Base Class for quasi-Newton acceleration schemes. More...

#include <BaseQNAcceleration.hpp>

Inheritance diagram for precice::acceleration::BaseQNAcceleration:

Collaboration diagram for precice::acceleration::BaseQNAcceleration:

Public Member Functions
	BaseQNAcceleration (double initialRelaxation, bool forceInitialRelaxation, int maxIterationsUsed, int timeWindowsReused, int filter, double singularityLimit, std::vector< int > dataIDs, impl::PtrPreconditioner preconditioner)

virtual	~BaseQNAcceleration ()
	Destructor, empty.

virtual std::vector< int >	getDataIDs () const
	Returns all IQN involved data IDs.

virtual void	initialize (const DataMap &cplData)
	Initializes the acceleration.

virtual void	performAcceleration (DataMap &cplData)
	Performs one acceleration step.

virtual void	iterationsConverged (const DataMap &cplData)
	Marks a iteration sequence as converged.

virtual void	exportState (io::TXTWriter &writer)
	Exports the current state of the acceleration to a file.

virtual void	importState (io::TXTReader &reader)
	Imports the last exported state of the acceleration from file.

virtual int	getDeletedColumns () const
	how many QN columns were deleted in this time window

virtual int	getDroppedColumns () const
	how many QN columns were dropped (went out of scope) in this time window

virtual int	getLSSystemCols () const
	: computes number of cols in least squares system, i.e, number of cols in _matrixV, _matrixW, _qrV, etc.. This is only necessary if some procs do not have any nodes on the coupling interface. In this case, the matrices are not constructed and we have no information about the number of cols. This info is needed for intra-participant communication. Number of its =! _cols in general.

Public Member Functions inherited from precice::acceleration::Acceleration
virtual	~Acceleration ()=default

Protected Member Functions
int	getLSSystemRows ()

virtual void	specializedIterationsConverged (const DataMap &cplData)=0
	Marks a iteration sequence as converged.

virtual void	updateDifferenceMatrices (const DataMap &cplData)
	Updates the V, W matrices (as well as the matrices for the secondary data)

virtual void	splitCouplingData (const DataMap &cplData)
	Splits up QN system vector back into the coupling data.

virtual void	applyFilter ()
	Applies the filter method for the least-squares system, defined in the configuration.

virtual void	computeUnderrelaxationSecondaryData (const DataMap &cplData)=0
	Computes underrelaxation for the secondary data.

virtual void	computeQNUpdate (const DataMap &cplData, Eigen::VectorXd &xUpdate)=0
	Computes the quasi-Newton update using the specified pp scheme (IQNIMVJ, IQNILS)

virtual void	removeMatrixColumn (int columnIndex)
	Removes one iteration from V,W matrices and adapts _matrixCols.

void	writeInfo (const std::string &s, bool allProcs=false)
	Wwrites info to the _infostream (also in parallel)

Protected Member Functions inherited from precice::acceleration::Acceleration
void	checkDataIDs (const DataMap &cplData) const
	Checks if all dataIDs are contained in cplData.

void	concatenateCouplingData (const DataMap &cplData, const std::vector< DataID > &dataIDs, Eigen::VectorXd &targetValues, Eigen::VectorXd &targetOldValues) const
	Concatenates all coupling data involved into a single vector.

Protected Attributes
logging::Logger	_log {"acceleration::BaseQNAcceleration"}

impl::PtrPreconditioner	_preconditioner
	Preconditioner for least-squares system if vectorial system is used.

double	_initialRelaxation
	Constant relaxation factor used for first iteration.

int	_maxIterationsUsed
	Maximum number of old data iterations kept.

int	_timeWindowsReused
	Maximum number of old time windows (with data values) kept.

std::vector< int >	_dataIDs
	Data IDs of data to be involved in the IQN algorithm.

std::vector< int >	_secondaryDataIDs
	Data IDs of data not involved in IQN coefficient computation.

bool	_firstIteration = true
	Indicates the first iteration, where constant relaxation is used.

bool	_firstTimeWindow = true

bool	_hasNodesOnInterface = true

bool	_forceInitialRelaxation

bool	_resetLS = false
	If true, the LS system has been modified (reset or recomputed) in such a way, that mere updating of matrices _Wtil, Q, R etc.. is not feasible any more and need to be recomputed.

