Preconditioner

In numerical mathematics preconditioning refers to a technique by which a problem is transformed so that the solution remains, however, there are positive features like better condition or faster convergence for the chosen numerical solution methods.

The most common form of the linear pre-conditioning, in which a linear equation system is transformed equivalent. This type of preconditioning is used in particular in solving the system of equations by means of Krylov subspace methods application. Another important form is produced by multiplying the time derivative term of a partial differential equation with a non-linear pre-conditioning. Here, the stationary solution of the equation is preserved.

Linear preconditioning

Here, one distinguishes between Linksvorkonditionierung, wherein the system of equations is multiplied from the left with a regular matrix: and Rechtsvorkonditionierung, wherein the equation system is solved with. The preconditioner should approximate the inverse of A as possible with the least possible effort. In principle, any iterative equation solution methods such as the Jacobi or the Gauss- Seidel method is used as a preconditioner, while the matrix for preconditioning the matrix as described in the article -splitting procedure.

In the context of Krylov subspace methods such as the CG method, it is advantageous if the system matrix is ​​a low condition, or in particular, has a " good " eigenvalue distribution. Here is the main application of preconditioners to find, as the rate of convergence of Krylov subspace methods can thus be significantly improved.

In addition to the already above-mentioned iterative method are incomplete LU factorization, called ILU factorization of particular interest. These are calculated using the Gaussian algorithm, a faulty decomposition of the system matrix A can be calculated using only fixed elements to save time and memory.

Since the 1990s, multilevel process gain as algebraic multigrid methods is becoming increasingly important.

A simple example is the equilibration, that is, the scaling of the rows or columns of the equation system with individual factors, such that all columns or rows of the matrix, then have the same column or row sum norm.

Nonlinear preconditioner

The computation of stationary solutions of a partial differential equation can be made more efficient by means of nonlinear preconditioning. For this purpose, the time derivative multiplied by a preconditioner, that is, the time is slow or fast for certain cells or variable. This is done to avoid the CFL condition for stiff problems especially.