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Stokhos::CompletePolynomialBasis< ordinal_type, value_type > Class Template Reference

Multivariate orthogonal polynomial basis generated from a total-order complete-polynomial tensor product of univariate polynomials. More...

#include <Stokhos_CompletePolynomialBasis.hpp>

Inheritance diagram for Stokhos::CompletePolynomialBasis< ordinal_type, value_type >:
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Collaboration diagram for Stokhos::CompletePolynomialBasis< ordinal_type, value_type >:
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Public Member Functions

 CompletePolynomialBasis (const Teuchos::Array< Teuchos::RCP< const OneDOrthogPolyBasis< ordinal_type, value_type > > > &bases, const value_type &sparse_tol=1.0e-12, bool use_old_cijk_alg=false, const Teuchos::RCP< Teuchos::Array< value_type > > &deriv_coeffs=Teuchos::null)
 Constructor.
virtual ~CompletePolynomialBasis ()
 Destructor.
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.
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.
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.
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.
virtual ordinal_type index (const MultiIndex< ordinal_type > &term) const
 Get index of the multivariate polynomial given orders of each coordinate.
Teuchos::Array< Teuchos::RCP< const OneDOrthogPolyBasis< ordinal_type, value_type > > > getCoordinateBases () const
 Return coordinate bases.
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.
Public Member Functions inherited from Stokhos::DerivBasis< ordinal_type, value_type >
 DerivBasis ()
 Constructor.
virtual ~DerivBasis ()
 Destructor.

Implementation of Stokhos::DerivBasis methods

typedef Stokhos::CompletePolynomialBasisUtils< ordinal_type, value_type > CPBUtils
typedef Stokhos::Sparse3Tensor< ordinal_type, value_type > Cijk_type
 Short-hand for Cijk.
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.
Teuchos::Array< ordinal_type > basis_orders
 Array storing order of each basis.
value_type sparse_tol
 Tolerance for computing sparse Cijk.
bool use_old_cijk_alg
 Use old algorithm for computing Cijk.
Teuchos::RCP< Teuchos::Array< value_type > > deriv_coeffs
 Coefficients for derivative.
Teuchos::Array< value_type > norms
 Norms.
Teuchos::Array< MultiIndex< ordinal_type > > terms
 2-D array of basis terms
Teuchos::Array< ordinal_type > num_terms
 Number of terms up to each order.
Teuchos::Array< Teuchos::Array< value_type > > basis_eval_tmp
 Temporary array used in basis evaluation.
virtual Teuchos::RCP< Stokhos::Dense3Tensor< ordinal_type, value_type > > computeDerivTripleProductTensor (const Teuchos::RCP< const Teuchos::SerialDenseMatrix< ordinal_type, value_type > > &Bij, const Teuchos::RCP< const Stokhos::Sparse3Tensor< ordinal_type, value_type > > &Cijk) const
 Compute triple product tensor $D_{ijk} = \langle\Psi_i\Psi_j D_v\Psi_k\rangle$ where $D_v\Psi_k$ represents the derivative of $\Psi_k$ in the direction $v$.
virtual Teuchos::RCP< Teuchos::SerialDenseMatrix< ordinal_type, value_type > > computeDerivDoubleProductTensor () const
 Compute double product tensor $B_{ij} = \langle \Psi_i D_v\Psi_j\rangle$ where $D_v\Psi_j$ represents the derivative of $\Psi_j$ in the direction $v$.
virtual Teuchos::RCP< Stokhos::Sparse3Tensor< ordinal_type, value_type > > computeTripleProductTensorOld (ordinal_type order) const
 Compute triple product tensor using old algorithm.
virtual Teuchos::RCP< Stokhos::Sparse3Tensor< ordinal_type, value_type > > computeTripleProductTensorNew (ordinal_type order) const
 Compute triple product tensor using new algorithm.

Detailed Description

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

Multivariate orthogonal polynomial basis generated from a total-order complete-polynomial tensor product of univariate polynomials.

The multivariate polynomials are given by

\‍[ \Psi_i(x) = \psi_{i_1}(x_1)\dots\psi_{i_d}(x_d) \‍]

where $d$ is the dimension of the basis and $i_1+\dots+ i_d\leq p$, where $p$ is the order of the basis. The size of the basis is given by $(d+p)!/(d!p!)$.

NOTE: Currently all coordinate bases must be of the samer order $p$.

Constructor & Destructor Documentation

◆ CompletePolynomialBasis()

template<typename ordinal_type, typename value_type>
Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::CompletePolynomialBasis ( const Teuchos::Array< Teuchos::RCP< const OneDOrthogPolyBasis< ordinal_type, value_type > > > & bases,
const value_type & sparse_tol = 1.0e-12,
bool use_old_cijk_alg = false,
const Teuchos::RCP< Teuchos::Array< value_type > > & deriv_coeffs = Teuchos::null )

Constructor.

Parameters
basesarray of 1-D coordinate bases
sparse_toltolerance used to drop terms in sparse triple-product tensors
use_old_cijk_alguse old algorithm for computing the sparse triple product tensor (significantly slower, but simpler)
deriv_coeffsdirection used to define derivatives for derivative product tensors. Defaults to all one's if not supplied.

References bases, basis_eval_tmp, basis_orders, Stokhos::CompletePolynomialBasisUtils< ordinal_type, value_type >::compute_terms(), d, deriv_coeffs, getName(), name, norm_squared(), norms, num_terms, p, size(), sparse_tol, sz, terms, and use_old_cijk_alg.

