Two-resolution preconditioner. More...

#include <preconditioners.h>

Inheritance diagram for TwoLevelPreconditioner:

Collaboration diagram for TwoLevelPreconditioner:

Public Member Functions
	TwoLevelPreconditioner (Parallel const &par, InputFile const &inp, std::vector< std::pair< int, int >> const &ll, Bspline const &s_bspline, CommandLine const &cmd)

RadialIntegrals const &	rad () const
	Get radial integrals. More...

void	precondition (const cArrayView r, cArrayView z) const
	Precondition the equation. More...

void	setup ()
	Initialize the preconditioner. More...

void	update (double E)
	Update the preconditioner for the next energy. More...

void	rhs (cArrayView chi, int ienergy, int instate) const
	Return the right-hand side. More...

void	multiply (const cArrayView p, cArrayView q) const
	Multiply by the matrix equation. More...

virtual void	CG_prec (int iblock, const cArrayView r, cArrayView z) const

Public Member Functions inherited from SSORCGPreconditioner
	SSORCGPreconditioner (Parallel const &par, InputFile const &inp, std::vector< std::pair< int, int >> const &ll, Bspline const &bspline, CommandLine const &cmd)

Public Member Functions inherited from CGPreconditioner
	CGPreconditioner (Parallel const &par, InputFile const &inp, std::vector< std::pair< int, int >> const &ll, Bspline const &bspline, CommandLine const &cmd)

virtual void	CG_mmul (int iblock, const cArrayView p, cArrayView q) const

Public Member Functions inherited from NoPreconditioner
	NoPreconditioner (Parallel const &par, InputFile const &inp, std::vector< std::pair< int, int >> const &ll, Bspline const &bspline, CommandLine const &cmd)

Static Public Attributes
static const std::string	name

static const std::string	description

Static Public Attributes inherited from SSORCGPreconditioner
static const std::string	name = "SSOR"

static const std::string	description = "Block inversion using SSOR-preconditioned conjugate gradients."

Static Public Attributes inherited from CGPreconditioner
static const std::string	name = "cg"

static const std::string	description = "Block inversion using plain conjugate gradients."

Static Public Attributes inherited from NoPreconditioner
static const std::string	name = "none"

static const std::string	description = "Preconditioning by the identity matrix."

Protected Member Functions
void	computeSigma_ ()

Complex	computeSigma_iknot_ (int qord, int is, int iknots, int ip, int iknotp) const

Protected Attributes
RowMatrix< Complex >	spSigma_

RowMatrix< Complex >	psSigma_

ColMatrix< Complex >	spSigma
	DEBUG. More...

ColMatrix< Complex >	SspSigma

ColMatrix< Complex >	psSigma

ColMatrix< Complex >	SpsSigma

Bspline	p_bspline_

RadialIntegrals	p_rad_

std::vector< CsrMatrix >	p_csr_

std::vector< CsrMatrix::LUft >	p_lu_

SymDiaMatrix	p_half_D_minus_Mm1_tr_kron_S_

SymDiaMatrix	p_S_kron_half_D_minus_Mm1_tr_

SymDiaMatrix	p_Mm2_kron_S_

SymDiaMatrix	p_S_kron_Mm2_

SymDiaMatrix	p_S_kron_S_

GaussLegendre	g_

Protected Attributes inherited from SSORCGPreconditioner
std::vector< SymDiaMatrix >	SSOR_

Protected Attributes inherited from NoPreconditioner
CommandLine const &	cmd_

Parallel const &	par_

InputFile const &	inp_

std::vector< std::pair< int, int > > const &	l1_l2_

std::vector< SymDiaMatrix >	dia_blocks_

Bspline	s_bspline_

RadialIntegrals	s_rad_

SymDiaMatrix	S_kron_S_

SymDiaMatrix	S_kron_Mm1_tr_

SymDiaMatrix	S_kron_Mm2_

SymDiaMatrix	Mm1_tr_kron_S_

SymDiaMatrix	Mm2_kron_S_

SymDiaMatrix	half_D_minus_Mm1_tr_

SymDiaMatrix	half_D_minus_Mm1_tr_kron_S_

SymDiaMatrix	S_kron_half_D_minus_Mm1_tr_

Detailed Description

This preconditioner class is inspired by the method of multi-resolution in the finite difference computations. A high-order differential operator that gives rise to the multi-diagonal matrix is initially solved in low order. The result is then used as a preconditioner for the full, high-order method.

