DGtal::DirichletConditions< TLinearAlgebraBackend > Class Template Reference

Aim: A helper class to solve a system with Dirichlet boundary conditions. More...

#include <DGtal/math/linalg/DirichletConditions.h>

Public Types
typedef TLinearAlgebraBackend	LinearAlgebraBackend
typedef LinearAlgebraBackend::DenseVector::Index	Index
typedef LinearAlgebraBackend::DenseVector::Scalar	Scalar
typedef LinearAlgebraBackend::DenseVector	DenseVector
typedef LinearAlgebraBackend::IntegerVector	IntegerVector
typedef LinearAlgebraBackend::DenseMatrix	DenseMatrix
typedef LinearAlgebraBackend::SparseMatrix	SparseMatrix
typedef LinearAlgebraBackend::Triplet	Triplet

Public Member Functions
	BOOST_CONCEPT_ASSERT ((concepts::CDynamicVector< DenseVector >))
	BOOST_CONCEPT_ASSERT ((concepts::CDynamicMatrix< DenseMatrix >))
	BOOST_CONCEPT_ASSERT ((concepts::CDynamicMatrix< SparseMatrix >))
	BOOST_CONCEPT_ASSERT ((concepts::CLinearAlgebra< DenseVector, SparseMatrix >))
	BOOST_CONCEPT_ASSERT ((concepts::CLinearAlgebra< DenseVector, DenseMatrix >))

Static Public Member Functions
static SparseMatrix	dirichletOperator (const SparseMatrix &A, const IntegerVector &p)
static DenseVector	dirichletVector (const SparseMatrix &A, const DenseVector &b, const IntegerVector &p, const DenseVector &u)
static DenseVector	dirichletSolution (const DenseVector &xd, const IntegerVector &p, const DenseVector &u)
static IntegerVector	nullBoundaryVector (const DenseVector &b)

Detailed Description

template<typename TLinearAlgebraBackend>
class DGtal::DirichletConditions< TLinearAlgebraBackend >

Aim: A helper class to solve a system with Dirichlet boundary conditions.

Description of template class 'DirichletConditions'

Typically you have a system of the form \( A x = b \), where you wish to set up Dirichlet boundary conditions \( u \) at some places \( p \) where \( p=1 \) ( \( p=0 \) within the domain). In this case, you solve a modified smaller system \( A_d x_d = b_d \).

A typical usage is:

typedef DirichletConditions< EigenLinearAlgebraBackend > DC;
DC::SparseMatrix  A = ...; // the matrix of the linear system
DC::DenseVector   b = ...; // contains right-hand side and boundary conditions
// set up boundary locations and values
DC::IntegerVector p = IntegerVector::Zero( b.rows() );
DC::DenseVector   u = DenseVector::Zero( b.rows() );
std::vector<DC::Index>  boundary_nodes  = { 12, 17, 25, 38, ... };
std::vector<DC::Scalar> boundary_values = { 10.0, 20.0, 15.0, 7.0, ... };
for ( int i = 0; i < boundary_nodes.size(); i++ )
{ 
   auto j = boundary_nodes[ i ];
   p[ j ] = 1;
   u[ j ] = boundary_values[ i ];
}
DC::SparseMatrix A_d = DC::dirichletOperator( A, p );
DC::DenseVector  b_d = DC::dirichletVector  ( A, b, p, u );
solver.compute( A_d ); // prefactorization
ASSERT(solver.info()==Eigen::Success);
DC::DenseVector x_d = solver.solve( b_d ); // solve Dirichlet system
DC::DenseVector x   = DC::dirichletSolution( x_d, p, u ); // get whole solution

It is a model of boost::CopyConstructible, boost::DefaultConstructible, boost::Assignable.

Template Parameters

TLinearAlgebraBackend

linear algebra backend used (i.e. EigenLinearAlgebraBackend).

Definition at line 97 of file DirichletConditions.h.

Member Typedef Documentation

DenseMatrix

template<typename TLinearAlgebraBackend >

typedef LinearAlgebraBackend::DenseMatrix DGtal::DirichletConditions< TLinearAlgebraBackend >::DenseMatrix

Definition at line 106 of file DirichletConditions.h.

DenseVector

template<typename TLinearAlgebraBackend >

typedef LinearAlgebraBackend::DenseVector DGtal::DirichletConditions< TLinearAlgebraBackend >::DenseVector

Definition at line 104 of file DirichletConditions.h.

