Coarsening Strategies¶
A coarsening strategy defines various options for creating coarse systems in the AMG hierarchy. A coarsening strategy takes the system matrix \(A\) at the current level, and returns prolongation operator \(P\) and the corresponding restriction operator \(R\).
RugeStuben¶

template<class
Backend
>
classamgcl::coarsening
::
ruge_stuben
¶ Include
<amgcl/coarsening/ruge_stuben>
The classic RugeStuben coarsening with direct interpolation [Stue99].

class
params
¶ 
float
eps_strong
= 0.25¶ Parameter \(\varepsilon_{str}\) defining strong couplings.
Variable \(i\) is defined to be strongly negatively coupled to another variable, \(j\), if
\[a_{ij} \geq \varepsilon_{str}\max\limits_{a_{ik}<0}a_{ik}\quad \text{with fixed} \quad 0 < \varepsilon_{str} < 1.\]In practice, a value of \(\varepsilon_{str}=0.25\) is usually taken.

bool
do_trunc
= true¶ Prolongation operator truncation. Interpolation operators, and, hence coarse operators may increase substantially towards coarser levels. Without truncation, this may become too costly. Truncation ignores all interpolatory connections which are smaller (in absolute value) than the largest one by a factor of \(\varepsilon_{tr}\). The remaining weights are rescaled so that the total sum remains unchanged. In practice, a value of \(\varepsilon_{tr}=0.2\) is usually taken.

float
eps_trunc
= 0.2¶ Truncation parameter \(\varepsilon_{tr}\).

float

class
Aggregationbased coarsening¶
The aggregationbase class of coarsening methods split the nodes at the fine grid into disjoint sets of nodes, the socalled aggregates that act as nodes on the coarse grid. The prolongation operators are then built by first constructing a tentative prolongator using the knowledge of zero energy modes of the principal part of the differential operator with natural boundary conditions (e.g., near nullspace vectors, or rigid body modes for elasticity). In case of smoothed aggregation the prolongation operator is then smoothed by a carefully selected iteration.
All of the aggregation based methods take the aggregation and the nullspace parameters:

class
amgcl::coarsening
::
pointwise_aggregates
¶ Pointwise aggregation. When the system matrix has a block structure, it is converted to a poinwise matrix (single value per block), and the aggregates are created for this reduced matrix instead.

class
params
¶ The aggregation parameters.

float
eps_strong
= 0.08¶ Parameter \(\varepsilon_{strong}\) defining strong couplings. Connectivity is defined in a symmetric way, that is, two variables \(i\) and \(j\) are considered to be connected to each other if \(\frac{a_{ij}^2}{a_{ii}a_{jj}} > \varepsilon_{strong}\) with fixed \(0 < \varepsilon_{strong} < 1\).

int
block_size
= 1¶ The block size in case the system matrix has a block structure.

float

class

class
amgcl::coarsening
::
nullspace_params
¶ The nullspace parameters.

int
cols
= 0¶ The number of near nullspace vectors.

std::vector<double>
B
¶ The near nullspace vectors. The vectors are represented as columns of a 2D matrix stored in rowmajor order.

int
Aggregation¶

template<class
Backend
>
classamgcl::coarsening
::
aggregation
¶ Include
<amgcl/coarsening/aggregation.hpp>
The nonsmoothed aggregation coarsening [Stue99].

class
params
¶ The aggregation coarsening parameters

amgcl::coarsening::pointwise_aggregates::params aggr;
The aggregation parameters

amgcl::coarsening::nullspace_params
nullspace
¶ The near nullspace parameters

float
over_interp
= 1.5¶ Overinterpolation factor \(\alpha\) [Stue99]. In case of aggregation coarsening, coarsegrid correction of smooth error, and by this the overall convergence, can often be substantially improved by using “overinterpolation”, that is, by multiplying the actual correction (corresponding to piecewise constant interpolation) by some factor \(\alpha > 1\). Equivalently, this means that the coarselevel Galerkin operator is rescaled by \(1/\alpha\):
\[I_h^HA_hI_H^h \to \frac{1}{\alpha}I_h^HA_hI_H^h.\]


class
Smoothed Aggregation¶

template<class
Backend
>
classamgcl::coarsening
::
smoothed_aggregation
¶ Include
<amgcl/coarsening/smoothed_aggregation.hpp>
The smoothed aggregation coarsening [VaMB96].

class
params
¶ The smoothed aggregation coarsening parameters

amgcl::coarsening::pointwise_aggregates::params aggr;
The aggregation parameters

amgcl::coarsening::nullspace_params
nullspace
¶ The near nullspace parameters

float
relax
= 1.0¶ The relaxation factor \(r\). Used as a scaling for the damping factor \(\omega\). When
estimate_spectral_radius
is set, then\[\omega = r * (4/3) / \rho.\]where \(\rho\) is the spectral radius of the system matrix. Otherwise
\[\omega = r * (2/3).\]The tentative prolongation \(\tilde P\) from the nonsmoothed aggregation is improved by smoothing to get the final prolongation matrix \(P\). Simple Jacobi smoother is used here, giving the prolongation matrix
\[P = \left( I  \omega D^{1} A^F \right) \tilde P.\]Here \(A^F = (a_{ij}^F)\) is the filtered matrix given by
\[\begin{split}\begin{aligned} a_{ij}^F &= \begin{cases} a_{ij} \quad \text{if} \; j \in N_i\\ 0 \quad \text{otherwise} \end{cases}, \quad \text{if}\; i \neq j, \\ \quad a_{ii}^F &= a_{ii}  \sum\limits_{j=1,j\neq i}^n \left(a_{ij}  a_{ij}^F \right), \end{aligned}\end{split}\]where \(N_i\) is the set of variables strongly coupled to variable \(i\), and \(D\) denotes the diagonal of \(A^F\).

bool
estimate_spectral_radius
= false¶ Estimate the matrix spectral radius. This usually improves the convergence rate and results in faster solves, but costs some time during setup.

int
power_iters
= 0¶ The number of power iterations to apply for the spectral radius estimation. Use Gershgorin disk theorem when
power_iters = 0
.


class
Smoothed Aggregation with Energy Minimization¶

template<class
Backend
>
classamgcl::coarsening
::
smoothed_aggr_emin
¶ Include
<amgcl/coarsening/smoothed_aggr_emin.hpp>
The smoothed aggregation with energy minimization coarsening [SaTu08].

class
params
¶ The smoothed aggregation with energy minimization coarsening parameters

amgcl::coarsening::pointwise_aggregates::params aggr;
The aggregation parameters

amgcl::coarsening::nullspace_params
nullspace
¶ The near nullspace parameters


class
Block to Scalar convertor¶

template<template<class> class
Coarsening
>
classamgcl::coarsening
::
as_scalar
¶ Include
<amgcl/coarsening/as_scalar.hpp>
Wrapper for the specified coarsening. Converts the input matrix from block to scalar format, applies the base coarsening, converts the resulting transfer operators back to block format. See the Using near nullspace vectors tutorial.

template <class Backend>

class
type
¶ The resulting coarsening class.
