::EigenVector::add(), DBUG(), NumLib::ExtrapolatableElementCollection::getIntegrationPointValues(), NumLib::ExtrapolatableElementCollection::getShapeMatrix(), OGS_FATAL, MathLib::toMatrix(), and MathLib::toVector().

Referenced by extrapolate().

getElementResiduals()

GlobalVector const & NumLib::LocalLinearLeastSquaresExtrapolator::getElementResiduals ( ) const

inlineoverridevirtual

Returns the extrapolation residuals.

Todo

Maybe write directly to a MeshProperty.

Implements NumLib::Extrapolator.

Definition at line 70 of file LocalLinearLeastSquaresExtrapolator.h.

    {
        return *_residuals;
    }

References _residuals.

getNodalValues()

GlobalVector const & NumLib::LocalLinearLeastSquaresExtrapolator::getNodalValues ( ) const

inlineoverridevirtual

Returns the extrapolated nodal values.

Todo

Maybe write directly to a MeshProperty.

Implements NumLib::Extrapolator.

Definition at line 65 of file LocalLinearLeastSquaresExtrapolator.h.

    {
        return *_nodal_values;
    }

References _nodal_values.

Member Data Documentation

_dof_table_single_component

NumLib::LocalToGlobalIndexMap const& NumLib::LocalLinearLeastSquaresExtrapolator::_dof_table_single_component

private

DOF table used for writing to global vectors.

Definition at line 97 of file LocalLinearLeastSquaresExtrapolator.h.

Referenced by LocalLinearLeastSquaresExtrapolator(), calculateResidualElement(), calculateResiduals(), extrapolate(), and extrapolateElement().

_integration_point_values_cache

std::vector<double> NumLib::LocalLinearLeastSquaresExtrapolator::_integration_point_values_cache

private

Avoids frequent reallocations.

Definition at line 100 of file LocalLinearLeastSquaresExtrapolator.h.

Referenced by calculateResidualElement(), and extrapolateElement().

_nodal_values

std::unique_ptr<GlobalVector> NumLib::LocalLinearLeastSquaresExtrapolator::_nodal_values

private

extrapolated nodal values

Definition at line 93 of file LocalLinearLeastSquaresExtrapolator.h.

Referenced by calculateResidualElement(), extrapolate(), extrapolateElement(), and getNodalValues().

_qr_decomposition_cache

std::map<std::pair<unsigned, unsigned>, CachedData> NumLib::LocalLinearLeastSquaresExtrapolator::_qr_decomposition_cache

private

Maps (#nodes, #int_pts) to (N_0, QR decomposition), where N_0 is the shape matrix of the first integration point.

Note

It is assumed that the pair (#nodes, #int_pts) uniquely identifies the set of all shape matrices N for a mesh element (i.e., only N, not dN/dx etc.).

Todo

Add the element dimension as identifying criterion, or change to typeid.

Definition at line 122 of file LocalLinearLeastSquaresExtrapolator.h.

Referenced by calculateResidualElement(), and extrapolateElement().

_residuals

std::unique_ptr<GlobalVector> NumLib::LocalLinearLeastSquaresExtrapolator::_residuals

private

extrapolation residuals

Definition at line 94 of file LocalLinearLeastSquaresExtrapolator.h.

Referenced by calculateResidualElement(), calculateResiduals(), and getElementResiduals().

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The documentation for this class was generated from the following files:

• LocalLinearLeastSquaresExtrapolator.h
• LocalLinearLeastSquaresExtrapolator.cpp