In the case of B-spline expansion, that is being used in Hex, the bandwidth of the resulting matrix is given by the order of the B-splines. Analogously to the just metioned method, we compute the solution in a sharper base (lower order), which is computationally less demanding. The algorithm works as follows:

In every step we are solving the equation \( \mathbf{M}^{(s)}\mathbf{z}^{(s)} = \mathbf{r}^{(s)} \).
The matrix \( \mathbf{M}^{(s)} \) is the diagonal block \( E\mathbf{S}^{(s)}\otimes\mathbf{S}^{(s)} - \mathbf{H}^{(s)} \). The symbol "s" stands for solution B-spline basis.
But we want just to compute the simplified system \( \mathbf{M}^{(p)}\mathbf{z}^{(p)} = \mathbf{r}^{(p)} \). The symbol "p" stands for preconditioner B-spline basis.
The matrices contained in \( \mathbf{M}^{(p)} \) are computed in a low-order B-spline basis; these are the matrices \( \mathbf{S}^{(p)} \), \( \mathbf{D}^{(p)} \) etc., whereas fhe original full-order matrices are \( \mathbf{S}^{(s)} \), \( \mathbf{D}^{(s)} \) etc.
Before and after the solution of the simplified system we need to map the solutions between the bases. This is done by the equations
\[ \mathbf{S}^{(p)}\otimes\mathbf{S}^{(p)} \mathbf{r}^{(p)} = \mathbb{\Sigma}^{(p,q)} \otimes \mathbb{\Sigma}^{(p,q)} \mathbf{r}^{(q)} \]
Together we have the sequence
\[ \mathbf{S}^{(p)}\otimes\mathbf{S}^{(p)} \mathbf{r}^{(p)} = \mathbb{\Sigma}^{(p,s)} \otimes \mathbb{\Sigma}^{(p,s)} \mathbf{r}^{(s)} \]

\[ \mathbf{M}^{(p)}\mathbf{z}^{(p)} = \mathbf{r}^{(p)} \]

\[ \mathbf{S}^{(s)}\otimes\mathbf{S}^{(s)} \mathbf{z}^{(s)} = \mathbb{\Sigma}^{(s,p)} \otimes \mathbb{\Sigma}^{(s,p)} \mathbf{z}^{(p)} \]
instead of the single equation
\[ \mathbf{M}^{(s)}\mathbf{z}^{(s)} = \mathbf{r}^{(s)} \ . \]

The transition overlap matrix \( \mathbb{\Sigma}^{(p,q)} \) is defined in components as

\[ \Sigma_{ij}^{(p,q)} = \int B_i^{(p)}(r) B_j^{(q)}(r) \mathrm{d}r \ . \]

The solution of transition equations can be recast into a matrix form of Kronecker product,

\[ \mathbf{S}^{(p)} \mathbf{R}^{(p)} \mathbf{S}^{(p)T} = \mathbb{\Sigma}^{(p,q)}\mathrm{Matrix}(\mathbf{r}^{(q)})\mathbb{\Sigma}^{(p,q)^T} \qquad\Rightarrow\qquad \mathbf{R}^{(p)} = \mathbf{S}^{(p)^{-1}} \mathbb{\Sigma}^{(p,q)} \mathrm{Matrix}(\mathbf{r}^{(q)}) \mathbb{\Sigma}^{(p,q)^T} \mathbf{S}^{(p)^{-T}} \]

where the word "Matrix" indicates that we want to split long vector into a square matrix. The resulting square matrix \( \mathbf{R} \) will then have the same structure.

