greencross

Algorithms and functions to perform the Green cross approximation method. More...

Data Structures
struct	_greencross
	Main container object for performing the Green cross approximation method. More...

Typedefs
typedef struct _greencross	greencross
	greencross is just an abbreviation for the struct _greencross. More...
typedef greencross *	pgreencross
	Pointer to a greencross object. More...
typedef const greencross *	pcgreencross
	Pointer to a constant greencross object. More...

Functions
INLINE_PREFIX uint	get_max_lvl_clusterbasis (pclusterbasis cb, const uint level)
	Retrieve the maximum level of a cluster basis. More...
INLINE_PREFIX void	set_on_lvl_info_clusterbasis (pclusterbasis cb, const uint level, uint *on_level)
	Retrieve the number of cluster basis for each level from a given root cluster basis. More...
INLINE_PREFIX void	set_per_level_clusterbasis (pclusterbasis cb, const uint level, uint *lvl_idx, pclusterbasis **cb_per_level)
	Retrieve the cluster bases of a given root cluster basis divided into the level they are on. More...
INLINE_PREFIX pclustergeometry	build_clustergeometry_greencross (const void *data, const uint dim, uint **idx)
	Creats a clustergeometry object from a triangular mesh. More...
HEADER_PREFIX void	init_greencross (pgreencross gc, uint dim)
	Initilize components related to the dimension of the problem and zero initializes all other components of the given greencross object gc. More...
HEADER_PREFIX void	uninit_greencross (pgreencross gc)
	Uninitilize a greencross object. More...
HEADER_PREFIX pgreencross	new_laplace2d_greencross (pcurve2d c2d, uint res, void *eta, uint q, uint m)
	Prepares a greencross object for approximating the integral equation \( \int\limits_{\Omega} -\frac{1}{2 \pi} \log{\left \Vert x - y \right \Vert_2} u(y) dy = f(x) \), where \(\Omega \subset \mathbb{R}^2 \). More...
HEADER_PREFIX void	del_greencross (pgreencross gc)
	Free memory and set pointer to NULL of the corresponding _greencross greencross object gc. More...
HEADER_PREFIX void	nearfield_greencross (pcgreencross gc, const uint rsize, const uint *ridx, const uint csize, const uint *cidx, pamatrix G)
	Constructs the matrix resulting from the Galerkin discretization of a variational formulation described in gc. More...
HEADER_PREFIX void	fill_green_left_greencross (pcgreencross gc, pccluster t, pamatrix A)
HEADER_PREFIX void	fill_green_right_greencross (pcgreencross gc, pccluster t, pccluster s, pamatrix B)
HEADER_PREFIX void	green_cross_approximation (pcgreencross gc, ph2matrix H2)
	Constructs a Green cross approximation of a problem given by a greencross, represented as an \(\mathcal{H}^{2}\)-Matrix [1]. More...

Variables
const uint	greencross_min_dim = 2
const uint	greencross_max_dim = 3

Detailed Description

Algorithms and functions to perform the Green cross approximation method.

Typedef Documentation

greencross

typedef struct _greencross greencross

greencross is just an abbreviation for the struct _greencross.

pcgreencross

typedef const greencross* pcgreencross

Pointer to a constant greencross object.

pgreencross

typedef greencross* pgreencross

Pointer to a greencross object.

Function Documentation

build_clustergeometry_greencross()

pclustergeometry build_clustergeometry_greencross	(	const void *	data,
		const uint	dim,
		uint **	idx
	)

Creats a clustergeometry object from a triangular mesh.

The geometry is taken from data interpreted as different objects depending on the dimension of the problem dim.

Attention

For dim = 2, data is expected to be an object of type curve2d.

Parameters

[in]	data	Pointer to an object containing the informations to construct the cluster geometry.
[in]	dim	Dimension of the cluster geometry.
[out]	idx	Index set which will be allocated by this method to index each triangle in data. The entries of idx will be 0, 1, ..., N-1, where N is equal to the number of triangles.

Returns

A valid cluster geometry object which can be used to construct a cluster tree along with the index set idx.

del_greencross()

HEADER_PREFIX void del_greencross

(

pgreencross

gc

)

Free memory and set pointer to NULL of the corresponding _greencross greencross object gc.

