Reference documentation for deal.II version 9.4.0

#include <deal.II/lac/precondition.h>
Classes  
struct  AdditionalData 
struct  EigenvalueInformation 
Public Types  
using  size_type = types::global_dof_index 
Public Member Functions  
PreconditionChebyshev ()  
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 
EigenvalueInformation  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 
Subscriptor functionality  
Classes derived from Subscriptor provide a facility to subscribe to this object. This is mostly used by the SmartPointer class.  
std::atomic< unsigned int >  counter 
std::map< std::string, unsigned int >  counter_map 
std::vector< std::atomic< bool > * >  validity_pointers 
const std::type_info *  object_info 
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) 
using  map_value_type = decltype(counter_map)::value_type 
using  map_iterator = decltype(counter_map)::iterator 
static ::ExceptionBase &  ExcInUse (int arg1, std::string arg2, std::string arg3) 
static ::ExceptionBase &  ExcNoSubscriber (std::string arg1, std::string arg2) 
void  check_no_subscribers () const noexcept 
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 threeterm recurrence:
\[ x^{n+1} = x^{n} + \alpha^n_0 (x^{n}  x^{n1}) + \alpha^n_1 P^{1} (bAx^n) \quad\text{with}\quad \alpha^0_0 := 0,\; \alpha^0_1 := \frac{2\rho_0}{\lambda_{\max}\lambda_{\min}}\; \alpha^n_0 := \rho_n \rho_{n1},\;\text{and}\; \alpha^n_1 := \frac{4\rho_n}{\lambda_{\max}\lambda_{\min}}, \]
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 \(\rho_n = \left(2\frac{\lambda_{\max{}}+\lambda_{\min{}}} {\lambda_{\max{}}\lambda_{\min{}}}  \rho_{n1}\right)^{1}\). 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 blockJacobi 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 matrixfree computations. In that context, this class can be used as a multigrid smoother that is trivially parallel (assuming that matrixvector products are parallel and the inner preconditioner is parallel). Its use is demonstrated in the step37 and step59 tutorial programs.
The Chebyshev method relies on an estimate of the eigenvalues of the matrix which are computed during the first invocation of vmult(). This class offers several algorithms to this end, see PreconditionChebyshev::AdditionalData::EigenvalueAlgorithm. The default algorithm invokes the Lanczos method via the SolverCG class, which requires symmetry and positive definiteness of the (preconditioned) matrix system are required. Also note that deal.II needs to be configured with LAPACK support to use this option. The eigenvalue algorithm can be controlled by PreconditionChebyshev::AdditionalData::eig_cg_n_iterations specifying how many iterations should be performed. For all algorithms, the iterative process is 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 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 highfrequency content again.
The computation of eigenvalues happens the first time one of the vmult(), Tvmult(), step() or Tstep() functions is called or when estimate_eigenvalues() is called directly. In the latter case, it is necessary to provide a temporary vector of the same layout as the source and destination vectors used during application of the preconditioner.
The estimates for minimum and maximum eigenvalue are taken from the underlying solver or eigenvalue algorithm in the given number of iterations, even if the solver did not converge in the requested number of iterations. Finally, the maximum eigenvalue is multiplied by a safety factor of 1.2.
Due to the cost of the eigenvalue estimate, 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.
In some contexts, the automatic eigenvalue computation of this class may result in a bad quality, e.g. when the polynomial basis or numbering of unknowns is such that the initial vector described above is a bad choice. 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
.
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. Note that this currently only works for symmetric positive definite matrices with the eigenvalue algorithm set to the conjugate gradient algorithm. 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
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 matrixvector 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.
MatrixType
argumentThis class enables to embed the vector updates into the matrixvector product in case the MatrixType
supports this. To this end, the VectorType
needs to be of type LinearAlgebra::distributed::Vector, the PreconditionerType
needs to be DiagonalMatrix, and the class MatrixType
needs to provide a function with the signature
where the two given functions run before and after the matrixvector product, respectively. They take as arguments a subrange among the locally owned elements of the vector, defined as halfopen intervals. The intervals are designed to be scheduled close to the time the matrixvector product touches upon the entries in the src
and dst
vectors, respectively, with the requirement that
src
or dst
once the operation_before_matrix_vector_product
has been run on that vector entry; operation_after_matrix_vector_product
may run on a range of entries [i,j)
once the matrixvector product does not access the entries [i,j)
in src
and dst
any more. The motivation for this function is to increase data locality and hence cache usage. For the example of a class similar to the one in the step37 tutorial program, the implementation is
In terms of the Chebyshev iteration, the operation before the loop will set dst
to zero, whereas the operation after the loop performs the iteration leading to \(x^{n+1}\) described above, modifying the dst
and src
vectors.
Definition at line 1594 of file precondition.h.