Generate a basis from a given set of PCE expansions that is orthogonal with respect to the product measure induced by these expansions. More...

#include <Stokhos_ProductLanczosPCEBasis.hpp>

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

[legend]

Public Member Functions
	ProductLanczosPCEBasis (ordinal_type p, const Teuchos::Array< Stokhos::OrthogPolyApprox< ordinal_type, value_type > > &pce, const Teuchos::RCP< const Stokhos::Quadrature< ordinal_type, value_type > > &quad, const Teuchos::RCP< const Stokhos::Sparse3Tensor< ordinal_type, value_type > > &Cijk, const Teuchos::ParameterList &params=Teuchos::ParameterList())
	Constructor. More...

virtual	~ProductLanczosPCEBasis ()
	Destructor. More...

Public Member Functions inherited from Stokhos::ProductBasis< ordinal_type, value_type >
	ProductBasis ()
	Constructor. More...

virtual	~ProductBasis ()
	Destructor. More...

Public Member Functions inherited from Stokhos::OrthogPolyBasis< ordinal_type, value_type >
	OrthogPolyBasis ()
	Constructor. More...

virtual	~OrthogPolyBasis ()
	Destructor. More...

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

virtual	~ReducedPCEBasis ()
	Destructor. More...

Protected Types
typedef Teuchos::SerialDenseVector < ordinal_type, value_type >	SDV

typedef Teuchos::SerialDenseMatrix < ordinal_type, value_type >	SDM

Protected Member Functions
ordinal_type	isInvariant (const Stokhos::OrthogPolyApprox< ordinal_type, value_type > &pce) const

Protected Attributes
std::string	name
	Name of basis. More...

Teuchos::ParameterList	params
	Algorithm parameters. More...

Teuchos::RCP < Stokhos::CompletePolynomialBasis < ordinal_type, value_type > >	tensor_lanczos_basis
	Product Lanczos basis. More...

SDM	Phi
	Values of transformed basis at quadrature points. More...

SDM	A
	Transition matrix from reduced basis to original. More...

SDM	Ainv
	Projection matrix from original matrix to reduced. More...

Teuchos::RCP< const Stokhos::Quadrature < ordinal_type, value_type > >	reduced_quad
	Reduced quadrature object. More...

Stokhos::OrthogPolyApprox < ordinal_type, value_type >	tmp_pce
	Temporary pce used in invariant subspace calculations. More...

Private Member Functions
	ProductLanczosPCEBasis (const ProductLanczosPCEBasis &)

ProductLanczosPCEBasis &	operator= (const ProductLanczosPCEBasis &)

Implementation of Stokhos::OrthogPolyBasis methods
ordinal_type	order () const
	Return order of basis. More...

ordinal_type	dimension () const
	Return dimension of basis. More...

virtual ordinal_type	size () const
	Return total size of basis. 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::Sparse3Tensor < ordinal_type, value_type > >	computeTripleProductTensor () const
	Compute triple product tensor. More...

virtual Teuchos::RCP < Stokhos::Sparse3Tensor < ordinal_type, value_type > >	computeLinearTripleProductTensor () const
	Compute linear triple product tensor where k = 0,1,..,d. More...

virtual value_type	evaluateZero (ordinal_type i) const
	Evaluate basis polynomial `i` at zero. More...

virtual void	evaluateBases (const Teuchos::ArrayView< const value_type > &point, Teuchos::Array< value_type > &basis_vals) const
	Evaluate basis polynomials 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...

Implementation of Stokhos::ProductBasis methods
virtual const MultiIndex < ordinal_type > &	term (ordinal_type i) const
	Get orders of each coordinate polynomial given an index `i`. More...

virtual ordinal_type	index (const MultiIndex< ordinal_type > &term) const
	Get index of the multivariate polynomial given orders of each coordinate. More...

Teuchos::Array< Teuchos::RCP < const OneDOrthogPolyBasis < ordinal_type, value_type > > >	getCoordinateBases () const
	Return coordinate bases. More...

virtual MultiIndex< ordinal_type >	getMaxOrders () const
	Return maximum order allowable for each coordinate basis. More...

Implementation of Stokhos::ReducedPCEBasis methods
virtual void	transformToOriginalBasis (const value_type in, value_type out, ordinal_type ncol=1, bool transpose=false) const
	Transform coefficients to original basis from this basis. More...

virtual void	transformFromOriginalBasis (const value_type in, value_type out, ordinal_type ncol=1, bool transpose=false) const
	Transform coefficients from original basis to this basis. More...

virtual Teuchos::RCP< const Stokhos::Quadrature < ordinal_type, value_type > >	getReducedQuadrature () const
	Get reduced quadrature object. More...

Detailed Description

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

Generate a basis from a given set of PCE expansions that is orthogonal with respect to the product measure induced by these expansions.

Given the PCE expansions, first build a an orthogonal basis for each compnent, then form the multivariate basis by a total-order tensor product. The resulting basis is not necessarily orthogonal with respect to the full measure.

Definition at line 71 of file Stokhos_ProductLanczosPCEBasis.hpp.

Member Typedef Documentation

template<typename ordinal_type , typename value_type >

typedef Teuchos::SerialDenseVector<ordinal_type,value_type> Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::SDV

protected

Definition at line 223 of file Stokhos_ProductLanczosPCEBasis.hpp.

template<typename ordinal_type , typename value_type >

typedef Teuchos::SerialDenseMatrix<ordinal_type,value_type> Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::SDM

protected

Definition at line 224 of file Stokhos_ProductLanczosPCEBasis.hpp.

