![]() |
Reference documentation for deal.II version 8.1.0
|
Macros | |
#define | DEAL_II_DEPRECATED __attribute__((deprecated)) |
Enumerations | |
enum | SparseBlockVanka< number >::BlockingStrategy { index_intervals, adaptive } |
Variables | |
PreconditionLACSolver | DEAL_II_DEPRECATED |
Preconditioners are used to accelerate the iterative solution of linear systems. Typical preconditioners are Jacobi, Gauss-Seidel, or SSOR, but the library also supports more complex ones such as Vanka or incomplete LU decompositions (ILU). In addition, sparse direct solvers can be used as preconditioners when available.
Broadly speaking, preconditioners are operators, which are multiplied with a matrix to improve conditioning. The idea is, that the preconditioned system P-1Ax = P-1b is much easier to solve than the original system Ax = b.What this means exactly depends on the structure of the matrix and cannot be discussed here in generality. For symmetric, positive definite matrices A and P, it means that the spectral condition number (the quotient of greatest and smallest eigenvalue) of P-1A is much smaller than the one of A.
At hand of the simplest example, Richardson iteration, implemented in SolverRichardson, the preconditioned iteration looks like
Accordingly, preconditioning amounts to applying a linear operator to the residual, and consequently, the action of the preconditioner P-1 is implemented as vmult()
. The generic interface is like for matrices
It is implemented in all the preconditioner classes in this module.
When used in Krylov space methods, it is up to the method, whether it simply replaces multiplications with A by those with P-1A (for instance SolverBicgstab), or does more sophisticated things. SolverCG for instance uses P-1 to define an inner product, which is the reason why it requires a symmetric, positive definite operator P.
Many preconditioners rely on an additive splitting A = P - N into two matrices. In this case, the iteration step of the Richardson method above can be simplified to
thus avoiding multiplication with A completely. We call operators mapping the previous iterate xk to the next iterate in this way relaxation operators. Their generic interface is
The classes with names starting with Relaxation
in this module implement this interface, as well as the preconditioners PreconditionJacobi, PreconditionSOR, PreconditionSSORP, reconditionBlockJacobi, PreconditionBlockSOR, and PreconditionBlockSSOR.
In this section, we discuss the interface preconditioners usually have to provide to work inside the deal.II library.
In order to be able to be stored in containers, all preconditioners have a constructor with no arguments. Since this will typically produce a useless object, all preconditioners have a function
This function receives the matrix to be preconditioned as well as additional required parameters and sets up the internal structures of the preconditioner.
Preconditioners in deal.II are just considered linear operators. Therefore, in order to be used by deal.II solvers, preconditioners must conform to the standard matrix interface, namely the functions
These functions apply the preconditioning operator to the source vector and return the result in
as
or
. Preconditioned iterative dolvers use these
vmult()
functions of the preconditioner. Some solvers may also use Tvmult()
.
Additional to the interface described above, some preconditioners like SOR and Jacobi have been known as iterative methods themselves. For them, an additional interface exists, consisting of the functions
Here, is a residual vector and
is the iterate that is supposed to be updated. In other words, the operation performed by these functions is
and
. The functions are called this way because they perform one step of a fixed point (Richardson) iteration. Note that preconditioners store a reference to the original matrix
during initialization.
enum SparseBlockVanka::BlockingStrategy |
Enumeration of the different methods by which the DoFs are distributed to the blocks on which we are to work.
Definition at line 531 of file sparse_vanka.h.