Generates three-term recurrence using the Discretized Stieltjes procedure applied to a polynomial chaos expansion in another basis. More...

#include <Stokhos_StieltjesPCEBasis.hpp>

Inheritance diagram for Stokhos::StieltjesPCEBasis< ordinal_type, value_type >:

[legend]

Public Member Functions
	StieltjesPCEBasis (ordinal_type p, const Teuchos::RCP< const Stokhos::OrthogPolyApprox< ordinal_type, value_type > > &pce, const Teuchos::RCP< const Stokhos::Quadrature< ordinal_type, value_type > > &quad, bool use_pce_quad_points, bool normalize=false, bool project_integrals=false, const Teuchos::RCP< const Stokhos::Sparse3Tensor< ordinal_type, value_type > > &Cijk=Teuchos::null)
	Constructor. More...

	~StieltjesPCEBasis ()
	Destructor. More...

void	transformCoeffsFromStieltjes (const value_type in, value_type out) const
	Map expansion coefficients from this basis to original. More...

Public Member Functions inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
virtual	~RecurrenceBasis ()
	Destructor. More...

virtual void	getRecurrenceCoefficients (Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &delta, Teuchos::Array< value_type > &gamma) const
	Return recurrence coefficients defined by above formula. More...

virtual void	evaluateBasesAndDerivatives (const value_type &point, Teuchos::Array< value_type > &vals, Teuchos::Array< value_type > &derivs) const
	Evaluate basis polynomials and their derivatives at given point `point`. More...

virtual void	setQuadZeroTol (value_type tol)
	Set tolerance for zero in quad point generation. More...

virtual ordinal_type	order () const
	Return order of basis (largest monomial degree ). More...

virtual ordinal_type	size () const
	Return total size of basis (given by order() + 1). More...

virtual const Teuchos::Array < value_type > &	norm_squared () const
	Return array storing norm-squared of each basis polynomial. More...

virtual const value_type &	norm_squared (ordinal_type i) const
	Return norm squared of basis polynomial `i`. More...

virtual Teuchos::RCP < Stokhos::Dense3Tensor < ordinal_type, value_type > >	computeTripleProductTensor () const
	Compute triple product tensor. More...

virtual Teuchos::RCP < Stokhos::Sparse3Tensor < ordinal_type, value_type > >	computeSparseTripleProductTensor (ordinal_type order) const
	Compute triple product tensor. More...

virtual Teuchos::RCP < Teuchos::SerialDenseMatrix < ordinal_type, value_type > >	computeDerivDoubleProductTensor () const
	Compute derivative double product tensor. More...

virtual void	evaluateBases (const value_type &point, Teuchos::Array< value_type > &basis_pts) const
	Evaluate each basis polynomial at given point `point`. More...

virtual value_type	evaluate (const value_type &point, ordinal_type order) const
	Evaluate basis polynomial given by order `order` at given point `point`. More...

virtual void	print (std::ostream &os) const
	Print basis to stream `os`. More...

virtual const std::string &	getName () const
	Return string name of basis. More...

virtual ordinal_type	quadDegreeOfExactness (ordinal_type n) const

virtual ordinal_type	coefficientGrowth (ordinal_type n) const
	Evaluate coefficient growth rule for Smolyak-type bases. More...

virtual ordinal_type	pointGrowth (ordinal_type n) const
	Evaluate point growth rule for Smolyak-type bases. More...

virtual LevelToOrderFnPtr	getSparseGridGrowthRule () const
	Get sparse grid level_to_order mapping function. More...

virtual void	setSparseGridGrowthRule (LevelToOrderFnPtr ptr)
	Set sparse grid rule. More...

Public Member Functions inherited from Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >
	OneDOrthogPolyBasis ()
	Default constructor. More...

virtual	~OneDOrthogPolyBasis ()
	Destructor. More...

virtual void	setSparseGridGrowthRule (LevelToOrderFnPtr ptr)=0
	Set sparse grid rule. More...

Protected Types
typedef Stokhos::Sparse3Tensor < ordinal_type, value_type >	Cijk_type
	Short-hand for triple product. More...

