Active Set method. More...

#include <active_set.h>

Public Types
typedef TAlgebra	algebra_type
	Type of algebra.

typedef GridFunction< TDomain, TAlgebra >	function_type
	Type of grid function.

typedef algebra_type::matrix_type	matrix_type
	Type of algebra matrix.

typedef domain_traits< TDomain::dim >::grid_base_object	TBaseElem
	base element type of associated domain

typedef vector_type::value_type	value_type
	Type of algebra value.

typedef algebra_type::vector_type	vector_type
	Type of algebra vector.

Public Member Functions
bool	active_index (function_type &u, function_type &rhs, function_type &lagrangeMult, function_type &gap)
	determines the active indices, stores them in a vector and sets dirichlet values in rhs for active indices

template<typename TElem , typename TIterator >
void	active_index_elem (TIterator iterBegin, TIterator iterEnd, function_type &u, function_type &rhs, function_type &lagrangeMult)
	checks the distance to the prescribed obstacle/constraint

	ActiveSet ()
	constructor

bool	check_conv (function_type &u, const function_type &lambda, const size_t step)
	checks if all constraints are fulfilled & the activeSet remained unchanged

template<typename TElem , typename TIterator >
bool	check_conv_elem (TIterator iterBegin, TIterator iterEnd, function_type &u, const function_type &lambda)

bool	check_inequ (const matrix_type &mat, const vector_type &u, const vector_type &lagrangeMult, const vector_type &rhs)
	checks if all inequalities are fulfilled

void	lagrange_mat_inv (matrix_type &lagrangeMatInv)

template<typename TElem , typename TIterator >
void	lagrange_mat_inv_elem (TIterator iterBegin, TIterator iterEnd, matrix_type &lagrangeMatInv)

void	lagrange_multiplier (function_type &lagrangeMult, const function_type &u)
	computes the lagrange multiplier for a given disc

void	prepare (function_type &u)

void	residual_lagrange_mult (vector_type &lagMult, const matrix_type &mat, const vector_type &u, vector_type &rhs)

void	set_dirichlet_rows (matrix_type &mat)

void	set_lagrange_multiplier_disc (SmartPtr< ILagrangeMultiplierDisc< TDomain, function_type > > lagMultDisc)
	sets a discretization in order to compute the lagrange multiplier

void	set_obstacle (ConstSmartPtr< function_type > obs)
	sets obstacle gridfunction, which limits the solution u

Static Public Attributes
static const int	dim = TDomain::dim
	domain dimension

Private Attributes
bool	m_bObs

SmartPtr< DoFDistribution >	m_spDD
	pointer to the DofDistribution on the whole domain

SmartPtr< TDomain >	m_spDom

SmartPtr< ILagrangeMultiplierDisc< TDomain, function_type > >	m_spLagMultDisc
	pointer to a lagrangeMultiplier-Disc

ConstSmartPtr< function_type >	m_spObs
	number of functions

vector< DoFIndex >	m_vActiveSetGlob
	vector of the current active set of global DoFIndices

vector< DoFIndex >	m_vActiveSetGlobOld
	vector remembering the active set of global DoFIndices

vector< int >	m_vActiveSubsets
	vector of possible active subsets

Detailed Description

template<typename TDomain, typename TAlgebra>
class ug::ActiveSet< TDomain, TAlgebra >

Active Set method.

The active Set method is a well-known method in constrained optimization theory. A general formulation for these problems reads

\begin{eqnarray*} min_{x \in \mathbb{R}^n} f(x) \end{eqnarray*}

s.t.

\begin{eqnarray*} c_i(x) = 0, \qquad i \in E \\ c_i(x) \ge 0 \qquad i \in I, \end{eqnarray*}

where \( f \), \( c_i \) are smooth, real-valued functions on a subset of \( \mathbb{R}^n \). \( I \) (set of inequality constraints) and \( E \) (set of equality constraints) are two finite sets of indices. \( f \) is called objective function.

The active Set \( A \) is defined as:

\begin{eqnarray*} A(x) := E \cup \{ i \in I | c_i(x) = 0 \}, \end{eqnarray*}

i.e. for \( i \in I \) the inequality constraint is said to be active, if \( c_i(x) = 0 \). Otherwise ( \( c_i(x) > 0 \)) it is called inactive.