Eigen::VectorXd	_oldXTilde
	Solver output from last iteration.

Eigen::VectorXd	_residuals
	Current iteration residuals of IQN data. Temporary.

std::map< int, Eigen::VectorXd >	_secondaryResiduals
	Current iteration residuals of secondary data.

Eigen::MatrixXd	_matrixV
	Stores residual deltas.

Eigen::MatrixXd	_matrixW
	Stores x tilde deltas, where x tilde are values computed by solvers.

impl::QRFactorization	_qrV
	Stores the current QR decomposition ov _matrixV, can be updated via deletion/insertion of columns.

int	_filter
	filter method that is used to maintain good conditioning of the least-squares system Either of two types: QR1FILTER or QR2Filter

double	_singularityLimit
	Determines sensitivity when two matrix columns are considered equal.

std::deque< int >	_matrixCols
	Indices (of columns in W, V matrices) of 1st iterations of time windows.

std::vector< int >	_dimOffsets
	Stores the local dimensions, i.e., the offsets in _invJacobian for all processors.

std::ostringstream	_infostringstream
	write some debug/acceleration info to file

std::fstream	_infostream

int	its = 0

int	tWindows = 0

Private Attributes
Eigen::VectorXd	_values
	Concatenation of all coupling data involved in the QN system.

Eigen::VectorXd	_oldValues
	Concatenation of all (old) coupling data involved in the QN system.

Eigen::VectorXd	_oldResiduals
	Difference between solver input and output from last time window.

Eigen::MatrixXd	_matrixVBackup
	backup of the V,W and matrixCols data structures. Needed for the skipping of initial relaxation, if previous time window converged within one iteration i.e., V and W are empty – in this case restore V and W with time window t-2.

Eigen::MatrixXd	_matrixWBackup

std::deque< int >	_matrixColsBackup

int	_nbDelCols = 0
	Number of filtered out columns in this time window.

int	_nbDropCols = 0
	Number of dropped columns in this time window (old time window out of scope)

Additional Inherited Members
Public Types inherited from precice::acceleration::Acceleration
using	DataMap = std::map<int, cplscheme::PtrCouplingData>
	Map from data ID to data values.

Static Public Attributes inherited from precice::acceleration::Acceleration
static const int	NOFILTER = 0

static const int	QR1FILTER = 1

static const int	QR1FILTER_ABS = 2

static const int	QR2FILTER = 3

static const int	PODFILTER = 4

Static Protected Member Functions inherited from precice::acceleration::Acceleration
static void	applyRelaxation (double omega, DataMap &cplData)
	performs a relaxation given a relaxation factor omega

Detailed Description

Base Class for quasi-Newton acceleration schemes.

Definition at line 63 of file BaseQNAcceleration.hpp.

Constructor & Destructor Documentation

◆ BaseQNAcceleration()

precice::acceleration::BaseQNAcceleration::BaseQNAcceleration	(	double	initialRelaxation,
		bool	forceInitialRelaxation,
		int	maxIterationsUsed,
		int	timeWindowsReused,
		int	filter,
		double	singularityLimit,
		std::vector< int >	dataIDs,
		impl::PtrPreconditioner	preconditioner )

Definition at line 33 of file BaseQNAcceleration.cpp.

◆ ~BaseQNAcceleration()

virtual precice::acceleration::BaseQNAcceleration::~BaseQNAcceleration ( )

inlinevirtual

Destructor, empty.

Definition at line 78 of file BaseQNAcceleration.hpp.

Member Function Documentation

◆ applyFilter()

void precice::acceleration::BaseQNAcceleration::applyFilter ( )

protectedvirtual

Applies the filter method for the least-squares system, defined in the configuration.

Definition at line 416 of file BaseQNAcceleration.cpp.

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◆ computeQNUpdate()

virtual void precice::acceleration::BaseQNAcceleration::computeQNUpdate	(	const DataMap &	cplData,
		Eigen::VectorXd &	xUpdate )

protectedpure virtual

Computes the quasi-Newton update using the specified pp scheme (IQNIMVJ, IQNILS)

Implemented in precice::acceleration::IQNILSAcceleration, and precice::acceleration::IQNIMVJAcceleration.

