Vorbedingungsoperatoren

Vorbedingungsoperatoren, often referred to as preconditioners, are auxiliary operators used in numerical linear algebra to improve the convergence rate of iterative methods for solving systems of linear equations. When a system Ax = b is ill-conditioned, meaning the condition number of A is large, standard iterative solvers like the conjugate gradient method can converge very slowly. A preconditioner M is an operator that is applied to the system in such a way that the preconditioned system M⁻¹Ax = M⁻¹b or AM⁻¹x = b has a much smaller condition number than the original system. The goal is to find a preconditioner M such that M⁻¹A (or AM⁻¹) is close to the identity matrix, or at least has eigenvalues clustered around 1.

The effectiveness of a preconditioner depends on its ability to approximate the inverse of A (or a significant part of it) while being computationally inexpensive to apply. Ideally, the preconditioner M should be easy to invert, so that solving Mx = y is much faster than solving Ax = b. Common types of preconditioners include diagonal preconditioning (M is the diagonal of A), incomplete LU factorization (ILU), and SOR (Successive Over-Relaxation). The choice of preconditioner often depends on the properties of the matrix A, such as its sparsity, structure, and condition number.

Iterative methods, when combined with a suitable preconditioner, can solve large-scale linear systems much more efficiently than direct methods. The preconditioned system is then solved iteratively. For example, in preconditioned conjugate gradient, each iteration involves a matrix-vector product with A and a solve with M. The success of preconditioning lies in transforming an ill-conditioned problem into a well-conditioned one, thereby accelerating the convergence of the iterative solver.