terra::linalg::solvers::FGMRES< OperatorT, PreconditionerT > Class Template Reference

Flexible GMRES (FGMRES) iterative solver for nonsymmetric linear systems. More...

#include <fgmres.hpp>

Public Types
using	OperatorType = OperatorT
	Operator type to be solved.
using	SolutionVectorType = SrcOf< OperatorType >
	Solution vector type.
using	RHSVectorType = DstOf< OperatorType >
	Right-hand side vector type.
using	ScalarType = typename SolutionVectorType::ScalarType
	Scalar type for computations.

Public Member Functions
	FGMRES (const std::vector< SolutionVectorType > &tmp, const FGMRESOptions< ScalarType > &options={}, const std::shared_ptr< util::Table > &statistics=nullptr, const PreconditionerT preconditioner=IdentitySolver< OperatorT >())
	Construct an FGMRES solver with a custom preconditioner.
void	set_tag (const std::string &tag)
	Set a tag string for statistics output identification.
void	set_restart (int m)
	Set the number of inner iterations before restart (FGMRES(m)).
void	solve_impl (OperatorType &A, SolutionVectorType &x, const RHSVectorType &b)
	Solve the linear system \( Ax = b \) using flexible GMRES with restarts.

Detailed Description

template<OperatorLike OperatorT, SolverLike PreconditionerT = IdentitySolver< OperatorT >>
class terra::linalg::solvers::FGMRES< OperatorT, PreconditionerT >

Flexible GMRES (FGMRES) iterative solver for nonsymmetric linear systems.

FGMRES allows for varying (flexible) preconditioning at each iteration, making it suitable for use with inexact or variable preconditioners.

Reference:

Saad, Y. (1993).
A flexible inner-outer preconditioned GMRES algorithm.
SIAM Journal on Scientific Computing, 14(2), 461-469.

Satisfies the SolverLike concept (see solver.hpp). Supports optional right preconditioning via the flexible framework.

Template Parameters

OperatorT	Operator type (must satisfy OperatorLike).
PreconditionerT	Preconditioner type (must satisfy SolverLike, defaults to IdentitySolver).

Member Typedef Documentation

OperatorType

template<OperatorLike OperatorT, SolverLike PreconditionerT = IdentitySolver< OperatorT >>

using terra::linalg::solvers::FGMRES< OperatorT, PreconditionerT >::OperatorType = OperatorT

Operator type to be solved.

RHSVectorType

template<OperatorLike OperatorT, SolverLike PreconditionerT = IdentitySolver< OperatorT >>

using terra::linalg::solvers::FGMRES< OperatorT, PreconditionerT >::RHSVectorType = DstOf< OperatorType >

Right-hand side vector type.

ScalarType

template<OperatorLike OperatorT, SolverLike PreconditionerT = IdentitySolver< OperatorT >>

using terra::linalg::solvers::FGMRES< OperatorT, PreconditionerT >::ScalarType = typename SolutionVectorType::ScalarType

Scalar type for computations.

SolutionVectorType

template<OperatorLike OperatorT, SolverLike PreconditionerT = IdentitySolver< OperatorT >>

using terra::linalg::solvers::FGMRES< OperatorT, PreconditionerT >::SolutionVectorType = SrcOf< OperatorType >

Solution vector type.

Constructor & Destructor Documentation

FGMRES()

template<OperatorLike OperatorT, SolverLike PreconditionerT = IdentitySolver< OperatorT >>

terra::linalg::solvers::FGMRES< OperatorT, PreconditionerT >::FGMRES ( const std::vector< SolutionVectorType > &  tmp, const FGMRESOptions< ScalarType > &  options = {}, const std::shared_ptr< util::Table > &  statistics = nullptr, const PreconditionerT  preconditioner = IdentitySolver< OperatorT >()  )

inline

Construct an FGMRES solver with a custom preconditioner.

Parameters

tmp	Temporary vectors for workspace. Must contain at least 2*restart + 4 vectors: tmp[0]: residual vector r tmp[1 .. restart+1]: Arnoldi basis vectors V_0..V_restart tmp[restart+2 .. 2*restart+1]: preconditioned directions Z_0..Z_{restart-1} tmp[2*restart+2]: work vector w for A*z tmp[2*restart+3]: accumulator vector for solution update
options	FGMRES solver parameters (restart, tolerances, max iterations).
statistics	Shared pointer to statistics table for logging iteration progress (optional).
preconditioner	Preconditioner solver (defaults to identity).

Member Function Documentation

set_restart()

template<OperatorLike OperatorT, SolverLike PreconditionerT = IdentitySolver< OperatorT >>

void terra::linalg::solvers::FGMRES< OperatorT, PreconditionerT >::set_restart ( int  m )

inline

Set the number of inner iterations before restart (FGMRES(m)).

Parameters

m	Restart parameter (must be at least 1).

set_tag()

template<OperatorLike OperatorT, SolverLike PreconditionerT = IdentitySolver< OperatorT >>

void terra::linalg::solvers::FGMRES< OperatorT, PreconditionerT >::set_tag ( const std::string &  tag )

inline

Set a tag string for statistics output identification.

Parameters

tag	Tag string to identify this solver instance in statistics.

solve_impl()

template<OperatorLike OperatorT, SolverLike PreconditionerT = IdentitySolver< OperatorT >>

void terra::linalg::solvers::FGMRES< OperatorT, PreconditionerT >::solve_impl ( OperatorType &  A, SolutionVectorType &  x, const RHSVectorType &  b  )

inline

Solve the linear system \( Ax = b \) using flexible GMRES with restarts.

Uses right preconditioning: solves \( A M^{-1} y = b \) where \( x = M^{-1} y \). The preconditioner M can vary at each iteration (flexibility).

The method builds an Arnoldi basis via modified Gram-Schmidt orthogonalization, applies Givens rotations to minimize the least-squares residual in the Krylov subspace, and updates the solution. If convergence is not achieved within the restart window, the process restarts with the updated solution until max_iterations is reached or convergence criteria are satisfied.

Parameters

A	Operator (matrix) to solve with.
x	Solution vector (input: initial guess, output: final solution).
b	Right-hand side vector (input).

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

• fgmres.hpp