Index

template<typename TLinearAlgebraBackend >

typedef LinearAlgebraBackend::DenseVector::Index DGtal::DirichletConditions< TLinearAlgebraBackend >::Index

Definition at line 102 of file DirichletConditions.h.

IntegerVector

template<typename TLinearAlgebraBackend >

typedef LinearAlgebraBackend::IntegerVector DGtal::DirichletConditions< TLinearAlgebraBackend >::IntegerVector

Definition at line 105 of file DirichletConditions.h.

LinearAlgebraBackend

template<typename TLinearAlgebraBackend >

typedef TLinearAlgebraBackend DGtal::DirichletConditions< TLinearAlgebraBackend >::LinearAlgebraBackend

Definition at line 100 of file DirichletConditions.h.

Scalar

template<typename TLinearAlgebraBackend >

typedef LinearAlgebraBackend::DenseVector::Scalar DGtal::DirichletConditions< TLinearAlgebraBackend >::Scalar

Definition at line 103 of file DirichletConditions.h.

SparseMatrix

template<typename TLinearAlgebraBackend >

typedef LinearAlgebraBackend::SparseMatrix DGtal::DirichletConditions< TLinearAlgebraBackend >::SparseMatrix

Definition at line 107 of file DirichletConditions.h.

Triplet

template<typename TLinearAlgebraBackend >

typedef LinearAlgebraBackend::Triplet DGtal::DirichletConditions< TLinearAlgebraBackend >::Triplet

Definition at line 108 of file DirichletConditions.h.

Member Function Documentation

BOOST_CONCEPT_ASSERT() [1/5]

template<typename TLinearAlgebraBackend >

DGtal::DirichletConditions< TLinearAlgebraBackend >::BOOST_CONCEPT_ASSERT

(

(concepts::CDynamicMatrix< DenseMatrix >)

)

BOOST_CONCEPT_ASSERT() [2/5]

template<typename TLinearAlgebraBackend >

DGtal::DirichletConditions< TLinearAlgebraBackend >::BOOST_CONCEPT_ASSERT

(

(concepts::CDynamicMatrix< SparseMatrix >)

)

BOOST_CONCEPT_ASSERT() [3/5]

template<typename TLinearAlgebraBackend >

DGtal::DirichletConditions< TLinearAlgebraBackend >::BOOST_CONCEPT_ASSERT

(

(concepts::CDynamicVector< DenseVector >)

)

BOOST_CONCEPT_ASSERT() [4/5]

template<typename TLinearAlgebraBackend >

DGtal::DirichletConditions< TLinearAlgebraBackend >::BOOST_CONCEPT_ASSERT

(

(concepts::CLinearAlgebra< DenseVector, DenseMatrix >)

)

BOOST_CONCEPT_ASSERT() [5/5]

template<typename TLinearAlgebraBackend >

DGtal::DirichletConditions< TLinearAlgebraBackend >::BOOST_CONCEPT_ASSERT

(

(concepts::CLinearAlgebra< DenseVector, SparseMatrix >)

)

dirichletOperator()

template<typename TLinearAlgebraBackend >

static SparseMatrix DGtal::DirichletConditions< TLinearAlgebraBackend >::dirichletOperator ( const SparseMatrix & A, const IntegerVector & p )

inlinestatic

Typically you have a system of the form \( A x = b \), where you wish to set up Dirichlet boundary conditions \( u \) at some places \( p \) where \( p=1 \) ( \( p=0 \) within the domain). In this case, you solve a modified smaller system \( A_d x_d = b_d \).

Parameters

A	the matrix representing the linear operator in Ax=b.
p	the vector such that p[i]=1 whenever the i-th node is a boundary node, p[i]=0 otherwise.

Precondition

#row(A) = #col(A) = #col(p)

Returns

the (smaller) linear matrix \( A_d \) to prefactor.

Definition at line 130 of file DirichletConditions.h.

