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 >:

Collaboration 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.

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

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

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

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)

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

ordinal_type	dimension () const
	Return dimension of basis.

virtual ordinal_type	size () const
	Return total size of basis.

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`.

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.

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

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`.

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

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.

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

virtual	~ProductBasis ()
	Destructor.

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

virtual	~OrthogPolyBasis ()
	Destructor.

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.

ordinal_type	p
	Total order of basis.

ordinal_type	d
	Total dimension of basis.

ordinal_type	sz
	Total size of basis.

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

value_type	sparse_tol
	Tolerance for computing sparse Cijk.

coeff_type	max_orders
	Maximum orders for each dimension.

coeff_set_type	basis_set
	Basis set.

coeff_map_type	basis_map
	Basis map.

Teuchos::Array< ordinal_type >	smolyak_coeffs
	Smolyak coefficients.

Teuchos::Array< value_type >	norms
	Norms.

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

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

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

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.

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

Member Function Documentation

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 >.

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 >.

Referenced by Stokhos::SmolyakPseudoSpectralOperator< ordinal_t, value_t, point_compare_type >::SmolyakPseudoSpectralOperator().

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 >.

Referenced by Stokhos::SmolyakPseudoSpectralOperator< ordinal_t, value_t, point_compare_type >::SmolyakPseudoSpectralOperator().

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 >.

Referenced by Stokhos::SmolyakPseudoSpectralOperator< ordinal_t, value_t, point_compare_type >::SmolyakPseudoSpectralOperator().

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 >.

Referenced by Stokhos::SmolyakPseudoSpectralOperator< ordinal_t, value_t, point_compare_type >::SmolyakPseudoSpectralOperator().

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 >.

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

Stokhos_SmolyakBasis.hpp
Stokhos_SmolyakBasisImp.hpp

Classes

Public Types

Public Member Functions

Protected Types

Protected Attributes

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 >

Constructor & Destructor Documentation

Member Function 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 >