◆ computeUnderrelaxationSecondaryData()

virtual void precice::acceleration::BaseQNAcceleration::computeUnderrelaxationSecondaryData ( const DataMap & cplData )

protectedpure virtual

Computes underrelaxation for the secondary data.

Implemented in precice::acceleration::IQNILSAcceleration, and precice::acceleration::IQNIMVJAcceleration.

◆ exportState()

void precice::acceleration::BaseQNAcceleration::exportState ( io::TXTWriter & writer )

virtual

Exports the current state of the acceleration to a file.

Reimplemented from precice::acceleration::Acceleration.

Definition at line 580 of file BaseQNAcceleration.cpp.

◆ getDataIDs()

virtual std::vector< int > precice::acceleration::BaseQNAcceleration::getDataIDs ( ) const

inlinevirtual

Returns all IQN involved data IDs.

Implements precice::acceleration::Acceleration.

Definition at line 91 of file BaseQNAcceleration.hpp.

◆ getDeletedColumns()

int precice::acceleration::BaseQNAcceleration::getDeletedColumns ( ) const

virtual

how many QN columns were deleted in this time window

Reimplemented from precice::acceleration::Acceleration.

Definition at line 590 of file BaseQNAcceleration.cpp.

◆ getDroppedColumns()

int precice::acceleration::BaseQNAcceleration::getDroppedColumns ( ) const

virtual

how many QN columns were dropped (went out of scope) in this time window

Reimplemented from precice::acceleration::Acceleration.

Definition at line 595 of file BaseQNAcceleration.cpp.

◆ getLSSystemCols()

int precice::acceleration::BaseQNAcceleration::getLSSystemCols ( ) const

virtual

: computes number of cols in least squares system, i.e, number of cols in _matrixV, _matrixW, _qrV, etc.. This is only necessary if some procs do not have any nodes on the coupling interface. In this case, the matrices are not constructed and we have no information about the number of cols. This info is needed for intra-participant communication. Number of its =! _cols in general.

Reimplemented from precice::acceleration::Acceleration.

Definition at line 600 of file BaseQNAcceleration.cpp.

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◆ getLSSystemRows()

int precice::acceleration::BaseQNAcceleration::getLSSystemRows ( )

protected

Definition at line 614 of file BaseQNAcceleration.cpp.

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◆ importState()

void precice::acceleration::BaseQNAcceleration::importState ( io::TXTReader & reader )

virtual

Imports the last exported state of the acceleration from file.

Is empty at the moment!!!

Reimplemented from precice::acceleration::Acceleration.

Definition at line 585 of file BaseQNAcceleration.cpp.

◆ initialize()

void precice::acceleration::BaseQNAcceleration::initialize ( const DataMap & cplData )

virtual

Initializes the acceleration.

initialize()

@brief: Initializes all the needed variables and data

make dimensions public to all procs, last entry _dimOffsets[IntraComm::getSize()] holds the global dimension, global,n

provide vertex offset information for all processors mesh->getVertexOffsets() provides an array that stores the number of mesh vertices on each processor This information needs to be gathered for all meshes. To get the number of respective unknowns of a specific processor we need to multiply the number of vertices with the dimensionality of the vector-valued data for each coupling data.

Implements precice::acceleration::Acceleration.

Reimplemented in precice::acceleration::IQNILSAcceleration, and precice::acceleration::IQNIMVJAcceleration.

Definition at line 76 of file BaseQNAcceleration.cpp.

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◆ iterationsConverged()

void precice::acceleration::BaseQNAcceleration::iterationsConverged ( const DataMap & cplData )

virtual

Marks a iteration sequence as converged.

: Is called when the convergence criterion for the coupling is fulfilled and finalizes the quasi Newton acceleration. Stores new differences in F and C, clears or

Since convergence measurements are done outside the acceleration, this method has to be used to signalize convergence to the acceleration.

iterationsConverged()

updates F and C according to the number of reused time windows

pending deletion (after first iteration of next time window Using the matrices from the old time window for the first iteration is better than doing underrelaxation as first iteration of every time window

Implements precice::acceleration::Acceleration.

Definition at line 462 of file BaseQNAcceleration.cpp.

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◆ performAcceleration()

void precice::acceleration::BaseQNAcceleration::performAcceleration ( DataMap & cplData )

virtual

Performs one acceleration step.

