Multivariate orthogonal polynomial basis generated from a Smolyak sparse grid. More...

#include <Stokhos_SmolyakBasis.hpp>

Inheritance diagram for Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >:

Classes
struct	SmolyakPredicate
	Predicate functor for building sparse triple products. More...

Public Types
typedef MultiIndex< ordinal_type >	coeff_type

typedef TensorProductBasis < ordinal_type, value_type, LexographicLess< coeff_type > >	tensor_product_basis_type

Public Member Functions
template<typename index_set_type >
	SmolyakBasis (const Teuchos::Array< Teuchos::RCP< const OneDOrthogPolyBasis< ordinal_type, value_type > > > &bases, const index_set_type &index_set, const value_type &sparse_tol=1.0e-12, const coeff_compare_type &coeff_compare=coeff_compare_type())
	Constructor. More...

virtual	~SmolyakBasis ()
	Destructor. More...

ordinal_type	getNumSmolyakTerms () const
	Return number of terms in Smolyak formula. More...

Teuchos::RCP< const tensor_product_basis_type >	getTensorProductBasis (ordinal_type i) const
	Return ith tensor product basis. More...

ordinal_type	getSmolyakCoefficient (ordinal_type i) const
	Return ith smolyak coefficient. More...

template<typename index_set_type >
	SmolyakBasis (const Teuchos::Array< Teuchos::RCP< const OneDOrthogPolyBasis< ordinal_type, value_type > > > &bases_, const index_set_type &index_set, const value_type &sparse_tol_, const ordering_type &coeff_compare)

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...

Protected Types
typedef std::map< coeff_type, ordinal_type, coeff_compare_type >	coeff_set_type

typedef Teuchos::Array < coeff_type >	coeff_map_type

typedef MultiIndex< ordinal_type >	multiindex_type

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

ordinal_type	p
	Total order of basis. More...

ordinal_type	d
	Total dimension of basis. More...

ordinal_type	sz
	Total size of basis. More...

Teuchos::Array< Teuchos::RCP < const OneDOrthogPolyBasis < ordinal_type, value_type > > >	bases
	Array of bases. More...

value_type	sparse_tol
	Tolerance for computing sparse Cijk. More...

coeff_type	max_orders
	Maximum orders for each dimension. More...

coeff_set_type	basis_set
	Basis set. More...

coeff_map_type	basis_map
	Basis map. More...

Teuchos::Array< ordinal_type >	smolyak_coeffs
	Smolyak coefficients. More...

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

Teuchos::Array< Teuchos::Array < value_type > >	basis_eval_tmp
	Temporary array used in basis evaluation. More...

Teuchos::Array< Teuchos::RCP < tensor_product_basis_type > >	tp_bases
	Tensor product bases comprising Smolyak set. More...

SmolyakPredicate < TensorProductPredicate < ordinal_type > >	sm_pred
	Predicate for building sparse triple products. More...

Private Member Functions
	SmolyakBasis (const SmolyakBasis &)

SmolyakBasis &	operator= (const SmolyakBasis &b)

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...

Detailed Description

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>
class Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >

Multivariate orthogonal polynomial basis generated from a Smolyak sparse grid.

Definition at line 62 of file Stokhos_SmolyakBasis.hpp.

Member Typedef Documentation

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

typedef MultiIndex<ordinal_type> Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::coeff_type

Definition at line 172 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

typedef TensorProductBasis<ordinal_type,value_type, LexographicLess<coeff_type> > Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::tensor_product_basis_type

Definition at line 175 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

typedef std::map<coeff_type,ordinal_type,coeff_compare_type> Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::coeff_set_type

protected

Definition at line 199 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

typedef Teuchos::Array<coeff_type> Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::coeff_map_type

protected

Definition at line 200 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

typedef MultiIndex<ordinal_type> Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::multiindex_type

protected

Definition at line 201 of file Stokhos_SmolyakBasis.hpp.

Constructor & Destructor Documentation

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

template<typename index_set_type >

Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::SmolyakBasis	(	const Teuchos::Array< Teuchos::RCP< const OneDOrthogPolyBasis< ordinal_type, value_type > > > &	bases,
		const index_set_type &	index_set,
		const value_type &	sparse_tol = `1.0e-12`,
		const coeff_compare_type &	coeff_compare = `coeff_compare_type()`
	)

Constructor.

Parameters

bases	array of 1-D coordinate bases
sparse_tol	tolerance used to drop terms in sparse triple-product tensors

template<typename ordinal_type , typename value_type , typename ordering_type >

Stokhos::SmolyakBasis< ordinal_type, value_type, ordering_type >::~SmolyakBasis ( )

virtual

Destructor.

Definition at line 213 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::SmolyakBasis ( const SmolyakBasis< ordinal_type, value_type, coeff_compare_type > & )

private

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

template<typename index_set_type >

Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::SmolyakBasis	(	const Teuchos::Array< Teuchos::RCP< const OneDOrthogPolyBasis< ordinal_type, value_type > > > &	bases_,
		const index_set_type &	index_set,
		const value_type &	sparse_tol_,
		const ordering_type &	coeff_compare
	)

Definition at line 48 of file Stokhos_SmolyakBasisImp.hpp.