A common approach to treat the inequality constraints is its reformulation as equations by using so called complementarity functions, see e.g. C.Hager und B. I. Wohlmuth: "Hindernis- und Kontaktprobleme" for a simple introduction into this topic. By means of complementarity functions, constraints of the form

\begin{eqnarray*} a \ge 0, \, b \ge 0, \, a b = 0 \end{eqnarray*}

with \( a, b \in \mathbb{R}^n \) can be reformulated as

\begin{eqnarray*} C(a,b) = 0, \end{eqnarray*}

with \( C: \mathbb{R}^n x \mathbb{R}^n \to \mathbb{R}^n \) being an appropriate complementarity function. The value of \( C \) indicates, whether the index is active or inactive (see method 'active_index'). Thus, it determines in every step the set of active indices, for which the original system of equations needs to be adapted. The influence of these active indices on the original system of equations ( \( K u = f \), with \( K \): system-matrix; \( u, f \) vectors) can be modelled by means of a lagrange multiplier ' \( \lambda \)'. \( \lambda \) can either be defined by the residual ( \( \lambda := f - K u \), see method 'residual_lagrange_mult') or by a problem-dependent computation (see method 'lagrange_multiplier').

In every Active Set step a linear or linearized system is solved. The algorithm stops when the active and inactive Set remains unchanged (see method 'check_conv').

References:

J. Nocedal and S. J. Wright. Numerical optimization.(2000)

Template Parameters

TDomain	Domain type
TAlgebra	Algebra type

Member Typedef Documentation

◆ algebra_type

template<typename TDomain , typename TAlgebra >

typedef TAlgebra ug::ActiveSet< TDomain, TAlgebra >::algebra_type

Type of algebra.

◆ function_type

template<typename TDomain , typename TAlgebra >

typedef GridFunction<TDomain, TAlgebra> ug::ActiveSet< TDomain, TAlgebra >::function_type

Type of grid function.

◆ matrix_type

template<typename TDomain , typename TAlgebra >

typedef algebra_type::matrix_type ug::ActiveSet< TDomain, TAlgebra >::matrix_type

Type of algebra matrix.

◆ TBaseElem

template<typename TDomain , typename TAlgebra >

typedef domain_traits<TDomain::dim>::grid_base_object ug::ActiveSet< TDomain, TAlgebra >::TBaseElem

base element type of associated domain

◆ value_type

template<typename TDomain , typename TAlgebra >

typedef vector_type::value_type ug::ActiveSet< TDomain, TAlgebra >::value_type

Type of algebra value.

◆ vector_type

template<typename TDomain , typename TAlgebra >

typedef algebra_type::vector_type ug::ActiveSet< TDomain, TAlgebra >::vector_type

Type of algebra vector.

Constructor & Destructor Documentation

◆ ActiveSet()

template<typename TDomain , typename TAlgebra >

ug::ActiveSet< TDomain, TAlgebra >::ActiveSet ( )

inline

constructor

Member Function Documentation

◆ active_index()

template<typename TDomain , typename TAlgebra >

bool ug::ActiveSet< TDomain, TAlgebra >::active_index	(	function_type &	u,
		function_type &	rhs,
		function_type &	lagrangeMult,
		function_type &	gap
	)

determines the active indices, stores them in a vector and sets dirichlet values in rhs for active indices

References ug::DimensionOfSubset(), ug::DoFDistributionInfoProvider::is_def_in_subset(), ug::DoFDistributionInfoProvider::num_fct(), ug::GridFunction< TDomain, TAlgebra >::num_indices(), UG_LOG, and UG_THROW.

◆ active_index_elem()

template<typename TDomain , typename TAlgebra >

template<typename TElem , typename TIterator >

void ug::ActiveSet< TDomain, TAlgebra >::active_index_elem	(	TIterator	iterBegin,
		TIterator	iterEnd,
		function_type &	u,
		function_type &	rhs,
		function_type &	lagrangeMult
	)

checks the distance to the prescribed obstacle/constraint

References ug::BlockRef(), ug::CollectCornerCoordinates(), ug::LocalIndices::comp(), ug::GridFunction< TDomain, TAlgebra >::domain(), ug::ReferenceElementProvider::get(), ug::GetLocalVector(), ug::LocalIndices::index(), ug::GridFunction< TDomain, TAlgebra >::indices(), ug::LocalIndices::num_dof(), ug::LocalIndices::num_fct(), ug::LocalVector::resize(), UG_LOG, ug::VecDot(), and ug::VecLength().

