#include <WLS_fwd.hpp>

Detailed Description

template<class basis_t, class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>
class mm::WLS< basis_t, weight_t, scale_t, solver_t >

A class for generating approximations using Weighted Least Squares over local neighborhoods.

Template Parameters

basis_t	Type of basis used in approximation. Must satisfy the Basis function concept.
weight_t	Type of weight function used in approximation. Must satisfy the Weight function concept.
scale_t	Scale function to be used in approximation. Must satisfy the Scale function concept.
solver_t	Which linear solver to use in weight computations. Must satisfy the Linear solver concept. Default: Jacobi SVD.

This class satisfies the Approximation engine concept.

Usage example:

    WLS<Monomials<Vec2d>, NoWeight<Vec2d>, ScaleToFarthest, Eigen::PartialPivLU<Eigen::MatrixXd>>
        appr(Monomials<Vec2d>::tensorBasis(2));  // full type specification with scale and solver
    std::cout << appr << std::endl;
 
    WLS<Monomials<Vec2d>> appr2(2);  // using default parameters
 
    // Local neighbourhood
    double h = 0.1;
    Range<Vec2d> support = {{0, 0}, {0, h}, {h, 0}, {0, -h}, {-h, 0},
                            {-h, h}, {-h, -h}, {h, -h}, {h, h}};
 
    // compute approximations at point `{0.0, 0.0}`.
    appr.compute({0.0, 0.0}, support);
 
    // Get info about the computation.
    std::cout << appr.basis() << std::endl;
    std::cout << appr.weight() << std::endl;
    std::cout << appr.center() << std::endl;
    std::cout << appr.scale() << std::endl;
    std::cout << appr.localCoordinates() << std::endl;
    std::cout << appr.getMatrix() << std::endl;
    std::cout << appr.getWeights() << std::endl;
    Eigen::PartialPivLU<Eigen::MatrixXd> solver = appr.solver();
 
    // Get shape (stencil weights) for approximation of Laplacian.
    Eigen::VectorXd shape = appr.getShape(Lap<2>());
 
    // Get shape (stencil weights) for approximation of value
    shape = appr.getShape();

Examples: test/end2end/poisson_explicit.cpp.

Definition at line 49 of file WLS_fwd.hpp.

Collaboration diagram for mm::WLS< basis_t, weight_t, scale_t, solver_t >:

Public Member Functions
	WLS (const basis_t &basis)
	Construct WLS with given basis and default constructed weight. More...

	WLS (const basis_t &basis, const weight_t &weight)
	Hold information about the type of approximation we want to have for the domain. More...

void	compute (const vector_t &point, const std::vector< vector_t > &support)
	Computes, decomposes and stores matrix for shape calculation at given point with given neighbours. More...

Eigen::Matrix< scalar_t, Eigen::Dynamic, 1 >	getShape () const
	Returns weights for approximation of function value at point `point`, knowing values in its `support`, given by a previous call to compute. More...

template<class operator_t >
Eigen::Matrix< scalar_t, Eigen::Dynamic, 1 >	getShape (const operator_t&op) const
	Returns weights for approximation of differentiation operator, given by `der` at point `point`, knowing values in its `support`, given by a previous call to compute. More...

template<typename Derived >
Eigen::Matrix< scalar_t, Eigen::Dynamic, 1 >	getShapeFromRhs (const Eigen::MatrixBase< Derived > &Lb) const
	Returns weights for approximation with a given right hand side, which is expected to contains the values of an operator applied to your basis functions. More...

int	requiredSupportSize () const
	Returns minimal support size required for this approximation to be performed. More...

const basis_t &	basis () const
	Returns basis used in the computation. More...

const weight_t &	weight () const
	Returns weight function used in the computation. More...

vector_t	center () const
	Return the center point of the approximation. More...

scalar_t	scale () const
	Return the scaling factor. Only initialized after the first call to compute. More...

Range< vector_t >	localCoordinates () const
	Returns the local (translated and scaled) coordinates of the support nodes. More...

ei_matrix_t	getMatrix () const
	Returns the matrix used in the approximation. More...