Parameters

gc	The greencross object which needs to be deleted.

fill_green_left_greencross()

HEADER_PREFIX void fill_green_left_greencross	(	pcgreencross	gc,
		pccluster	t,
		pamatrix	A
	)

Constructs the first factor resulting of using Green's representation formula as a degenerated approximation of the Galerkin discretization [1].

Constructs the matrix \(A_{t} = (A_{t+} \ A_{t-}) \in \mathbb{R^{\hat{t} \times K}}\), where \(\hat{t}\) and K are given by t and gc respectively, and

\[ \begin{align*} a_{t+, i \nu} &:= \sqrt{w_{\nu}} \int\limits_{\Omega} \Phi_{i}(x) g(x, z_{\nu}) dx, \\ a_{t-, i \nu} &:= \delta_{t} \sqrt{w_{\nu}} \int\limits_{\Omega} \varphi_{i}(x) \frac{\partial g}{\partial n_{\iota}}(x, z_{\nu}) dx, \end{align*} \]

where \(w_{\nu}, \delta{t}, \Omega, \varphi_i, n_{\iota}, z_{\nu}\) and the quadrature rules can be obtained or calculated from gc and t.

Parameters

[in]	gc	The greencross contains the problem description and several data to approximate the integral.
[in]	t	Cluster containing the indices of the base functions needed to discretize the integral and mapping informations for the green quadrature points.
[out]	A	Matrix object which will be filled with \(A_{t}\).

fill_green_right_greencross()

HEADER_PREFIX void fill_green_right_greencross	(	pcgreencross	gc,
		pccluster	t,
		pccluster	s,
		pamatrix	B
	)

Constructs the second factor resulting of using Green's representation formula as a degenerated approximation of the Galerkin discretization [1].

Constructs the matrix \(B_{ts} = (B_{ts+} \ B_{ts-}) \in \mathbb{R^{\hat{s} \times K}}\), where \(\hat{s}\) and K are given by s and gc respectively, and

\[ \begin{align*} b_{t+, j \nu} &:= \sqrt{w_{\nu}} \int\limits_{\Omega} \varphi_{j}(y) \frac{\partial g}{\partial n_{\iota}}(z_{\nu}, y) dy, \\ b_{t-, j \nu} &:= -\frac{\sqrt{w_{\nu}}}{\delta_{t}}\int\limits_{\Omega} \Phi_{j}(y) g(z_{\nu}, y) dy, \\ \end{align*} \]

where \(w_{\nu}, \delta{t}, \Omega, \varphi_i, n_{\iota}, z_{\nu}\) and the quadrature rules can obtained or calculated from gc, t and s.

Parameters

[in]	gc	The greencross object contains the problem description and several data to approximate the integral.
[in]	t	Cluster containing mapping informations for the green quadrature points.
[in]	s	Cluster containing the indices of the base functions needed to discretize the integral.
[out]	B	Matrix object which will be filled with \(A_{t}\).

get_max_lvl_clusterbasis()

uint get_max_lvl_clusterbasis	(	pclusterbasis	cb,
		const uint	level
	)

Retrieve the maximum level of a cluster basis.

Attention

level should be 0 since it's only used insid the function to keep track of the level of the current viewed cluster basis.

Parameters

cb	The cluster basis of which we search the maximum level.
level	The level of the current viewed cluster basis.

green_cross_approximation()

HEADER_PREFIX void green_cross_approximation	(	pcgreencross	gc,
		ph2matrix	H2
	)

Constructs a Green cross approximation of a problem given by a greencross, represented as an \(\mathcal{H}^{2}\)-Matrix [1].

Parameters

[in]	gc	The greencross object contains the problem description and several data to perform the approximation.
[out]	H2	\(\mathcal{H}^{2}\)-Matrix object which will be filled according to the Green cross approximation method.

init_greencross()

HEADER_PREFIX void init_greencross	(	pgreencross	gc,
		uint	dim
	)

Initilize components related to the dimension of the problem and zero initializes all other components of the given greencross object gc.

Note

Should be called at the beginning of the initialization of each greencross object.

Should always be matched with a call of uninit_greencross.

Parameters

[in,out]	gc	The greencross object which needs to be initialized.
[in]	dim	The dimension of the problem.

nearfield_greencross()

HEADER_PREFIX void nearfield_greencross	(	pcgreencross	gc,
		const uint	rsize,
		const uint *	ridx,
		const uint	csize,
		const uint *	cidx,
		pamatrix	G
	)

Constructs the matrix resulting from the Galerkin discretization of a variational formulation described in gc.

