Solving linear systems by methods based on a probabilistic interpretation

Solving Linear Systems by Methods Based on a Probabilistic Interpretation. In this paper it is demonstrated how the probabilistic concept of a stopping time in a random process may be used to generate an iterative method for solving a system of linear equations. Actually all known iterative approximation methods for solving linear equations are generated by various choices of a stopping time e. g. the point and block Jacobi methods, the point and block Gauss-Seidel Methods and overrelaxation methods are covered. The probabilistic approach offers -- in a natural way -- the possibility of adapting the solution technique to the special structure of the problem, Moreover, posterior bounds for the solution are constructed, which lead to faster convergence of the approximations than with usual prior bounds. Die Loisung linearer Systeme mit Hilfe eines probabilistischen Konzepts. Es wird gezeigt, wie das probabilistische Konzept der Stoppzeit eines Zufallsprozesses fur eine iterative Methode zur Losung eines Systems linearer Gleichungen herangezogen werden kann. Alle bekannten iterativen Naherungsmethoden zur Losung linearer Gleichungssysteme, wie z. B. die Blockmethoden nach Jakobi und Gauss-Seidel und Uberrelaxationsmethoden, entsprechen verschieden gewahlten Stoppzeiten. Das probabilistische Verfahren bietet die Moglichkeit, Lasungstechniken ffir die spezielle Struktur des Problems zu adaptieren. Darfiber hinaus ffihren die vorgeschlagenen Methoden zu einer schnelleren Konvergenz.