#include <deal.II/lac/precondition.h>

Inheritance diagram for PreconditionChebyshev< MatrixType, VectorType, PreconditionerType >:

Classes
struct	AdditionalData

Public Types
using	size_type = types::global_dof_index

Public Member Functions
void	initialize (const MatrixType &matrix, const AdditionalData &additional_data=AdditionalData())

void	vmult (VectorType &dst, const VectorType &src) const

void	Tvmult (VectorType &dst, const VectorType &src) const

void	step (VectorType &dst, const VectorType &src) const

void	Tstep (VectorType &dst, const VectorType &src) const

void	clear ()

size_type	m () const

size_type	n () const

Public Member Functions inherited from Subscriptor
	Subscriptor ()

	Subscriptor (const Subscriptor &)

	Subscriptor (Subscriptor &&) noexcept

virtual	~Subscriptor ()

Subscriptor &	operator= (const Subscriptor &)

Subscriptor &	operator= (Subscriptor &&) noexcept

void	subscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

void	unsubscribe (std::atomic< bool > *const validity, const std::string &identifier="") const

unsigned int	n_subscriptions () const

template<typename StreamType >
void	list_subscribers (StreamType &stream) const

void	list_subscribers () const

template<class Archive >
void	serialize (Archive &ar, const unsigned int version)

Private Member Functions
void	estimate_eigenvalues (const VectorType &src) const

Private Attributes
SmartPointer< const MatrixType, PreconditionChebyshev< MatrixType, VectorType, PreconditionerType > >	matrix_ptr

VectorType	solution_old

VectorType	temp_vector1

VectorType	temp_vector2

AdditionalData	data

double	theta

double	delta

bool	eigenvalues_are_initialized

Threads::Mutex	mutex

Additional Inherited Members
Static Public Member Functions inherited from Subscriptor
static ::ExceptionBase &	ExcInUse (int arg1, std::string arg2, std::string arg3)

static ::ExceptionBase &	ExcNoSubscriber (std::string arg1, std::string arg2)

Detailed Description

template<typename MatrixType = SparseMatrix<double>, typename VectorType = Vector<double>, typename PreconditionerType = DiagonalMatrix<VectorType>>
class PreconditionChebyshev< MatrixType, VectorType, PreconditionerType >

Preconditioning with a Chebyshev polynomial for symmetric positive definite matrices. This preconditioner is based on an iteration of an inner preconditioner of type PreconditionerType with coefficients that are adapted to optimally cover an eigenvalue range between the largest eigenvalue \(\lambda_{\max{}}\) down to a given lower eigenvalue \(\lambda_{\min{}}\) specified by the optional parameter smoothing_range. The algorithm is based on the following three-term recurrence:

\[ x^{n+1} = x^{n} + \rho_n \rho_{n-1} (x^{n} - x^{n-1}) + \frac{\rho_n}{\lambda_{\max{}}-\lambda_{\min{}}} P^{-1} (b-Ax^n). \]

where the parameter \(rho_0\) is set to \(rho_0 = \frac{\lambda_{\max{}}-\lambda_{\min{}}}{\lambda_{\max{}}+\lambda_{\min{}}}\) for the maximal eigenvalue \(\lambda_{\max{}}\) and updated via \(\rho_n = \left(2\frac{\lambda_{\max{}}+\lambda_{\min{}}} {\lambda_{\max{}}-\lambda_{\min{}}} - \rho_{n-1}\right)^{-1}\). The Chebyshev polynomial is constructed to strongly damp the eigenvalue range between \(\lambda_{\min{}}\) and \(\lambda_{\max{}}\) and is visualized e.g. in Utilities::LinearAlgebra::chebyshev_filter().

The typical use case for the preconditioner is a Jacobi preconditioner specified through DiagonalMatrix, which is also the default value for the preconditioner. Note that if the degree variable is set to one, the Chebyshev iteration corresponds to a Jacobi preconditioner (or the underlying preconditioner type) with relaxation parameter according to the specified smoothing range.

Besides the default choice of a pointwise Jacobi preconditioner, this class also allows for more advanced types of preconditioners, for example iterating block-Jacobi preconditioners in DG methods.