The arrays ridx and cidx serve as the indices for the set of base functions described through the geometry of the problem greencross::geom. Further choices like quadrature for solving the integral are taken from gc.

Parameters

[in]	gc	Object containing the problem describtion like base functions and Quadratur choices.
[in]	rsize	Number of base functions used from greencross::geom for the row of the Galerkin discretization matrix.
[in]	ridx	Indices of the base functions used from greencross::geom for the row of the Galerkin discretization matrix.
[in]	csize	Number of base functions used from greencross::geom for the column of the Galerkin discretization matrix.
[in]	cidx	Indices of the base functions used from greencross::geom for the column of the Galerkin discretization matrix.
[out]	G	Matrix which will contain the Galerkin discretization.

new_laplace2d_greencross()

HEADER_PREFIX pgreencross new_laplace2d_greencross	(	pcurve2d	c2d,
		uint	res,
		void *	eta,
		uint	q,
		uint	m
	)

Prepares a greencross object for approximating the integral equation \( \int\limits_{\Omega} -\frac{1}{2 \pi} \log{\left \Vert x - y \right \Vert_2} u(y) dy = f(x) \), where \(\Omega \subset \mathbb{R}^2 \).

Given a subspace \(\Omega \subset \mathbb{R}^2 \) by c2d, an object to approximate the integral equation \( \int\limits_{\Omega} -\frac{1}{2 \pi} \log{\left \Vert x - y \right \Vert_2} u(y) dy = f(x) \) is given with an approximation order m, a quadrature order q, necessary data for the admissibility condition in the hierarchical clustering eta and a maximal size of leaf clusters res.

Parameters

[in]	c2d	Geometry describing the subspace \(\Omega \subset \mathbb{R}^2 \).
[in]	res	Maximal size of leaf clusters.
[in]	eta	Necessary data for the admissibility condition in the hierarchical clustering.
[in]	q	Quadratur order.
[in]	m	Approximation order.

Returns

A greencross object ready for approximating the integral equation \( \int\limits_{\Omega} -\frac{1}{2 \pi} \log{\left \Vert x - y \right \Vert_2} u(y) dy = f(x) \).

set_on_lvl_info_clusterbasis()

void set_on_lvl_info_clusterbasis	(	pclusterbasis	cb,
		const uint	level,
		uint *	on_level
	)

Retrieve the number of cluster basis for each level from a given root cluster basis.

After execution, each entry of on_level with index i will hold the number of cluster bases whose level are i.

Attention

level should be 0 since it's only used insid the function to keep track of the level of the current viewed cluster basis.

on_level should be an zero initialized array of length get_max_lvl_clusterbasis(cb, level) + 1.

Parameters

cb	The root cluster basis.
level	The level of the current viewed cluster basis.
on_level	An array whose entries will hold the number of cluster bases for each level.

set_per_level_clusterbasis()

void set_per_level_clusterbasis	(	pclusterbasis	cb,
		const uint	level,
		uint *	lvl_idx,
		pclusterbasis **	cb_per_level
	)

Retrieve the cluster bases of a given root cluster basis divided into the level they are on.

After execution, each entry of cb_per_level will hold the pointers to all cluster bases whose level is i, where i is the index of the entry inside cb_per_level.

Attention

level should be 0 since it's only used insid the function to keep track of the level of the current viewed cluster basis.

lvl_idx should be an zero initialized array of length get_max_lvl_clusterbasis(cb, level) + 1.

cb_per_level should be an initialized array of length get_max_lvl_clusterbasis(cb, level) + 1, where each entry with index should be an initialized array of length on_level[i], where on_level is retrieved by set_on_lvl_info_clusterbasis(cb, 0, on_level).

Parameters

cb	The root cluster basis.
level	The level of the current viewed cluster basis.
lvl_idx	The number of cluster bases already found on each level respectively.
cb_per_level	An arrays whose entries will hold pointers to all cluster bases of the corresponding level.

uninit_greencross()

HEADER_PREFIX void uninit_greencross

(

pgreencross

gc

)

Uninitilize a greencross object.

Invalidates member pointers, freeing corresponding storage if appropriate, and prepares the object for deletion.

Note

Should always be matched with a call of init_greencross.

Parameters

[out]

gc

The greencross object which needs to be initialized.

Variable Documentation

greencross_max_dim

const uint greencross_max_dim = 3

greencross_min_dim

const uint greencross_min_dim = 2