Constructor & Destructor Documentation

template<typename ordinal_type , typename value_type >

Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::ProductLanczosPCEBasis	(	ordinal_type	p,
		const Teuchos::Array< Stokhos::OrthogPolyApprox< ordinal_type, value_type > > &	pce,
		const Teuchos::RCP< const Stokhos::Quadrature< ordinal_type, value_type > > &	quad,
		const Teuchos::RCP< const Stokhos::Sparse3Tensor< ordinal_type, value_type > > &	Cijk,
		const Teuchos::ParameterList &	params = `Teuchos::ParameterList()`
	)

Constructor.

Parameters

p	order of the basis
pce	polynomial chaos expansions defining new measure
Cijk	sparse triple product tensor for basis defining pce

Definition at line 50 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

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

virtual

Destructor.

Definition at line 242 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

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

private

Member Function Documentation

template<typename ordinal_type , typename value_type >

ordinal_type Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::order ( ) const

virtual

Return order of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 249 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

ordinal_type Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::dimension ( ) const

virtual

Return dimension of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 257 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

ordinal_type Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::size ( ) const

virtual

Return total size of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 265 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

const Teuchos::Array< value_type > & Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::norm_squared ( ) const

virtual

Return array storing norm-squared of each basis polynomial.

Entry $l$ of returned array is given by $\langle\Psi_l^2\rangle$ for $l=0,\dots,P$ where $P$ is size()-1.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 273 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

const value_type & Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::norm_squared ( ordinal_type i ) const

virtual

Return norm squared of basis polynomial i.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 281 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

Teuchos::RCP< Stokhos::Sparse3Tensor< ordinal_type, value_type > > Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::computeTripleProductTensor ( ) const

virtual

Compute triple product tensor.

The $(i,j,k)$ entry of the tensor $C_{ijk}$ is given by $C_{ijk} = \langle\Psi_i\Psi_j\Psi_k\rangle$ where $\Psi_l$ represents basis polynomial $l$ and $i,j,k=0,\dots,P$ where $P$ is size()-1.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 289 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

Teuchos::RCP< Stokhos::Sparse3Tensor< ordinal_type, value_type > > Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::computeLinearTripleProductTensor ( ) const

virtual

Compute linear triple product tensor where k = 0,1,..,d.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 301 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

value_type Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::evaluateZero ( ordinal_type i ) const

virtual

Evaluate basis polynomial i at zero.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 313 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

void Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::evaluateBases	(	const Teuchos::ArrayView< const value_type > &	point,
		Teuchos::Array< value_type > &	basis_vals
	)		const

virtual

Evaluate basis polynomials at given point point.

Size of returned array is given by size(), and coefficients are ordered from order 0 up to size size()-1.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 321 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

void Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::print ( std::ostream & os ) const

virtual

Print basis to stream os.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 330 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

const std::string & Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::getName ( ) const

virtual

Return string name of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 338 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

const Stokhos::MultiIndex< ordinal_type > & Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::term ( ordinal_type i ) const

virtual

Get orders of each coordinate polynomial given an index i.

The returned array is of size $d$ , where $d$ is the dimension of the basis, and entry $l$ is given by $i_l$ where $\Psi_i(x) = \psi_{i_1}(x_1)\dots\psi_{i_d}(x_d)$ .

Implements Stokhos::ProductBasis< ordinal_type, value_type >.

Definition at line 346 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

ordinal_type Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::index ( const MultiIndex< ordinal_type > & term ) const

virtual

Get index of the multivariate polynomial given orders of each coordinate.

Given the array term storing $i_1,\dots,\i_d$ , returns the index $i$ such that $\Psi_i(x) = \psi_{i_1}(x_1)\dots\psi_{i_d}(x_d)$ .

Implements Stokhos::ProductBasis< ordinal_type, value_type >.

Definition at line 354 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

Teuchos::Array< Teuchos::RCP< const Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type > > > Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::getCoordinateBases ( ) const

virtual

Return coordinate bases.

Array is of size dimension().

Implements Stokhos::ProductBasis< ordinal_type, value_type >.

Definition at line 362 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

Stokhos::MultiIndex< ordinal_type > Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::getMaxOrders ( ) const

virtual

Return maximum order allowable for each coordinate basis.

Implements Stokhos::ProductBasis< ordinal_type, value_type >.

Definition at line 370 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

void Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::transformToOriginalBasis	(	const value_type *	in,
		value_type *	out,
		ordinal_type	ncol = `1`,
		bool	transpose = `false`
	)		const

virtual

Transform coefficients to original basis from this basis.

Implements Stokhos::ReducedPCEBasis< ordinal_type, value_type >.

Definition at line 378 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

void Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::transformFromOriginalBasis	(	const value_type *	in,
		value_type *	out,
		ordinal_type	ncol = `1`,
		bool	transpose = `false`
	)		const

virtual

Transform coefficients from original basis to this basis.

Implements Stokhos::ReducedPCEBasis< ordinal_type, value_type >.

Definition at line 402 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

Teuchos::RCP< const Stokhos::Quadrature< ordinal_type, value_type > > Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::getReducedQuadrature ( ) const

virtual

Get reduced quadrature object.

Implements Stokhos::ReducedPCEBasis< ordinal_type, value_type >.

Definition at line 426 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

ordinal_type Stokhos::ProductLanczosPCEBasis< ordinal_type, value_type >::isInvariant ( const Stokhos::OrthogPolyApprox< ordinal_type, value_type > & pce ) const

protected

Definition at line 434 of file Stokhos_ProductLanczosPCEBasisImp.hpp.

template<typename ordinal_type , typename value_type >

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

private