Protected Member Functions
void	stieltjes (ordinal_type nstart, ordinal_type nfinish, const Teuchos::Array< value_type > &weights, const Teuchos::Array< value_type > &points, Teuchos::Array< value_type > &a, Teuchos::Array< value_type > &b, Teuchos::Array< value_type > &nrm, Teuchos::Array< Teuchos::Array< value_type > > &phi_vals) const
	Compute 3-term recurrence using Stieljtes procedure. More...

void	integrateBasisSquared (ordinal_type k, const Teuchos::Array< value_type > &a, const Teuchos::Array< value_type > &b, const Teuchos::Array< value_type > &weights, const Teuchos::Array< value_type > &points, Teuchos::Array< Teuchos::Array< value_type > > &phi_vals, value_type &val1, value_type &val2) const
	Compute $\int\psi^2_k(t) d\lambda(t)$ and $\int t\psi^2_k(t) d\lambda(t)$ . More...

void	integrateBasisSquaredProj (ordinal_type k, const Teuchos::Array< value_type > &a, const Teuchos::Array< value_type > &b, const Teuchos::Array< value_type > &weights, const Teuchos::Array< value_type > &points, Teuchos::Array< Teuchos::Array< value_type > > &phi_vals, value_type &val1, value_type &val2) const
	Compute $\int\psi^2_k(t) d\lambda(t)$ and $\int t\psi^2_k(t) d\lambda(t)$ by projecting onto original PCE basis. More...

void	evaluateRecurrence (ordinal_type k, const Teuchos::Array< value_type > &a, const Teuchos::Array< value_type > &b, const Teuchos::Array< value_type > &points, Teuchos::Array< Teuchos::Array< value_type > > &values) const
	Evaluate polynomials via 3-term recurrence. More...

	StieltjesPCEBasis (ordinal_type p, const StieltjesPCEBasis &basis)
	Copy constructor with specified order. More...

Protected Member Functions inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
	RecurrenceBasis (const std::string &name, ordinal_type p, bool normalize, GrowthPolicy growth=SLOW_GROWTH)
	Constructor to be called by derived classes. More...

	RecurrenceBasis (ordinal_type p, const RecurrenceBasis &basis)
	Copy constructor with specified order. More...

void	normalizeRecurrenceCoefficients (Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &delta, Teuchos::Array< value_type > &gamma) const
	Normalize coefficients. More...

Protected Attributes
Teuchos::RCP< const Stokhos::OrthogPolyApprox < ordinal_type, value_type > >	pce
	PC expansion. More...

Teuchos::RCP< const Stokhos::Quadrature < ordinal_type, value_type > >	quad
	Quadrature object. More...

const Teuchos::Array < value_type > &	pce_weights
	PCE quadrature weights. More...

const Teuchos::Array < Teuchos::Array< value_type > > &	basis_values
	Values of PCE basis functions at quadrature points. More...

Teuchos::Array< value_type >	pce_vals
	Values of PCE at quadrature points. More...

Teuchos::Array< Teuchos::Array < value_type > >	phi_vals
	Values of generated polynomials at PCE quadrature points. More...

bool	use_pce_quad_points
	Use underlying pce's quadrature data. More...

Teuchos::SerialDenseMatrix < ordinal_type, value_type >	fromStieltjesMat
	Matrix mapping coefficients in Stieltjes basis back to original basis. More...

bool	project_integrals
	Project Stieltjes integrals to original PCE basis. More...

Teuchos::RCP< const Stokhos::OrthogPolyBasis < ordinal_type, value_type > >	basis
	PCE basis (needed for integral projection method) More...

Teuchos::RCP< const Cijk_type >	Cijk
	Triple product tensor (needed for integral projection method) More...

Teuchos::Array< value_type >	phi_pce_coeffs
	Array store PC expansion of generated orthogonal polynomials. More...

Protected Attributes inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
std::string	name
	Name of basis. More...

ordinal_type	p
	Order of basis. More...

bool	normalize
	Normalize basis. More...

