Asymptotic Mean Squared Error of Noisy Periodical Successive Over-Relaxation

Chebyshev-periodical successive over-relaxation was recently proposed as a method of accelerating the convergence speed of fixed-point iterations. If a PSOR iteration is influenced by stochastic disturbances, such as Gaussian noise, then the behavior of the PSOR iteration deviates from the predicted behavior of the noiseless iterations, i.e., the convergence behavior of the Chebyshev-PSOR is highly sensitive to the noises. This paper presents a concise formula for the asymptotic mean squared error (AMSE) of the noisy PSOR iterations. A PSOR iteration can be regarded as a stochastic difference equation and spectral decomposition plays a key role to reveal the asymptotic behaviors of the error covariance. Based on the AMSE formula, a noise mitigation method is developed to reduce the effects of the stochastic disturbance.