◆ check_conv()

template<typename TDomain , typename TAlgebra >

bool ug::ActiveSet< TDomain, TAlgebra >::check_conv	(	function_type &	u,
		const function_type &	lambda,
		const size_t	step
	)

checks if all constraints are fulfilled & the activeSet remained unchanged

References ug::DimensionOfSubset(), UG_LOG, and UG_THROW.

◆ check_conv_elem()

template<typename TDomain , typename TAlgebra >

template<typename TElem , typename TIterator >

bool ug::ActiveSet< TDomain, TAlgebra >::check_conv_elem	(	TIterator	iterBegin,
		TIterator	iterEnd,
		function_type &	u,
		const function_type &	lambda
	)

References ug::CollectCornerCoordinates(), ug::GridFunction< TDomain, TAlgebra >::domain(), ug::ReferenceElementProvider::get(), ug::GetLocalVector(), ug::GridFunction< TDomain, TAlgebra >::indices(), ug::LocalIndices::num_dof(), ug::LocalIndices::num_fct(), ug::LocalVector::resize(), ug::VecDot(), and ug::VecLength().

◆ check_inequ()

template<typename TDomain , typename TAlgebra >

bool ug::ActiveSet< TDomain, TAlgebra >::check_inequ	(	const matrix_type &	mat,
		const vector_type &	u,
		const vector_type &	lagrangeMult,
		const vector_type &	rhs
	)

checks if all inequalities are fulfilled

References ug::MatMult(), ug::MatMultDirect(), UG_LOG, and UG_THROW.

◆ lagrange_mat_inv()

template<typename TDomain , typename TAlgebra >

void ug::ActiveSet< TDomain, TAlgebra >::lagrange_mat_inv ( matrix_type & lagrangeMatInv )

References ug::DimensionOfSubset(), UG_LOG, and UG_THROW.

◆ lagrange_mat_inv_elem()

template<typename TDomain , typename TAlgebra >

template<typename TElem , typename TIterator >

void ug::ActiveSet< TDomain, TAlgebra >::lagrange_mat_inv_elem	(	TIterator	iterBegin,
		TIterator	iterEnd,
		matrix_type &	lagrangeMatInv
	)

References ug::AddLocalMatrixToGlobal(), ug::CollectCornerCoordinates(), ug::ReferenceElementProvider::get(), ug::LFEID::LAGRANGE, ug::LocalMatrix::resize(), ug::ReferenceElement::size(), UG_CATCH_THROW, and UG_LOG.

◆ lagrange_multiplier()

template<typename TDomain , typename TAlgebra >

void ug::ActiveSet< TDomain, TAlgebra >::lagrange_multiplier	(	function_type &	lagrangeMult,
		const function_type &	u
	)

computes the lagrange multiplier for a given disc

References UG_LOG, and UG_THROW.

◆ prepare()

template<typename TDomain , typename TAlgebra >

void ug::ActiveSet< TDomain, TAlgebra >::prepare ( function_type & u )

References ug::GridFunction< TDomain, TAlgebra >::dof_distribution(), and ug::GridFunction< TDomain, TAlgebra >::domain().

◆ residual_lagrange_mult()

template<typename TDomain , typename TAlgebra >

void ug::ActiveSet< TDomain, TAlgebra >::residual_lagrange_mult	(	vector_type &	lagMult,
		const matrix_type &	mat,
		const vector_type &	u,
		vector_type &	rhs
	)

computes the lagrange multiplier by means of the residuum (lagMult = rhs - mat * u)

References ug::DoFRef(), ug::MatMult(), ug::MatMultDirect(), UG_LOG, and UG_THROW.

◆ set_dirichlet_rows()

template<typename TDomain , typename TAlgebra >

void ug::ActiveSet< TDomain, TAlgebra >::set_dirichlet_rows ( matrix_type & mat )

References ug::SetDirichletRow().