GrowthPolicy	growth
	Smolyak growth policy. More...

value_type	quad_zero_tol
	Tolerance for quadrature points near zero. More...

LevelToOrderFnPtr	sparse_grid_growth_rule
	Sparse grid growth rule (as determined by Pecos) More...

Teuchos::Array< value_type >	alpha
	Recurrence $\alpha$ coefficients. More...

Teuchos::Array< value_type >	beta
	Recurrence $\beta$ coefficients. More...

Teuchos::Array< value_type >	delta
	Recurrence $\delta$ coefficients. More...

Teuchos::Array< value_type >	gamma
	Recurrence $\gamma$ coefficients. More...

Teuchos::Array< value_type >	norms
	Norms. More...

Private Member Functions
	StieltjesPCEBasis (const StieltjesPCEBasis &)

StieltjesPCEBasis &	operator= (const StieltjesPCEBasis &b)

Implementation of Stokhos::OneDOrthogPolyBasis methods
virtual void	getQuadPoints (ordinal_type quad_order, Teuchos::Array< value_type > &points, Teuchos::Array< value_type > &weights, Teuchos::Array< Teuchos::Array< value_type > > &values) const
	Get Gauss quadrature points, weights, and values of basis at points. More...

virtual Teuchos::RCP < OneDOrthogPolyBasis < ordinal_type, value_type > >	cloneWithOrder (ordinal_type p) const
	Clone this object with the option of building a higher order basis. More...

Implementation of Stokhos::RecurrenceBasis methods
virtual bool	computeRecurrenceCoefficients (ordinal_type n, Teuchos::Array< value_type > &alpha, Teuchos::Array< value_type > &beta, Teuchos::Array< value_type > &delta, Teuchos::Array< value_type > &gamma) const
	Compute recurrence coefficients. More...

virtual void	setup ()
	Setup basis after computing recurrence coefficients. More...

Additional Inherited Members
Public Types inherited from Stokhos::RecurrenceBasis< ordinal_type, value_type >
typedef OneDOrthogPolyBasis < ordinal_type, value_type > ::LevelToOrderFnPtr	LevelToOrderFnPtr
	Function pointer needed for level_to_order mappings. More...

Public Types inherited from Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >
typedef int(*	LevelToOrderFnPtr )(int level, int growth)
	Function pointer needed for level_to_order mappings. More...

Detailed Description

template<typename ordinal_type, typename value_type>
class Stokhos::StieltjesPCEBasis< ordinal_type, value_type >

Generates three-term recurrence using the Discretized Stieltjes procedure applied to a polynomial chaos expansion in another basis.

Definition at line 30 of file Stokhos_StieltjesPCEBasis.hpp.

Member Typedef Documentation

template<typename ordinal_type, typename value_type>

typedef Stokhos::Sparse3Tensor<ordinal_type, value_type> Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::Cijk_type

protected

Short-hand for triple product.

Definition at line 190 of file Stokhos_StieltjesPCEBasis.hpp.

Constructor & Destructor Documentation

template<typename ordinal_type, typename value_type>

Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::StieltjesPCEBasis	(	ordinal_type	p,
		const Teuchos::RCP< const Stokhos::OrthogPolyApprox< ordinal_type, value_type > > &	pce,
		const Teuchos::RCP< const Stokhos::Quadrature< ordinal_type, value_type > > &	quad,
		bool	use_pce_quad_points,
		bool	normalize = `false`,
		bool	project_integrals = `false`,
		const Teuchos::RCP< const Stokhos::Sparse3Tensor< ordinal_type, value_type > > &	Cijk = `Teuchos::null`
	)

Constructor.

Parameters

p	order of the basis
pce	polynomial chaos expansion defining new density function
quad	quadrature data for basis of PC expansion
use_pce_quad_points	whether to use quad to define quadrature points for the new basis, or whether to use the Golub-Welsch system.
normalize	whether polynomials should be given unit norm

Definition at line 16 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::~StieltjesPCEBasis ( )

Destructor.

Definition at line 44 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type, typename value_type>

Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::StieltjesPCEBasis	(	ordinal_type	p,
		const StieltjesPCEBasis< ordinal_type, value_type > &	basis
	)

protected

Copy constructor with specified order.