    {
      ASSERT( A.cols() == A.rows() );
      ASSERT( p.rows() == A.rows() );
      const auto n = p.rows();
      std::vector< Index > relabeling( n );
      Index j = 0;
      for ( Index i = 0; i < p.rows(); i++ )
        relabeling[ i ] = ( p[ i ] == 0 ) ? j++ : n;
      // Building matrix
      SparseMatrix Ap( j, j );
      std::vector< Triplet > triplets;
      for ( int k = 0; k < A.outerSize(); ++k )
        for ( typename SparseMatrix::InnerIterator it( A, k ); it; ++it )
          {
            if ( ( relabeling[ it.row() ] != n ) && ( relabeling[ it.col() ] != n ) )
              triplets.push_back( { relabeling[ it.row() ], relabeling[ it.col() ],
                  it.value() } );
          }
      Ap.setFromTriplets( triplets.cbegin(), triplets.cend() );
      return Ap;
    }

Referenced by DGtal::GeodesicsInHeat< TPolygonalCalculus >::init(), and DGtal::VectorsInHeat< TPolygonalCalculus >::init().

dirichletSolution()

template<typename TLinearAlgebraBackend >

static DenseVector DGtal::DirichletConditions< TLinearAlgebraBackend >::dirichletSolution ( const DenseVector & xd, const IntegerVector & p, const DenseVector & u )

inlinestatic

Typically you have a system of the form \( A x = b \), where you wish to set up Dirichlet boundary conditions \( u \) at some places \( p \) where \( p=1 \) ( \( p=0 \) within the domain). In this case, you solve a modified smaller system \( A_d x_d = b_d \).

Parameters

xd	the solution to the smaller system \( A_d x_d = b_d \).
p	the vector such that p[i]=1 whenever the i-th node is a boundary node, p[i]=0 otherwise.
u	the vector giving the constrained Dirichlet values on places such that p[i]=1.

Returns

the solution \( x \) to the original system \( A x = b \),

Definition at line 203 of file DirichletConditions.h.

    {
      DenseVector  x( p.rows() );
      Index j = 0;
      for ( Index i = 0; i < p.rows(); i++ )
        x[ i ] = ( p[ i ] == 0 ) ? xd[ j++ ] : u[ i ];
      return x;
    }

Referenced by DGtal::GeodesicsInHeat< TPolygonalCalculus >::compute(), and DGtal::VectorsInHeat< TPolygonalCalculus >::compute().

dirichletVector()

template<typename TLinearAlgebraBackend >

static DenseVector DGtal::DirichletConditions< TLinearAlgebraBackend >::dirichletVector ( const SparseMatrix & A, const DenseVector & b, const IntegerVector & p, const DenseVector & u )

inlinestatic

Typically you have a system of the form \( A x = b \), where you wish to set up Dirichlet boundary conditions \( u \) at some places \( p \) where \( p=1 \) ( \( p=0 \) within the domain). In this case, you solve a modified smaller system \( A_d x_d = b_d \).

Parameters

A	the matrix representing the linear operator in \( Ax=b \).
b	the vector representing the vector to solve for in \( Ax=b \).
p	the vector such that p[i]=1 whenever the i-th node is a boundary node, p[i]=0 otherwise.
u	the vector giving the constrained Dirichlet values on places such that p[i]=1.

Precondition

#row(A) = #col(A) = #col(p) = #col(b) = #col(u)

Returns

the (smaller) form \( b_d \) to solve for

Definition at line 170 of file DirichletConditions.h.

    {
      ASSERT( A.cols() == A.rows() );
      ASSERT( p.rows() == A.rows() );
      const auto n = p.rows();
      DenseVector  up = p.array().template cast<double>() * u.array();
      DenseVector tmp = b.array() - (A * up).array();
      std::vector< Index > relabeling( n );
      Index j = 0;
      for ( Index i = 0; i < p.rows(); i++ )
        relabeling[ i ] = ( p[ i ] == 0 ) ? j++ : n;
      DenseVector  bp( j );
      for ( Index i = 0; i < p.rows(); i++ )
        if ( p[ i ] == 0 ) bp[ relabeling[ i ] ] = tmp[ i ];
      return bp;
    }

Referenced by DGtal::GeodesicsInHeat< TPolygonalCalculus >::compute(), and DGtal::VectorsInHeat< TPolygonalCalculus >::compute().

nullBoundaryVector()

template<typename TLinearAlgebraBackend >

static IntegerVector DGtal::DirichletConditions< TLinearAlgebraBackend >::nullBoundaryVector ( const DenseVector & b )

inlinestatic

Utility method to build a null vector that will be useful to define the boundary characteristic set for Dirichlet conditions.

Parameters

b	any vector of the size of your problem (e.g., \( b \) in your problem \( Ax = b \))

Returns

an integer vector of same size as b with zero everywhere.

Definition at line 223 of file DirichletConditions.h.

    {
      return IntegerVector::Zero( b.rows() );
    }

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

• DirichletConditions.h