Eigen::Matrix< scalar_t, Eigen::Dynamic, 1 >	getWeights () const
	Returns the current support weights. Only initialized after the first call to compute. More...

const solver_t &	solver () const
	Read access to the linear solver used for matrix decomposition. More...

solver_t &	solver ()
	Read-write access to the linear solver used for matrix decomposition. More...

template<typename Derived >
WLSApproximant< basis_t >	getApproximant (const vector_t &point, const std::vector< vector_t > &support, const Eigen::MatrixBase< Derived > &values) const
	Construct a WLS approximant anchored at `point` with given values in given support nodes. More...

Public Types
enum	{ dim = vector_t::dim }
	Store dimension of the domain. More...

typedef basis_t::scalar_t	scalar_t
	Numeric scalar data type, e.g. double. More...

typedef basis_t::vector_t	vector_t
	Vector data type. More...

typedef Eigen::Matrix< scalar_t, Eigen::Dynamic, Eigen::Dynamic >	ei_matrix_t
	Eigen matrix data type. More...

Friends
template<class B , class W , class S , class L >
std::ostream &	operator<< (std::ostream &os, const WLS< B, W, S, L > &wls)
	Output basic info about given approximation engine. More...

Private Member Functions
ei_matrix_t	getMatrix (const std::vector< vector_t > &local_coordinates) const
	Returns the collocation matrix. More...

Private Attributes
basis_t	basis_
	Basis functions. More...

weight_t	weight_
	Weight function. More...

Range< vector_t >	local_coordinates_
	Support points after translation and scaling. More...

vector_t	point_
	Saved center point. More...

scalar_t	scale_
	Saved scale. More...

solver_t	solver_
	Saved decomposition. More...

Member Enumeration Documentation

◆ anonymous enum

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

anonymous enum

Store dimension of the domain.

Enumerator
dim	Dimensionality of the domain.

Definition at line 59 of file WLS_fwd.hpp.

Constructor & Destructor Documentation

◆ WLS() [1/2]

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

mm::WLS< basis_t, weight_t, scale_t, solver_t >::WLS ( const basis_t & basis )

inlineexplicit

Construct WLS with given basis and default constructed weight.

Definition at line 74 of file WLS_fwd.hpp.

◆ WLS() [2/2]

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

mm::WLS< basis_t, weight_t, scale_t, solver_t >::WLS	(	const basis_t &	basis,
		const weight_t &	weight
	)

inline

Hold information about the type of approximation we want to have for the domain.

Parameters

basis	Basis to use in the algorithm.
weight	Weight function to use.

Definition at line 80 of file WLS_fwd.hpp.

Member Function Documentation

◆ basis()

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

const basis_t& mm::WLS< basis_t, weight_t, scale_t, solver_t >::basis ( ) const

inline

Returns basis used in the computation.

Definition at line 138 of file WLS_fwd.hpp.

◆ center()

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

vector_t mm::WLS< basis_t, weight_t, scale_t, solver_t >::center ( ) const

inline

Return the center point of the approximation.

Only initialized after the first call to compute.

Definition at line 146 of file WLS_fwd.hpp.

◆ compute()

template<class basis_t , class weight_t , class scale_t , class solver_t >

void mm::WLS< basis_t, weight_t, scale_t, solver_t >::compute	(	const vector_t &	point,
		const std::vector< vector_t > &	support
	)

Computes, decomposes and stores matrix for shape calculation at given point with given neighbours.

Given basis and weight function are used and evaluated in the local coordinate system with origin in point and scales by scale_. Decomposition given by solver_ is used. This function must be called before any calls to getShape are made for this point configuration.

Parameters

point	Point around which to prepare for shape computation.
support	Neighbouring points where function values are expected.

Definition at line 19 of file WLS.hpp.

◆ getApproximant()

template<class basis_t , class weight_t , class scale_t , class solver_t >

template<typename Derived >

WLSApproximant< basis_t > mm::WLS< basis_t, weight_t, scale_t, solver_t >::getApproximant	(	const vector_t &	point,
		const std::vector< vector_t > &	support,
		const Eigen::MatrixBase< Derived > &	values
	)		const

Construct a WLS approximant anchored at point with given values in given support nodes.