Definition at line 318 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type, typename value_type>

Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::StieltjesPCEBasis ( const StieltjesPCEBasis< ordinal_type, value_type > & )

private

Member Function Documentation

template<typename ordinal_type, typename value_type>

void Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::getQuadPoints	(	ordinal_type	quad_order,
		Teuchos::Array< value_type > &	points,
		Teuchos::Array< value_type > &	weights,
		Teuchos::Array< Teuchos::Array< value_type > > &	values
	)		const

virtual

Get Gauss quadrature points, weights, and values of basis at points.

Reimplemented from Stokhos::RecurrenceBasis< ordinal_type, value_type >.

Definition at line 51 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type, typename value_type >

Teuchos::RCP< Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type > > Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::cloneWithOrder ( ordinal_type p ) const

virtual

Clone this object with the option of building a higher order basis.

This method is following the Prototype pattern (see Design Pattern's textbook). The slight variation is that it allows the order of the polynomial to be modified, otherwise an exact copy is formed. The use case for this is creating basis functions for column indices in a spatially varying adaptive refinement context.

Implements Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type >.

Definition at line 311 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type>

void Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::transformCoeffsFromStieltjes	(	const value_type *	in,
		value_type *	out
	)		const

Map expansion coefficients from this basis to original.

Definition at line 300 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type, typename value_type>

bool Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::computeRecurrenceCoefficients	(	ordinal_type	n,
		Teuchos::Array< value_type > &	alpha,
		Teuchos::Array< value_type > &	beta,
		Teuchos::Array< value_type > &	delta,
		Teuchos::Array< value_type > &	gamma
	)		const

protectedvirtual

Compute recurrence coefficients.

Implements Stokhos::RecurrenceBasis< ordinal_type, value_type >.

Definition at line 99 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

void Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::setup ( )

protectedvirtual

Setup basis after computing recurrence coefficients.

Reimplemented from Stokhos::RecurrenceBasis< ordinal_type, value_type >.

Definition at line 126 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type, typename value_type>

void Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::stieltjes	(	ordinal_type	nstart,
		ordinal_type	nfinish,
		const Teuchos::Array< value_type > &	weights,
		const Teuchos::Array< value_type > &	points,
		Teuchos::Array< value_type > &	a,
		Teuchos::Array< value_type > &	b,
		Teuchos::Array< value_type > &	nrm,
		Teuchos::Array< Teuchos::Array< value_type > > &	phi_vals
	)		const

protected

Compute 3-term recurrence using Stieljtes procedure.

Definition at line 159 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type, typename value_type>

void Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::integrateBasisSquared	(	ordinal_type	k,
		const Teuchos::Array< value_type > &	a,
		const Teuchos::Array< value_type > &	b,
		const Teuchos::Array< value_type > &	weights,
		const Teuchos::Array< value_type > &	points,
		Teuchos::Array< Teuchos::Array< value_type > > &	phi_vals,
		value_type &	val1,
		value_type &	val2
	)		const

protected

Compute $\int\psi^2_k(t) d\lambda(t)$ and $\int t\psi^2_k(t) d\lambda(t)$ .

Definition at line 208 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type, typename value_type>

void Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::integrateBasisSquaredProj	(	ordinal_type	k,
		const Teuchos::Array< value_type > &	a,
		const Teuchos::Array< value_type > &	b,
		const Teuchos::Array< value_type > &	weights,
		const Teuchos::Array< value_type > &	points,
		Teuchos::Array< Teuchos::Array< value_type > > &	phi_vals,
		value_type &	val1,
		value_type &	val2
	)		const

protected

Compute $\int\psi^2_k(t) d\lambda(t)$ and $\int t\psi^2_k(t) d\lambda(t)$ by projecting onto original PCE basis.

Definition at line 251 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type, typename value_type>

void Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::evaluateRecurrence	(	ordinal_type	k,
		const Teuchos::Array< value_type > &	a,
		const Teuchos::Array< value_type > &	b,
		const Teuchos::Array< value_type > &	points,
		Teuchos::Array< Teuchos::Array< value_type > > &	values
	)		const

protected

Evaluate polynomials via 3-term recurrence.