Apart from the inner preconditioner object, this iteration does not need access to matrix entries, which makes it an ideal ingredient for matrix-free computations. In that context, this class can be used as a multigrid smoother that is trivially parallel (assuming that matrix-vector products are parallel and the inner preconditioner is parallel). Its use is demonstrated in the step-37 and step-59 tutorial programs.

Estimation of the eigenvalues

The Chebyshev method relies on an estimate of the eigenvalues of the matrix which are computed during the first invocation of vmult(). The algorithm invokes a conjugate gradient solver so symmetry and positive definiteness of the (preconditioned) matrix system are requirements. The eigenvalue algorithm can be controlled by PreconditionChebyshev::AdditionalData::eig_cg_n_iterations specifying how many iterations should be performed. The iterations are started from an initial vector that depends on the vector type. For the classes Vector or LinearAlgebra::distributed::Vector, which have fast element access, it is either a vector with entries (-5.5, -4.5, -3.5, -2.5, ..., 3.5, 4.5, 5.5) with appropriate epilogue and adjusted such that its mean is always zero, which works well for the Laplacian. This setup is stable in parallel in the sense that for a different number of processors but the same ordering of unknowns, the same initial vector and thus eigenvalue distribution will be computed, apart from roundoff errors. For other vector types, the initial vector contains all ones, scaled by the length of the vector, except for the very first entry that is zero, triggering high-frequency content again.

The computation of eigenvalues happens the first time one of the vmult(), Tvmult(), step() or Tstep() functions is called. This is because temporary vectors of the same layout as the source and destination vectors are necessary for these computations and this information gets only available through vmult().

Due to the cost of the eigenvalue estimate in the first vmult(), this class is most appropriate if it is applied repeatedly, e.g. in a smoother for a geometric multigrid solver, that can in turn be used to solve several linear systems.

Bypassing the eigenvalue computation

In some contexts, the automatic eigenvalue computation of this class may result in bad quality, or it may be unstable when used in parallel with different enumerations of the degrees of freedom, making computations strongly dependent on the parallel configuration. It is possible to bypass the automatic eigenvalue computation by setting AdditionalData::eig_cg_n_iterations to zero, and provide the variable AdditionalData::max_eigenvalue instead. The minimal eigenvalue is implicitly specified via max_eigenvalue/smoothing_range.

Using the PreconditionChebyshev as a solver

If the range [max_eigenvalue/smoothing_range, max_eigenvalue] contains all eigenvalues of the preconditioned matrix system and the degree (i.e., number of iterations) is high enough, this class can also be used as a direct solver. For an error estimation of the Chebyshev iteration that can be used to determine the number of iteration, see Varga (2009).

In order to use Chebyshev as a solver, set the degree to numbers::invalid_unsigned_int to force the automatic computation of the number of iterations needed to reach a given target tolerance. In this case, the target tolerance is read from the variable PreconditionChebyshev::AdditionalData::smoothing_range (it needs to be a number less than one to force any iterations obviously).

For details on the algorithm, see section 5.1 of

@book{Varga2009,
  Title     = {Matrix iterative analysis},
  Author    = {Varga, R. S.},
  Publisher = {Springer},
  Address   = {Berlin},
  Edition   = {2nd},
  Year      = {2009},
}

Requirements on the templated classes

The class MatrixType must be derived from Subscriptor because a SmartPointer to MatrixType is held in the class. In particular, this means that the matrix object needs to persist during the lifetime of PreconditionChebyshev. The preconditioner is held in a shared_ptr that is copied into the AdditionalData member variable of the class, so the variable used for initialization can safely be discarded after calling initialize(). Both the matrix and the preconditioner need to provide vmult() functions for the matrix-vector product and m() functions for accessing the number of rows in the (square) matrix. Furthermore, the matrix must provide el(i,i) methods for accessing the matrix diagonal in case the preconditioner type is DiagonalMatrix. Even though it is highly recommended to pass the inverse diagonal entries inside a separate preconditioner object for implementing the Jacobi method (which is the only possible way to operate this class when computing in parallel with MPI because there is no knowledge about the locally stored range of entries that would be needed from the matrix alone), there is a backward compatibility function that can extract the diagonal in case of a serial computation.

Author: Martin Kronbichler, 2009, 2016; extension for full compatibility with LinearOperator class: Jean-Paul Pelteret, 2015

Definition at line 1006 of file precondition.h.