Has to be called after every implicit coupling iteration.

performAcceleration()

@brief: performs one iteration of the quasi Newton acceleration.

update the difference matrices V,W includes: scaling of values computation of residuals appending the difference matrices

=== update and apply preconditioner ===

The preconditioner is only applied to the matrix V and the columns that are inserted into the QR-decomposition of V.

compute quasi-Newton update PRECONDITION: All objects are unscaled, except the matrices within the QR-dec of V. Thus, the pseudo inverse needs to be reverted before using it.

apply quasiNewton update

Implements precice::acceleration::Acceleration.

Definition at line 275 of file BaseQNAcceleration.cpp.

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◆ removeMatrixColumn()

void precice::acceleration::BaseQNAcceleration::removeMatrixColumn ( int columnIndex )

protectedvirtual

Removes one iteration from V,W matrices and adapts _matrixCols.

removeMatrixColumn()

@brief: removes a column from the least squares system, i. e., from the matrices F and C

Reimplemented in precice::acceleration::IQNILSAcceleration, and precice::acceleration::IQNIMVJAcceleration.

Definition at line 552 of file BaseQNAcceleration.cpp.

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◆ specializedIterationsConverged()

virtual void precice::acceleration::BaseQNAcceleration::specializedIterationsConverged ( const DataMap & cplData )

protectedpure virtual

Marks a iteration sequence as converged.

called by the iterationsConverged() method in the BaseQNAcceleration class handles the acceleration specific action after the convergence of one iteration

Implemented in precice::acceleration::IQNILSAcceleration, and precice::acceleration::IQNIMVJAcceleration.

◆ splitCouplingData()

void precice::acceleration::BaseQNAcceleration::splitCouplingData ( const DataMap & cplData )

protectedvirtual

Splits up QN system vector back into the coupling data.

Definition at line 438 of file BaseQNAcceleration.cpp.

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◆ updateDifferenceMatrices()

void precice::acceleration::BaseQNAcceleration::updateDifferenceMatrices ( const DataMap & cplData )

protectedvirtual

Updates the V, W matrices (as well as the matrices for the secondary data)

: computes the current residual and stores it, computes the differences and

updateDifferenceMatrices()

updates the difference matrices F and C.

Reimplemented in precice::acceleration::IQNILSAcceleration, and precice::acceleration::IQNIMVJAcceleration.

Definition at line 180 of file BaseQNAcceleration.cpp.

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◆ writeInfo()

void precice::acceleration::BaseQNAcceleration::writeInfo	(	const std::string &	s,
		bool	allProcs = false )

protected

Wwrites info to the _infostream (also in parallel)

Definition at line 622 of file BaseQNAcceleration.cpp.

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Member Data Documentation

◆ _dataIDs

std::vector<int> precice::acceleration::BaseQNAcceleration::_dataIDs

protected

Data IDs of data to be involved in the IQN algorithm.

Definition at line 159 of file BaseQNAcceleration.hpp.

◆ _dimOffsets

std::vector<int> precice::acceleration::BaseQNAcceleration::_dimOffsets

protected

Stores the local dimensions, i.e., the offsets in _invJacobian for all processors.

Definition at line 231 of file BaseQNAcceleration.hpp.

◆ _filter

int precice::acceleration::BaseQNAcceleration::_filter

protected

filter method that is used to maintain good conditioning of the least-squares system Either of two types: QR1FILTER or QR2Filter

Definition at line 209 of file BaseQNAcceleration.hpp.

◆ _firstIteration

bool precice::acceleration::BaseQNAcceleration::_firstIteration = true

protected

Indicates the first iteration, where constant relaxation is used.

Definition at line 165 of file BaseQNAcceleration.hpp.

◆ _firstTimeWindow

bool precice::acceleration::BaseQNAcceleration::_firstTimeWindow = true

protected

Definition at line 170 of file BaseQNAcceleration.hpp.

◆ _forceInitialRelaxation

bool precice::acceleration::BaseQNAcceleration::_forceInitialRelaxation

protected

Definition at line 181 of file BaseQNAcceleration.hpp.

◆ _hasNodesOnInterface

bool precice::acceleration::BaseQNAcceleration::_hasNodesOnInterface = true

protected

Definition at line 175 of file BaseQNAcceleration.hpp.