The approximation function \(\hat{u}(x) = \sum_{j=1}^m \alpha_j b_j\left(\frac{x - p}{s}\right)\) minimises

\[ J = \sum_{i=1}^n w\left( \frac{p_i-p}{s}\right)^2 (u_i - \hat{u}(p_i))^2 \]

Parameters

point	Center point \(p\).
support	Support values \(p_i\).
values	Function values \(u_i\) at \(p_i\).

Returns: An object representing the approximant.

Definition at line 123 of file WLS.hpp.

◆ getMatrix() [1/2]

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

ei_matrix_t mm::WLS< basis_t, weight_t, scale_t, solver_t >::getMatrix ( ) const

inline

Returns the matrix used in the approximation.

Only initialized after the first call to compute.

Definition at line 161 of file WLS_fwd.hpp.

◆ getMatrix() [2/2]

template<class basis_t , class weight_t , class scale_t , class solver_t >

WLS< basis_t, weight_t, scale_t, solver_t >::ei_matrix_t mm::WLS< basis_t, weight_t, scale_t, solver_t >::getMatrix ( const std::vector< vector_t > & local_coordinates ) const

private

Returns the collocation matrix.

Definition at line 37 of file WLS.hpp.

◆ getShape() [1/2]

template<class basis_t , class weight_t , class scale_t , class solver_t >

Eigen::Matrix< typename basis_t::scalar_t, Eigen::Dynamic, 1 > mm::WLS< basis_t, weight_t, scale_t, solver_t >::getShape

Returns weights for approximation of function value at point point, knowing values in its support, given by a previous call to compute.

Returns: Vector of nodal weights \(\chi\), such that \(u(p) \approx \sum_{i=1}^n \chi_i u(p_i)\) where \(p\) represents point and \(p_i\) are its support nodes.

Definition at line 54 of file WLS.hpp.

◆ getShape() [2/2]

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

template<class operator_t >

Eigen::Matrix<scalar_t, Eigen::Dynamic, 1> mm::WLS< basis_t, weight_t, scale_t, solver_t >::getShape ( const operator_t& op ) const

Returns weights for approximation of differentiation operator, given by der at point point, knowing values in its support, given by a previous call to compute.

Returns: Vector of nodal weights \(\chi\), such that \((\mathcal{L}u)(p) \approx \sum_{i=1}^n \chi_i u(p_i)\) where \(p\) represents point, \(p_i\) are its support nodes and \(\mathcal{L}\) is given by der as \(\mathcal{L} = \dpar{^{\sum_{i=1}^d m_i}}{x_1^{m_1} \cdots \partial x_d^{m_d}}\), where \(m_i\) are elements of der.

Parameters

op	Operator to be approximated.

◆ getShapeFromRhs()

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

template<typename Derived >

Eigen::Matrix<scalar_t, Eigen::Dynamic, 1> mm::WLS< basis_t, weight_t, scale_t, solver_t >::getShapeFromRhs ( const Eigen::MatrixBase< Derived > & Lb ) const

Returns weights for approximation with a given right hand side, which is expected to contains the values of an operator applied to your basis functions.

This method is useful if values of the operator are computed differently to the procedure used by getShape.

Parameters

Lb	The right hand side of the approximation system representing an values of an operator applied to basis functions at the point where compute was called.

See also: getShape

◆ getWeights()

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

Eigen::Matrix<scalar_t, Eigen::Dynamic, 1> mm::WLS< basis_t, weight_t, scale_t, solver_t >::getWeights ( ) const

Returns the current support weights. Only initialized after the first call to compute.

◆ localCoordinates()

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

Range<vector_t> mm::WLS< basis_t, weight_t, scale_t, solver_t >::localCoordinates ( ) const

inline

Returns the local (translated and scaled) coordinates of the support nodes.

Only initialized after the first call to compute.

Definition at line 155 of file WLS_fwd.hpp.

◆ requiredSupportSize()

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

int mm::WLS< basis_t, weight_t, scale_t, solver_t >::requiredSupportSize ( ) const

inline

Returns minimal support size required for this approximation to be performed.

