LinearSolver::Richardson::Richardson< FieldsTag, OptionsGroup, SourceTag, ArraySectionIdTag > Struct Template Reference

A simple Richardson scheme for solving a system of linear equations \(Ax=b\). More...

#include <Richardson.hpp>

Public Types
using	fields_tag = FieldsTag
using	options_group = OptionsGroup
using	source_tag = SourceTag
using	operand_tag = fields_tag
using	component_list = tmpl::list<>
using	observed_reduction_data_tags
using	initialize_element
using	amr_projectors = initialize_element
using	register_element
template<typename ApplyOperatorActions, typename Label = OptionsGroup>
using	solve

Detailed Description

template<typename FieldsTag, typename OptionsGroup, typename SourceTag = db::add_tag_prefix<::Tags::FixedSource, FieldsTag>, typename ArraySectionIdTag = void>
struct LinearSolver::Richardson::Richardson< FieldsTag, OptionsGroup, SourceTag, ArraySectionIdTag >

A simple Richardson scheme for solving a system of linear equations \(Ax=b\).

Warning

This linear solver is useful only for basic preconditioning of another linear solver or for testing purposes. See LinearSolver::cg::ConjugateGradient or LinearSolver::gmres::Gmres for more useful general-purpose linear solvers.

In each step the solution is updated from its initial state \(x_0\) as

\[x_{k+1} = x_k + \omega \left(b - Ax\right) \]

where \(\omega\) is a relaxation parameter that weights the residual.

The scheme converges if the spectral radius (i.e. the largest absolute eigenvalue) of the iteration operator \(G=1-\omega A\) is smaller than one. For symmetric positive definite (SPD) matrices \(A\) with largest eigenvalue \(\lambda_\mathrm{max}\) and smallest eigenvalue \(\lambda_\mathrm{min}\) choose

\[\omega_\mathrm{SPD,optimal} = \frac{2}{\lambda_\mathrm{max} + \lambda_\mathrm{min}} \]

for optimal convergence.

Array sections

This linear solver requires no synchronization between elements, so it runs on all elements in the array parallel component. Partitioning of the elements in sections is only relevant for observing residual norms. Pass the section ID tag for the ArraySectionIdTag template parameter if residual norms should be computed over a section. Pass void (default) to compute residual norms over all elements in the array.

Member Typedef Documentation

initialize_element

template<typename FieldsTag, typename OptionsGroup, typename SourceTag = db::add_tag_prefix<::Tags::FixedSource, FieldsTag>, typename ArraySectionIdTag = void>

using LinearSolver::Richardson::Richardson< FieldsTag, OptionsGroup, SourceTag, ArraySectionIdTag >::initialize_element

Initial value:

 
      async_solvers::InitializeElement<FieldsTag, OptionsGroup, SourceTag>

observed_reduction_data_tags

template<typename FieldsTag, typename OptionsGroup, typename SourceTag = db::add_tag_prefix<::Tags::FixedSource, FieldsTag>, typename ArraySectionIdTag = void>

using LinearSolver::Richardson::Richardson< FieldsTag, OptionsGroup, SourceTag, ArraySectionIdTag >::observed_reduction_data_tags

Initial value:

 observers::make_reduction_data_tags<
      tmpl::list<async_solvers::reduction_data>>

register_element

template<typename FieldsTag, typename OptionsGroup, typename SourceTag = db::add_tag_prefix<::Tags::FixedSource, FieldsTag>, typename ArraySectionIdTag = void>

using LinearSolver::Richardson::Richardson< FieldsTag, OptionsGroup, SourceTag, ArraySectionIdTag >::register_element

Initial value:

 
      async_solvers::RegisterElement<FieldsTag, OptionsGroup, SourceTag,
                                     ArraySectionIdTag>

solve

template<typename FieldsTag, typename OptionsGroup, typename SourceTag = db::add_tag_prefix<::Tags::FixedSource, FieldsTag>, typename ArraySectionIdTag = void>

template<typename ApplyOperatorActions, typename Label = OptionsGroup>

using LinearSolver::Richardson::Richardson< FieldsTag, OptionsGroup, SourceTag, ArraySectionIdTag >::solve

Initial value:

 
      tmpl::list<async_solvers::PrepareSolve<FieldsTag, OptionsGroup, SourceTag,
                                             Label, ArraySectionIdTag>,
                 detail::UpdateFields<FieldsTag, OptionsGroup, SourceTag>,
                 ApplyOperatorActions,
                 async_solvers::CompleteStep<FieldsTag, OptionsGroup, SourceTag,
                                             Label, ArraySectionIdTag>>

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

• src/ParallelAlgorithms/LinearSolver/Richardson/Richardson.hpp