◆ _infostream

std::fstream precice::acceleration::BaseQNAcceleration::_infostream

protected

Definition at line 235 of file BaseQNAcceleration.hpp.

◆ _infostringstream

std::ostringstream precice::acceleration::BaseQNAcceleration::_infostringstream

protected

write some debug/acceleration info to file

Definition at line 234 of file BaseQNAcceleration.hpp.

◆ _initialRelaxation

double precice::acceleration::BaseQNAcceleration::_initialRelaxation

protected

Constant relaxation factor used for first iteration.

Definition at line 150 of file BaseQNAcceleration.hpp.

◆ _log

logging::Logger precice::acceleration::BaseQNAcceleration::_log {"acceleration::BaseQNAcceleration"}

protected

Definition at line 144 of file BaseQNAcceleration.hpp.

◆ _matrixCols

std::deque<int> precice::acceleration::BaseQNAcceleration::_matrixCols

protected

Indices (of columns in W, V matrices) of 1st iterations of time windows.

When old time windows are reused (_timeWindowsReused > 0), the indices of the first iteration of each time window needs to be stored, such that, e.g., all iterations of the last time window, or one specific iteration that leads to a singular matrix in the QR decomposition can be removed and tracked.

Definition at line 226 of file BaseQNAcceleration.hpp.

◆ _matrixColsBackup

std::deque<int> precice::acceleration::BaseQNAcceleration::_matrixColsBackup

private

Definition at line 286 of file BaseQNAcceleration.hpp.

◆ _matrixV

Eigen::MatrixXd precice::acceleration::BaseQNAcceleration::_matrixV

protected

Stores residual deltas.

Definition at line 198 of file BaseQNAcceleration.hpp.

◆ _matrixVBackup

Eigen::MatrixXd precice::acceleration::BaseQNAcceleration::_matrixVBackup

private

backup of the V,W and matrixCols data structures. Needed for the skipping of initial relaxation, if previous time window converged within one iteration i.e., V and W are empty – in this case restore V and W with time window t-2.

Definition at line 284 of file BaseQNAcceleration.hpp.

◆ _matrixW

Eigen::MatrixXd precice::acceleration::BaseQNAcceleration::_matrixW

protected

Stores x tilde deltas, where x tilde are values computed by solvers.

Definition at line 201 of file BaseQNAcceleration.hpp.

◆ _matrixWBackup

Eigen::MatrixXd precice::acceleration::BaseQNAcceleration::_matrixWBackup

private

Definition at line 285 of file BaseQNAcceleration.hpp.

◆ _maxIterationsUsed

int precice::acceleration::BaseQNAcceleration::_maxIterationsUsed

protected

Maximum number of old data iterations kept.

Definition at line 153 of file BaseQNAcceleration.hpp.

◆ _nbDelCols

int precice::acceleration::BaseQNAcceleration::_nbDelCols = 0

private

Number of filtered out columns in this time window.

Definition at line 289 of file BaseQNAcceleration.hpp.

◆ _nbDropCols

int precice::acceleration::BaseQNAcceleration::_nbDropCols = 0

private

Number of dropped columns in this time window (old time window out of scope)

Definition at line 292 of file BaseQNAcceleration.hpp.

◆ _oldResiduals

Eigen::VectorXd precice::acceleration::BaseQNAcceleration::_oldResiduals

private

Difference between solver input and output from last time window.

Definition at line 278 of file BaseQNAcceleration.hpp.

◆ _oldValues

Eigen::VectorXd precice::acceleration::BaseQNAcceleration::_oldValues

private

Concatenation of all (old) coupling data involved in the QN system.

Definition at line 275 of file BaseQNAcceleration.hpp.

◆ _oldXTilde

Eigen::VectorXd precice::acceleration::BaseQNAcceleration::_oldXTilde

protected

Solver output from last iteration.

Definition at line 189 of file BaseQNAcceleration.hpp.

◆ _preconditioner

impl::PtrPreconditioner precice::acceleration::BaseQNAcceleration::_preconditioner

protected

Preconditioner for least-squares system if vectorial system is used.

Definition at line 147 of file BaseQNAcceleration.hpp.

◆ _qrV

impl::QRFactorization precice::acceleration::BaseQNAcceleration::_qrV

protected

Stores the current QR decomposition ov _matrixV, can be updated via deletion/insertion of columns.

