Symmetric Gauss-Seidel smoother for sparse systems. More...

#include <smoother.hpp>

Inheritance diagram for AMG::SparseGaussSeidel< EleType >:

Public Member Functions
	SparseGaussSeidel ()
	Construct a new Sparse Gauss Seidel object for pre/post smoother in AMG.

void	smooth (const Eigen::SparseMatrix< EleType > &A, Eigen::Matrix< EleType, -1, 1 > &u, const Eigen::Matrix< EleType, -1, 1 > &b)
	Must implement function that smooths `Au = b`.

Public Member Functions inherited from AMG::SmootherBase< EleType >
	SmootherBase ()

	SmootherBase (size_t n_iters_)

	SmootherBase (double tolerance_, size_t compute_error_every_n_iters_, size_t n_iters_)
	Construct a new Smoother Base object with iterative solver member data.

Private Member Functions
void	matvecprod (int col, EleType &rsum, EleType &diag, const EleType &z, const Eigen::SparseMatrix< EleType > &A, const Eigen::Matrix< EleType, -1, 1 > &u)
	Modified sparse matvec product of `Au` for Gauss-Seidel sweeps.

void	update (int col, const Eigen::SparseMatrix< EleType > &A, const Eigen::Matrix< EleType, -1, 1 > &b, Eigen::Matrix< EleType, -1, 1 > &u)
	Updates a single entry in `u` inplace using the definition of a Gauss-Seidel.

void	forwardsweep (const Eigen::SparseMatrix< EleType > &A, const Eigen::Matrix< EleType, -1, 1 > &b, Eigen::Matrix< EleType, -1, 1 > &u, const int &ncols)
	Gauss-Seidel iteration starting from the first row.

void	backwardsweep (const Eigen::SparseMatrix< EleType > &A, const Eigen::Matrix< EleType, -1, 1 > &b, Eigen::Matrix< EleType, -1, 1 > &u, const int &ncols)
	Gauss-Seidel iteration starting from the last row.

Additional Inherited Members
Public Attributes inherited from AMG::SmootherBase< EleType >
EleType	tolerance {1e-9}
	Tolerance below which a smoother is considered to have converged.

size_t	compute_error_every_n_iters {100}
	Compute the error every `n` iterations during smoothing.

size_t	n_iters {1}
	Maximum number of iterations before smoothing termination.

Detailed Description

template<class EleType>
class AMG::SparseGaussSeidel< EleType >

Symmetric Gauss-Seidel smoother for sparse systems.

The forward and backward sweep are not strictly both necessary, it seems like most numerical linear algebra books (E.g., Heath's "Scientific Computing") only propose one sweep as the algorithm. For completeness, however, Ref [2] does assert the justification of the so-called symmetric sweep.

References

[1] : smoother.jl in AlgebraicMultigrid.jl

[2] : Strang2006 MIT 18.086 Lecture: Iterative Methods

[3] : Julia Lang: Dot product as linear combination

Template Parameters

EleType

Constructor & Destructor Documentation

◆ SparseGaussSeidel()

template<class EleType >

AMG::SparseGaussSeidel< EleType >::SparseGaussSeidel ( )

inline

Construct a new Sparse Gauss Seidel object for pre/post smoother in AMG.

Member Function Documentation

◆ backwardsweep()

template<class EleType >

void AMG::SparseGaussSeidel< EleType >::backwardsweep	(	const Eigen::SparseMatrix< EleType > &	A,
		const Eigen::Matrix< EleType, -1, 1 > &	b,
		Eigen::Matrix< EleType, -1, 1 > &	u,
		const int &	ncols )

inlineprivate

Gauss-Seidel iteration starting from the last row.

Parameters

A	Coefficients matrix for discretized governing equations.
b	Right hand side of linear system `Au = b`.
u	Solution to linear system of equations.
nrows

◆ forwardsweep()

template<class EleType >

void AMG::SparseGaussSeidel< EleType >::forwardsweep	(	const Eigen::SparseMatrix< EleType > &	A,
		const Eigen::Matrix< EleType, -1, 1 > &	b,
		Eigen::Matrix< EleType, -1, 1 > &	u,
		const int &	ncols )

inlineprivate

Gauss-Seidel iteration starting from the first row.

Parameters

A	Coefficients matrix for discretized governing equations.
b	Right hand side of linear system `Au = b`.
u	Solution to linear system of equations.
nrows

◆ matvecprod()

template<class EleType >

void AMG::SparseGaussSeidel< EleType >::matvecprod	(	int	col,
		EleType &	rsum,
		EleType &	diag,
		const EleType &	z,
		const Eigen::SparseMatrix< EleType > &	A,
		const Eigen::Matrix< EleType, -1, 1 > &	u )

inlineprivate

Modified sparse matvec product of Au for Gauss-Seidel sweeps.

The results of this operation update rsum and diag inplace.

Parameters

col	Index of the column for which you desire sparse row indices and values.
rsum	Sum representing the matvec product.
diag	Value of diagonal for particular column of CSC matrix.
z	Zero value
A	Coefficients matrix for discretized governing equations.
u	Solution to linear system of equations.

◆ smooth()

template<class EleType >

void AMG::SparseGaussSeidel< EleType >::smooth	(	const Eigen::SparseMatrix< EleType > &	A,
		Eigen::Matrix< EleType, -1, 1 > &	u,
		const Eigen::Matrix< EleType, -1, 1 > &	b )

inlinevirtual

Must implement function that smooths Au = b.

Parameters

A	Coefficients matrix for discretized governing equations. Coeffcients matrix for linear system of equations.
u	Solution to linear system of equations.
b	Right hand side of linear system `Au = b`. Right hand side of linear system of equations.

Implements AMG::SmootherBase< EleType >.

◆ update()

template<class EleType >

void AMG::SparseGaussSeidel< EleType >::update	(	int	col,
		const Eigen::SparseMatrix< EleType > &	A,
		const Eigen::Matrix< EleType, -1, 1 > &	b,
		Eigen::Matrix< EleType, -1, 1 > &	u )

inlineprivate

Updates a single entry in u inplace using the definition of a Gauss-Seidel.

$u_{i} = \frac{b_i - \sum_{j \neq i} a_{ij} u_j }{a_{ii}}.$

Parameters

col
A	Coefficients matrix for discretized governing equations.
b	Right hand side of linear system `Au = b`.
u	Solution to linear system of equations.

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

include/amg/smoother.hpp

Public Member Functions

Private Member Functions

Additional Inherited Members

Detailed Description

Constructor & Destructor Documentation

◆ SparseGaussSeidel()

Member Function Documentation

◆ backwardsweep()

◆ forwardsweep()

◆ matvecprod()

◆ smooth()

◆ update()