sgpp::base::NakBsplineExtendedBasis< LT, IT > Class Template Reference

Extended Not-a-knot B-spline basis. More...

#include <NakBsplineExtendedBasis.hpp>

Inheritance diagram for sgpp::base::NakBsplineExtendedBasis< LT, IT >:

Public Member Functions
double	basisMean (LT l, IT i, size_t start, size_t stop, double offset, double hik, std::shared_ptr< sgpp::base::DataVector > quadCoordinates, std::shared_ptr< sgpp::base::DataVector > quadWeights, std::shared_ptr< sgpp::base::Distribution > pdf)
double	eval (LT l, IT i, double x) override
double	evalDx (LT l, IT i, double x) override
size_t	getDegree () const override
double	getIntegral (LT l, IT i) override
double	getMean (LT l, IT i, std::shared_ptr< sgpp::base::Distribution > pdf, std::shared_ptr< sgpp::base::DataVector > quadCoordinates, std::shared_ptr< sgpp::base::DataVector > quadWeights)
	Calculates the mean int b_i(x) rho(x) dx of a basis function b_i w.r.t.
	NakBsplineExtendedBasis ()
	Default constructor.
	NakBsplineExtendedBasis (size_t degree)
	Constructor.
	~NakBsplineExtendedBasis () override
	Destructor.
Public Member Functions inherited from sgpp::base::Basis< LT, IT >
virtual	~Basis ()
	Destructor.

Protected Attributes
NakBsplineBasis< LT, IT >	notAKnotBsplineBasis
	B-spline basis for B-spline evaluation.

Detailed Description

template<class LT, class IT>
class sgpp::base::NakBsplineExtendedBasis< LT, IT >

Extended Not-a-knot B-spline basis.

This basis is a nonboundary version for the not-a-knot Bspline basis which still can represent 1,x,...x^degree exact on [0,1]. This is achieved by adding the B-splines belonging to the left(0) and right boundary point (1) to (degree+1) inner B-splines with extension coefficients derived from SLE for the representation of 1,x,...x^degree.

Extension of the not hierarchical basis: For each of the outer B-splines b_0 and b_{2^l} a set of (degree+1) closest inner B-splines I(j) is chosen. I.e. I(0) = {1,..,degree+1}, I(2^l) = {2^l-(degree+1),...,2^l-1}. Now the functions 1,x,...,x^degree are interpolated with the not a knot B-splines basis resulting in coefficients a^1_i,...,a^{x^degree}_i, i=0,...,2^l Then the coefficients a^1_j,...,a^{x^degree}_j are represented as a linear combination of the coefficients of I(j) for j in {0,2^l}. The coefficients e_ij of these linear combinations are the extension coefficients

We build a hierarchical basis from this by choosing Lagrange polynomials on levels 1 (and 2 for degree 3 and 5) and then simply using the extended basis funtions from the nonhierarchical basis.

(Note that one could also use the Lagrange polynomials in the sets I(0),I(8). This leads to another spanned space. Also on can only extend once on level 3, then 1,x,...,x^degree are already part of the spanned space and not extend on levels >=3 again leading to another spanned space. ToDo (rehmemk) It might be interesting to compare the various possibilities of extended not a knot B-spline bases (Note: coefficients were calculated with ExtensionBspline.m))

Constructor & Destructor Documentation

NakBsplineExtendedBasis() [1/2]

template<class LT , class IT >

sgpp::base::NakBsplineExtendedBasis< LT, IT >::NakBsplineExtendedBasis ( )

inline

Default constructor.

NakBsplineExtendedBasis() [2/2]

template<class LT , class IT >

sgpp::base::NakBsplineExtendedBasis< LT, IT >::NakBsplineExtendedBasis ( size_t  degree )

inlineexplicit

Constructor.

Parameters

degree

B-spline degree, must be odd (if it's even, degree - 1 is used)

References sgpp::base::NakBsplineExtendedBasis< LT, IT >::getDegree().

~NakBsplineExtendedBasis()

template<class LT , class IT >

sgpp::base::NakBsplineExtendedBasis< LT, IT >::~NakBsplineExtendedBasis ( )

inlineoverride

Destructor.

Member Function Documentation

basisMean()

template<class LT , class IT >

double sgpp::base::NakBsplineExtendedBasis< LT, IT >::basisMean ( LT  l, IT  i, size_t  start, size_t  stop, double  offset, double  hik, std::shared_ptr< sgpp::base::DataVector >  quadCoordinates, std::shared_ptr< sgpp::base::DataVector >  quadWeights, std::shared_ptr< sgpp::base::Distribution >  pdf  )

inline

References sgpp::base::NakBsplineExtendedBasis< LT, IT >::eval().

Referenced by sgpp::base::NakBsplineExtendedBasis< LT, IT >::getMean().

eval()

template<class LT , class IT >

double sgpp::base::NakBsplineExtendedBasis< LT, IT >::eval ( LT  l, IT  i, double  x  )

inlineoverridevirtual

Parameters

l	level of basis function
i	index of basis function
x	evaluation point

Returns

value of basis function

Implements sgpp::base::Basis< LT, IT >.