Definition at line 204 of file BaseQNAcceleration.hpp.

◆ _resetLS

bool precice::acceleration::BaseQNAcceleration::_resetLS = false

protected

If true, the LS system has been modified (reset or recomputed) in such a way, that mere updating of matrices _Wtil, Q, R etc.. is not feasible any more and need to be recomputed.

Definition at line 186 of file BaseQNAcceleration.hpp.

◆ _residuals

Eigen::VectorXd precice::acceleration::BaseQNAcceleration::_residuals

protected

Current iteration residuals of IQN data. Temporary.

Definition at line 192 of file BaseQNAcceleration.hpp.

◆ _secondaryDataIDs

std::vector<int> precice::acceleration::BaseQNAcceleration::_secondaryDataIDs

protected

Data IDs of data not involved in IQN coefficient computation.

Definition at line 162 of file BaseQNAcceleration.hpp.

◆ _secondaryResiduals

std::map<int, Eigen::VectorXd> precice::acceleration::BaseQNAcceleration::_secondaryResiduals

protected

Current iteration residuals of secondary data.

Definition at line 195 of file BaseQNAcceleration.hpp.

◆ _singularityLimit

double precice::acceleration::BaseQNAcceleration::_singularityLimit

protected

Determines sensitivity when two matrix columns are considered equal.

When during the QR decomposition of the V matrix a pivot element smaller than the singularity limit is found, the matrix is considered to be singular and the corresponding (older) iteration is removed.

Definition at line 217 of file BaseQNAcceleration.hpp.

◆ _timeWindowsReused

int precice::acceleration::BaseQNAcceleration::_timeWindowsReused

protected

Maximum number of old time windows (with data values) kept.

Definition at line 156 of file BaseQNAcceleration.hpp.

◆ _values

Eigen::VectorXd precice::acceleration::BaseQNAcceleration::_values

private

Concatenation of all coupling data involved in the QN system.

Definition at line 272 of file BaseQNAcceleration.hpp.

◆ its

int precice::acceleration::BaseQNAcceleration::its = 0

protected

Definition at line 268 of file BaseQNAcceleration.hpp.

◆ tWindows

int precice::acceleration::BaseQNAcceleration::tWindows = 0

protected

Definition at line 268 of file BaseQNAcceleration.hpp.

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

src/acceleration/BaseQNAcceleration.hpp
src/acceleration/BaseQNAcceleration.cpp

Public Member Functions

Protected Member Functions

Protected Attributes

Private Attributes

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ BaseQNAcceleration()

◆ ~BaseQNAcceleration()

Member Function Documentation

◆ applyFilter()

◆ computeQNUpdate()

◆ computeUnderrelaxationSecondaryData()

◆ exportState()

◆ getDataIDs()

◆ getDeletedColumns()

◆ getDroppedColumns()

◆ getLSSystemCols()

◆ getLSSystemRows()

◆ importState()

◆ initialize()

@brief: Initializes all the needed variables and data

◆ iterationsConverged()

updates F and C according to the number of reused time windows

◆ performAcceleration()

@brief: performs one iteration of the quasi Newton acceleration.

◆ removeMatrixColumn()

@brief: removes a column from the least squares system, i. e., from the matrices F and C

◆ specializedIterationsConverged()

◆ splitCouplingData()

◆ updateDifferenceMatrices()

updates the difference matrices F and C.

◆ writeInfo()

Member Data Documentation

◆ _dataIDs

◆ _dimOffsets

◆ _filter

◆ _firstIteration

◆ _firstTimeWindow

◆ _forceInitialRelaxation

◆ _hasNodesOnInterface

◆ _infostream

◆ _infostringstream

◆ _initialRelaxation

◆ _log

◆ _matrixCols

◆ _matrixColsBackup

◆ _matrixV

◆ _matrixVBackup

◆ _matrixW

◆ _matrixWBackup

◆ _maxIterationsUsed

◆ _nbDelCols

◆ _nbDropCols

◆ _oldResiduals

◆ _oldValues

◆ _oldXTilde

◆ _preconditioner

◆ _qrV

◆ _resetLS

◆ _residuals

◆ _secondaryDataIDs

◆ _secondaryResiduals

◆ _singularityLimit

◆ _timeWindowsReused

◆ _values

◆ its

◆ tWindows