Member Function Documentation

◆ computeDerivDoubleProductTensor()

template<typename ordinal_type, typename value_type>
Teuchos::RCP< Teuchos::SerialDenseMatrix< ordinal_type, value_type > > Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::computeDerivDoubleProductTensor ( ) const
virtual

Compute double product tensor $B_{ij} = \langle \Psi_i D_v\Psi_j\rangle$ where $D_v\Psi_j$ represents the derivative of $\Psi_j$ in the direction $v$.

The definition of $v$ is defined by the deriv_coeffs constructor argument.

Implements Stokhos::DerivBasis< ordinal_type, value_type >.

References bases, computeDerivDoubleProductTensor(), d, sz, and terms.

Referenced by computeDerivDoubleProductTensor().

◆ computeDerivTripleProductTensor()

template<typename ordinal_type, typename value_type>
Teuchos::RCP< Stokhos::Dense3Tensor< ordinal_type, value_type > > Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::computeDerivTripleProductTensor ( const Teuchos::RCP< const Teuchos::SerialDenseMatrix< ordinal_type, value_type > > & Bij,
const Teuchos::RCP< const Stokhos::Sparse3Tensor< ordinal_type, value_type > > & Cijk ) const
virtual

Compute triple product tensor $D_{ijk} = \langle\Psi_i\Psi_j D_v\Psi_k\rangle$ where $D_v\Psi_k$ represents the derivative of $\Psi_k$ in the direction $v$.

The definition of $v$ is defined by the deriv_coeffs constructor argument.

Implements Stokhos::DerivBasis< ordinal_type, value_type >.

References norms, and sz.

◆ computeLinearTripleProductTensor()

template<typename ordinal_type, typename value_type>
Teuchos::RCP< Stokhos::Sparse3Tensor< ordinal_type, value_type > > Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::computeLinearTripleProductTensor ( ) const
virtual

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

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

References computeTripleProductTensorNew(), computeTripleProductTensorOld(), d, and use_old_cijk_alg.

◆ computeTripleProductTensor()

template<typename ordinal_type, typename value_type>
Teuchos::RCP< Stokhos::Sparse3Tensor< ordinal_type, value_type > > Stokhos::CompletePolynomialBasis< 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 >.

References computeTripleProductTensorNew(), computeTripleProductTensorOld(), sz, and use_old_cijk_alg.

Referenced by computeTripleProductTensorOld().

◆ dimension()

template<typename ordinal_type, typename value_type>
ordinal_type Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::dimension ( ) const
virtual

Return dimension of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

References d.

◆ evaluateBases()

template<typename ordinal_type, typename value_type>
void Stokhos::CompletePolynomialBasis< 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 >.

References bases, basis_eval_tmp, d, evaluateBases(), sz, and terms.

Referenced by evaluateBases().

◆ evaluateZero()

template<typename ordinal_type, typename value_type>
value_type Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::evaluateZero ( ordinal_type i) const
virtual

Evaluate basis polynomial i at zero.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

References bases, d, and terms.

◆ getCoordinateBases()

template<typename ordinal_type, typename value_type>
Teuchos::Array< Teuchos::RCP< const Stokhos::OneDOrthogPolyBasis< ordinal_type, value_type > > > Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::getCoordinateBases ( ) const
virtual

Return coordinate bases.

Array is of size dimension().

Implements Stokhos::ProductBasis< ordinal_type, value_type >.

References bases.

◆ getMaxOrders()

template<typename ordinal_type, typename value_type>
Stokhos::MultiIndex< ordinal_type > Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::getMaxOrders ( ) const
virtual

Return maximum order allowable for each coordinate basis.

Implements Stokhos::ProductBasis< ordinal_type, value_type >.

References basis_orders, and d.

◆ getName()

template<typename ordinal_type, typename value_type>
const std::string & Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::getName ( ) const
virtual

Return string name of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

References name.

Referenced by CompletePolynomialBasis().

◆ index()

template<typename ordinal_type, typename value_type>
ordinal_type Stokhos::CompletePolynomialBasis< 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 >.

References Stokhos::CompletePolynomialBasisUtils< ordinal_type, value_type >::compute_index(), num_terms, p, term(), and terms.

◆ norm_squared() [1/2]

template<typename ordinal_type, typename value_type>
const Teuchos::Array< value_type > & Stokhos::CompletePolynomialBasis< 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 >.

References norms.

Referenced by CompletePolynomialBasis().

◆ norm_squared() [2/2]

template<typename ordinal_type, typename value_type>
const value_type & Stokhos::CompletePolynomialBasis< 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 >.

References norms.

◆ order()

template<typename ordinal_type, typename value_type>
ordinal_type Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::order ( ) const
virtual

Return order of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

References p.

Referenced by computeTripleProductTensorNew(), and computeTripleProductTensorOld().

◆ print()

template<typename ordinal_type, typename value_type>
void Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::print ( std::ostream & os) const
virtual

Print basis to stream os.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

References bases, d, norms, p, and sz.

◆ size()

template<typename ordinal_type, typename value_type>
ordinal_type Stokhos::CompletePolynomialBasis< ordinal_type, value_type >::size ( ) const
virtual

Return total size of basis.

Implements Stokhos::OrthogPolyBasis< ordinal_type, value_type >.

References sz.

Referenced by CompletePolynomialBasis().

◆ term()

template<typename ordinal_type, typename value_type>
const Stokhos::MultiIndex< ordinal_type > & Stokhos::CompletePolynomialBasis< 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 >.

References terms.

Referenced by computeTripleProductTensorNew(), and index().

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