◆ set_lagrange_multiplier_disc()

template<typename TDomain , typename TAlgebra >

void ug::ActiveSet< TDomain, TAlgebra >::set_lagrange_multiplier_disc ( SmartPtr< ILagrangeMultiplierDisc< TDomain, function_type > > lagMultDisc )

inline

sets a discretization in order to compute the lagrange multiplier

References ug::ActiveSet< TDomain, TAlgebra >::m_spLagMultDisc.

◆ set_obstacle()

template<typename TDomain , typename TAlgebra >

void ug::ActiveSet< TDomain, TAlgebra >::set_obstacle ( ConstSmartPtr< function_type > obs )

inline

sets obstacle gridfunction, which limits the solution u

References ug::ActiveSet< TDomain, TAlgebra >::m_bObs, and ug::ActiveSet< TDomain, TAlgebra >::m_spObs.

Member Data Documentation

◆ dim

template<typename TDomain , typename TAlgebra >

const int ug::ActiveSet< TDomain, TAlgebra >::dim = TDomain::dim

static

domain dimension

◆ m_bObs

template<typename TDomain , typename TAlgebra >

bool ug::ActiveSet< TDomain, TAlgebra >::m_bObs

private

Referenced by ug::ActiveSet< TDomain, TAlgebra >::set_obstacle().

◆ m_spDD

template<typename TDomain , typename TAlgebra >

SmartPtr<DoFDistribution> ug::ActiveSet< TDomain, TAlgebra >::m_spDD

private

pointer to the DofDistribution on the whole domain

◆ m_spDom

template<typename TDomain , typename TAlgebra >

SmartPtr<TDomain> ug::ActiveSet< TDomain, TAlgebra >::m_spDom

private

◆ m_spLagMultDisc

template<typename TDomain , typename TAlgebra >

SmartPtr<ILagrangeMultiplierDisc<TDomain, function_type> > ug::ActiveSet< TDomain, TAlgebra >::m_spLagMultDisc

private

pointer to a lagrangeMultiplier-Disc

Referenced by ug::ActiveSet< TDomain, TAlgebra >::set_lagrange_multiplier_disc().

◆ m_spObs

template<typename TDomain , typename TAlgebra >

ConstSmartPtr<function_type> ug::ActiveSet< TDomain, TAlgebra >::m_spObs

private

number of functions

pointer to a gridfunction describing an obstacle-constraint

Referenced by ug::ActiveSet< TDomain, TAlgebra >::set_obstacle().

◆ m_vActiveSetGlob

template<typename TDomain , typename TAlgebra >

vector<DoFIndex> ug::ActiveSet< TDomain, TAlgebra >::m_vActiveSetGlob

private

vector of the current active set of global DoFIndices

◆ m_vActiveSetGlobOld

template<typename TDomain , typename TAlgebra >

vector<DoFIndex> ug::ActiveSet< TDomain, TAlgebra >::m_vActiveSetGlobOld

private

vector remembering the active set of global DoFIndices

◆ m_vActiveSubsets

template<typename TDomain , typename TAlgebra >

vector<int> ug::ActiveSet< TDomain, TAlgebra >::m_vActiveSubsets

private

vector of possible active subsets

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

ugbase/lib_algebra/active_set/active_set.h
ugbase/lib_algebra/active_set/active_set_impl.h

Public Types

Public Member Functions

Static Public Attributes

Private Attributes

Detailed Description

Member Typedef Documentation

◆ algebra_type

◆ function_type

◆ matrix_type

◆ TBaseElem

◆ value_type

◆ vector_type

Constructor & Destructor Documentation

◆ ActiveSet()

Member Function Documentation

◆ active_index()

◆ active_index_elem()

◆ check_conv()

◆ check_conv_elem()

◆ check_inequ()

◆ lagrange_mat_inv()

◆ lagrange_mat_inv_elem()

◆ lagrange_multiplier()

◆ prepare()

◆ residual_lagrange_mult()

◆ set_dirichlet_rows()

◆ set_lagrange_multiplier_disc()

◆ set_obstacle()

Member Data Documentation

◆ dim

◆ m_bObs

◆ m_spDD

◆ m_spDom

◆ m_spLagMultDisc

◆ m_spObs

◆ m_vActiveSetGlob

◆ m_vActiveSetGlobOld

◆ m_vActiveSubsets