References sgpp::base::NakBsplineExtendedBasis< LT, IT >::getDegree(), and sgpp::base::NakBsplineExtendedBasis< LT, IT >::notAKnotBsplineBasis.

Referenced by sgpp::base::NakBsplineExtendedBasis< LT, IT >::basisMean(), python.uq.analysis.asgc.ASGCAnalysis.ASGCAnalysis::estimateDensity(), sgpp::base::OperationEvalNakBsplineExtendedNaive::eval(), sgpp::base::OperationEvalNakBsplineExtendedNaive::eval(), sgpp::base::OperationEvalGradientNakBsplineExtendedNaive::evalGradient(), sgpp::base::OperationEvalGradientNakBsplineExtendedNaive::evalGradient(), sgpp::base::OperationMultipleEvalNakBsplineExtendedNaive::mult(), and sgpp::base::OperationMultipleEvalNakBsplineExtendedNaive::multTranspose().

evalDx()

template<class LT , class IT >

double sgpp::base::NakBsplineExtendedBasis< LT, IT >::evalDx ( LT  l, IT  i, double  x  )

inlineoverridevirtual

Parameters

l	level of basis function
i	index of basis function
x	evaluation point

Returns

value of basis function

Implements sgpp::base::Basis< LT, IT >.

References sgpp::base::NakBsplineExtendedBasis< LT, IT >::getDegree(), and sgpp::base::NakBsplineExtendedBasis< LT, IT >::notAKnotBsplineBasis.

Referenced by sgpp::base::OperationEvalGradientNakBsplineExtendedNaive::evalGradient(), and sgpp::base::OperationEvalGradientNakBsplineExtendedNaive::evalGradient().

getDegree()

template<class LT , class IT >

size_t sgpp::base::NakBsplineExtendedBasis< LT, IT >::getDegree ( ) const

inlineoverridevirtual

Returns

B-spline degree

Implements sgpp::base::Basis< LT, IT >.

References sgpp::base::NakBsplineExtendedBasis< LT, IT >::notAKnotBsplineBasis.

Referenced by sgpp::base::NakBsplineExtendedBasis< LT, IT >::eval(), sgpp::base::NakBsplineExtendedBasis< LT, IT >::evalDx(), sgpp::base::NakBsplineExtendedBasis< LT, IT >::getIntegral(), sgpp::base::NakBsplineExtendedBasis< LT, IT >::getMean(), and sgpp::base::NakBsplineExtendedBasis< LT, IT >::NakBsplineExtendedBasis().

getIntegral()

template<class LT , class IT >

double sgpp::base::NakBsplineExtendedBasis< LT, IT >::getIntegral ( LT  l, IT  i  )

inlineoverridevirtual

Parameters

l	level of basis function
i	index of basis function

Returns

integral of basis function

Implements sgpp::base::Basis< LT, IT >.

References sgpp::base::NakBsplineExtendedBasis< LT, IT >::getDegree(), and sgpp::base::NakBsplineExtendedBasis< LT, IT >::notAKnotBsplineBasis.

Referenced by sgpp::base::OperationQuadratureNakBsplineExtended::doQuadrature().

getMean()

template<class LT , class IT >

double sgpp::base::NakBsplineExtendedBasis< LT, IT >::getMean ( LT  l, IT  i, std::shared_ptr< sgpp::base::Distribution >  pdf, std::shared_ptr< sgpp::base::DataVector >  quadCoordinates, std::shared_ptr< sgpp::base::DataVector >  quadWeights  )

inline

Calculates the mean int b_i(x) rho(x) dx of a basis function b_i w.r.t.

the probability density function rho

Parameters

l	level of basis function
i	index of basis function
pdf	probability density function
quadCoordinates	coordinates of the quadrature rule to be used
quadWeights	weights of the quadrature rule to be used

Returns

mean of basis function

References sgpp::base::NakBsplineExtendedBasis< LT, IT >::basisMean(), and sgpp::base::NakBsplineExtendedBasis< LT, IT >::getDegree().

Referenced by sgpp::base::OperationWeightedQuadratureNakBsplineExtended::doWeightedQuadrature().

Member Data Documentation

notAKnotBsplineBasis

template<class LT , class IT >

NakBsplineBasis<LT, IT> sgpp::base::NakBsplineExtendedBasis< LT, IT >::notAKnotBsplineBasis

protected

B-spline basis for B-spline evaluation.

Referenced by sgpp::base::NakBsplineExtendedBasis< LT, IT >::eval(), sgpp::base::NakBsplineExtendedBasis< LT, IT >::evalDx(), sgpp::base::NakBsplineExtendedBasis< LT, IT >::getDegree(), and sgpp::base::NakBsplineExtendedBasis< LT, IT >::getIntegral().

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

• base/src/sgpp/base/operation/hash/common/basis/NakBsplineExtendedBasis.hpp