Definition at line 229 of file Stokhos_StieltjesPCEBasisImp.hpp.

template<typename ordinal_type, typename value_type>

StieltjesPCEBasis& Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::operator= ( const StieltjesPCEBasis< ordinal_type, value_type > & b )

private

Member Data Documentation

template<typename ordinal_type, typename value_type>

Teuchos::RCP<const Stokhos::OrthogPolyApprox<ordinal_type, value_type> > Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::pce

protected

PC expansion.

Definition at line 160 of file Stokhos_StieltjesPCEBasis.hpp.

template<typename ordinal_type, typename value_type>

Teuchos::RCP<const Stokhos::Quadrature<ordinal_type, value_type> > Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::quad

protected

Quadrature object.

Definition at line 163 of file Stokhos_StieltjesPCEBasis.hpp.

template<typename ordinal_type, typename value_type>

const Teuchos::Array<value_type>& Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::pce_weights

protected

PCE quadrature weights.

Definition at line 166 of file Stokhos_StieltjesPCEBasis.hpp.

template<typename ordinal_type, typename value_type>

const Teuchos::Array< Teuchos::Array<value_type> >& Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::basis_values

protected

Values of PCE basis functions at quadrature points.

Definition at line 169 of file Stokhos_StieltjesPCEBasis.hpp.

template<typename ordinal_type, typename value_type>

Teuchos::Array<value_type> Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::pce_vals

mutableprotected

Values of PCE at quadrature points.

Definition at line 172 of file Stokhos_StieltjesPCEBasis.hpp.

template<typename ordinal_type, typename value_type>

Teuchos::Array< Teuchos::Array<value_type> > Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::phi_vals

mutableprotected

Values of generated polynomials at PCE quadrature points.

Definition at line 175 of file Stokhos_StieltjesPCEBasis.hpp.

template<typename ordinal_type, typename value_type>

bool Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::use_pce_quad_points

protected

Use underlying pce's quadrature data.

Definition at line 178 of file Stokhos_StieltjesPCEBasis.hpp.

template<typename ordinal_type, typename value_type>

Teuchos::SerialDenseMatrix<ordinal_type, value_type> Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::fromStieltjesMat

protected

Matrix mapping coefficients in Stieltjes basis back to original basis.

Definition at line 181 of file Stokhos_StieltjesPCEBasis.hpp.

template<typename ordinal_type, typename value_type>

bool Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::project_integrals

protected

Project Stieltjes integrals to original PCE basis.

Definition at line 184 of file Stokhos_StieltjesPCEBasis.hpp.

template<typename ordinal_type, typename value_type>

Teuchos::RCP<const Stokhos::OrthogPolyBasis<ordinal_type, value_type> > Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::basis

protected

PCE basis (needed for integral projection method)

Definition at line 187 of file Stokhos_StieltjesPCEBasis.hpp.

template<typename ordinal_type, typename value_type>

Teuchos::RCP<const Cijk_type> Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::Cijk

protected

Triple product tensor (needed for integral projection method)

Definition at line 193 of file Stokhos_StieltjesPCEBasis.hpp.

template<typename ordinal_type, typename value_type>

Teuchos::Array<value_type> Stokhos::StieltjesPCEBasis< ordinal_type, value_type >::phi_pce_coeffs

mutableprotected

Array store PC expansion of generated orthogonal polynomials.

Definition at line 196 of file Stokhos_StieltjesPCEBasis.hpp.

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

Public Member Functions

Protected Types

Protected Member Functions

Protected Attributes

Private Member Functions

Implementation of Stokhos::OneDOrthogPolyBasis methods

Implementation of Stokhos::RecurrenceBasis methods

Additional Inherited Members

Detailed Description

template<typename ordinal_type, typename value_type> class Stokhos::StieltjesPCEBasis< ordinal_type, value_type >

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<typename ordinal_type, typename value_type>
class Stokhos::StieltjesPCEBasis< ordinal_type, value_type >