Member Function Documentation

template<typename ordinal_type , typename value_type , typename ordering_type >

ordinal_type Stokhos::SmolyakBasis< ordinal_type, value_type, ordering_type >::order ( ) const

virtual

Return order of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 220 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

ordinal_type Stokhos::SmolyakBasis< ordinal_type, value_type, ordering_type >::dimension ( ) const

virtual

Return dimension of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 228 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

ordinal_type Stokhos::SmolyakBasis< ordinal_type, value_type, ordering_type >::size ( ) const

virtual

Return total size of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 236 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

const Teuchos::Array< value_type > & Stokhos::SmolyakBasis< ordinal_type, value_type, ordering_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 244 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

const value_type & Stokhos::SmolyakBasis< ordinal_type, value_type, ordering_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 252 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

Teuchos::RCP< Stokhos::Sparse3Tensor< ordinal_type, value_type > > Stokhos::SmolyakBasis< ordinal_type, value_type, ordering_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 260 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

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

virtual

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

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 273 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

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

virtual

Evaluate basis polynomial i at zero.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 292 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

void Stokhos::SmolyakBasis< ordinal_type, value_type, ordering_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 306 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

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

virtual

Print basis to stream os.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 324 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

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

virtual

Return string name of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

Definition at line 358 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

const Stokhos::MultiIndex< ordinal_type > & Stokhos::SmolyakBasis< ordinal_type, value_type, ordering_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 339 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

ordinal_type Stokhos::SmolyakBasis< ordinal_type, value_type, ordering_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 347 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

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

virtual

Return coordinate bases.

Array is of size dimension().

Implements Stokhos::ProductBasis< ordinal_type, value_type >.

Definition at line 366 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type , typename value_type , typename ordering_type >

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

virtual

Return maximum order allowable for each coordinate basis.

Implements Stokhos::ProductBasis< ordinal_type, value_type >.

Definition at line 374 of file Stokhos_SmolyakBasisImp.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

ordinal_type Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::getNumSmolyakTerms ( ) const

inline

Return number of terms in Smolyak formula.

Definition at line 178 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

Teuchos::RCP<const tensor_product_basis_type> Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::getTensorProductBasis ( ordinal_type i ) const

inline

Return ith tensor product basis.

Definition at line 182 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

ordinal_type Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::getSmolyakCoefficient ( ordinal_type i ) const

inline

Return ith smolyak coefficient.

Definition at line 185 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

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

private

Member Data Documentation

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

std::string Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::name

protected

Name of basis.

Definition at line 204 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

ordinal_type Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::p

protected

Total order of basis.

Definition at line 207 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

ordinal_type Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::d

protected

Total dimension of basis.

Definition at line 210 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

ordinal_type Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::sz

protected

Total size of basis.

Definition at line 213 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

Teuchos::Array< Teuchos::RCP<const OneDOrthogPolyBasis<ordinal_type, value_type> > > Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::bases

protected

Array of bases.

Definition at line 216 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

value_type Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::sparse_tol

protected

Tolerance for computing sparse Cijk.

Definition at line 219 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

coeff_type Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::max_orders

protected

Maximum orders for each dimension.

Definition at line 222 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

coeff_set_type Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::basis_set

protected

Basis set.

Definition at line 225 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

coeff_map_type Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::basis_map

protected

Basis map.

Definition at line 228 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

Teuchos::Array<ordinal_type> Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::smolyak_coeffs

protected

Smolyak coefficients.

Definition at line 231 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

Teuchos::Array<value_type> Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::norms

protected

Norms.

Definition at line 234 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

Teuchos::Array< Teuchos::Array<value_type> > Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::basis_eval_tmp

mutableprotected

Temporary array used in basis evaluation.

Definition at line 237 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

Teuchos::Array< Teuchos::RCP< tensor_product_basis_type > > Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::tp_bases

protected

Tensor product bases comprising Smolyak set.

Definition at line 240 of file Stokhos_SmolyakBasis.hpp.

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>

SmolyakPredicate< TensorProductPredicate<ordinal_type> > Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >::sm_pred

protected

Predicate for building sparse triple products.

Definition at line 258 of file Stokhos_SmolyakBasis.hpp.

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

Classes

Public Types

Public Member Functions

Protected Types

Protected Attributes

Private Member Functions

Implementation of Stokhos::OrthogPolyBasis methods

Implementation of Stokhos::ProductBasis methods

Detailed Description

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >> class Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >

Member Typedef Documentation

Constructor & Destructor Documentation

Member Function Documentation

Member Data Documentation

template<typename ordinal_type, typename value_type, typename coeff_compare_type = TotalOrderLess<MultiIndex<ordinal_type> >>
class Stokhos::SmolyakBasis< ordinal_type, value_type, coeff_compare_type >