Definition at line 136 of file WLS_fwd.hpp.

◆ scale()

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

scalar_t mm::WLS< basis_t, weight_t, scale_t, solver_t >::scale ( ) const

inline

Return the scaling factor. Only initialized after the first call to compute.

Definition at line 149 of file WLS_fwd.hpp.

◆ solver() [1/2]

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

solver_t& mm::WLS< basis_t, weight_t, scale_t, solver_t >::solver ( )

inline

Read-write access to the linear solver used for matrix decomposition.

Can be used to set additional solver parameters. Only initialized after the first call to compute.

Definition at line 177 of file WLS_fwd.hpp.

◆ solver() [2/2]

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

const solver_t& mm::WLS< basis_t, weight_t, scale_t, solver_t >::solver ( ) const

inline

Read access to the linear solver used for matrix decomposition.

Only initialized after the first call to compute.

Definition at line 170 of file WLS_fwd.hpp.

◆ weight()

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

const weight_t& mm::WLS< basis_t, weight_t, scale_t, solver_t >::weight ( ) const

inline

Returns weight function used in the computation.

Definition at line 140 of file WLS_fwd.hpp.

Friends And Related Function Documentation

◆ operator<<

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

template<class B , class W , class S , class L >

std::ostream& operator<<	(	std::ostream &	os,
		const WLS< B, W, S, L > &	wls
	)

friend

Output basic info about given approximation engine.

Member Data Documentation

◆ basis_

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

basis_t mm::WLS< basis_t, weight_t, scale_t, solver_t >::basis_

private

Basis functions.

Definition at line 64 of file WLS_fwd.hpp.

◆ local_coordinates_

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

Range<vector_t> mm::WLS< basis_t, weight_t, scale_t, solver_t >::local_coordinates_

private

Support points after translation and scaling.

Definition at line 67 of file WLS_fwd.hpp.

◆ point_

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

vector_t mm::WLS< basis_t, weight_t, scale_t, solver_t >::point_

private

Saved center point.

Definition at line 68 of file WLS_fwd.hpp.

◆ scale_

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

scalar_t mm::WLS< basis_t, weight_t, scale_t, solver_t >::scale_

private

Saved scale.

Definition at line 69 of file WLS_fwd.hpp.

◆ solver_

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

solver_t mm::WLS< basis_t, weight_t, scale_t, solver_t >::solver_

private

Saved decomposition.

Definition at line 70 of file WLS_fwd.hpp.

◆ weight_

template<class basis_t , class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>

weight_t mm::WLS< basis_t, weight_t, scale_t, solver_t >::weight_

private

Weight function.

Definition at line 65 of file WLS_fwd.hpp.

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

include/medusa/bits/approximations/WLS_fwd.hpp
include/medusa/bits/approximations/WLS.hpp

Detailed Description

template<class basis_t, class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>> class mm::WLS< basis_t, weight_t, scale_t, solver_t >

Public Member Functions

Public Types

Friends

Private Member Functions

Private Attributes

Member Enumeration Documentation

◆ anonymous enum

Constructor & Destructor Documentation

◆ WLS() [1/2]

◆ WLS() [2/2]

Member Function Documentation

◆ basis()

◆ center()

◆ compute()

◆ getApproximant()

◆ getMatrix() [1/2]

◆ getMatrix() [2/2]

◆ getShape() [1/2]

◆ getShape() [2/2]

◆ getShapeFromRhs()

◆ getWeights()

◆ localCoordinates()

◆ requiredSupportSize()

◆ scale()

◆ solver() [1/2]

◆ solver() [2/2]

◆ weight()

Friends And Related Function Documentation

◆ operator<<

Member Data Documentation

◆ basis_

◆ local_coordinates_

◆ point_

◆ scale_

◆ solver_

◆ weight_

template<class basis_t, class weight_t = NoWeight<typename basis_t::vector_t>, class scale_t = NoScale, class solver_t = JacobiSVDWrapper<typename basis_t::scalar_t, Eigen::ColPivHouseholderQRPreconditioner>>
class mm::WLS< basis_t, weight_t, scale_t, solver_t >