Constructor & Destructor Documentation

TwoLevelPreconditioner::TwoLevelPreconditioner	(	Parallel const &	par,
		InputFile const &	inp,
		std::vector< std::pair< int, int >> const &	ll,
		Bspline const &	s_bspline,
		CommandLine const &	cmd
	)

inline

Member Function Documentation

virtual void TwoLevelPreconditioner::CG_prec	(	int	iblock,
		const cArrayView	r,
		cArrayView	z
	)		const

virtual

Reimplemented from SSORCGPreconditioner.

void TwoLevelPreconditioner::computeSigma_ ( )

protected

Complex TwoLevelPreconditioner::computeSigma_iknot_	(	int	qord,
		int	is,
		int	iknots,
		int	ip,
		int	iknotp
	)		const

protected

void TwoLevelPreconditioner::multiply	(	const cArrayView	p,
		cArrayView	q
	)		const

virtual

This function implements matrix multiplication by the matrix of the set of equations that is to be solved.

Reimplemented from SSORCGPreconditioner.

void TwoLevelPreconditioner::precondition	(	const cArrayView	r,
		cArrayView	z
	)		const

inlinevirtual

This function preconditions the equation, solving the preconditioner equation

\[ \mathbf{M}\mathbf{z} = \mathbf{r} \ . \]

It may use the MPI environment.

Reimplemented from SSORCGPreconditioner.

RadialIntegrals const& TwoLevelPreconditioner::rad ( ) const

inlinevirtual

Reimplemented from SSORCGPreconditioner.

void TwoLevelPreconditioner::rhs	(	cArrayView	chi,
		int	ienergy,
		int	instate
	)		const

virtual

Reimplemented from SSORCGPreconditioner.

void TwoLevelPreconditioner::setup ( )

virtual

This function contains all computation intensive preparations for the preconditioner, e.g. computation of radial integrals. It may use only SMP environment.

Reimplemented from SSORCGPreconditioner.

void TwoLevelPreconditioner::update ( double E )

virtual

This function updates the preconditioner for another right hand side. It may use the MPI environment.

Reimplemented from SSORCGPreconditioner.

Field Documentation

const std::string TwoLevelPreconditioner::description

static

GaussLegendre TwoLevelPreconditioner::g_

protected

const std::string TwoLevelPreconditioner::name

static

Bspline TwoLevelPreconditioner::p_bspline_

protected

std::vector<CsrMatrix> TwoLevelPreconditioner::p_csr_

protected

SymDiaMatrix TwoLevelPreconditioner::p_half_D_minus_Mm1_tr_kron_S_

protected

std::vector<CsrMatrix::LUft> TwoLevelPreconditioner::p_lu_

protected

SymDiaMatrix TwoLevelPreconditioner::p_Mm2_kron_S_

protected

RadialIntegrals TwoLevelPreconditioner::p_rad_

protected

SymDiaMatrix TwoLevelPreconditioner::p_S_kron_half_D_minus_Mm1_tr_

protected

SymDiaMatrix TwoLevelPreconditioner::p_S_kron_Mm2_

protected

SymDiaMatrix TwoLevelPreconditioner::p_S_kron_S_

protected

ColMatrix<Complex> TwoLevelPreconditioner::psSigma

protected

RowMatrix<Complex> TwoLevelPreconditioner::psSigma_

protected

ColMatrix<Complex> TwoLevelPreconditioner::spSigma

protected

RowMatrix<Complex> TwoLevelPreconditioner::spSigma_

protected

ColMatrix<Complex> TwoLevelPreconditioner::SpsSigma

protected

ColMatrix<Complex> TwoLevelPreconditioner::SspSigma

protected

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

src/preconditioners.h

Public Member Functions

Static Public Attributes

Protected Member Functions

Protected Attributes

Detailed Description

Constructor & Destructor Documentation

Member Function Documentation

Field Documentation