Abstract: The mechanical quadrature methods for solving the boundary integral equations of the anisotropic Darcy-s equations with Dirichlet conditions in smooth domains are presented. By applying the collectively compact theory, we prove the convergence and stability of approximate solutions. The asymptotic expansions for the error show that the methods converge with the order O (h3), where h is the mesh size. Based on these analysis, extrapolation methods can be introduced to achieve a higher convergence rate O (h5). An a posterior asymptotic error representation is derived in order to construct self-adaptive algorithms. Finally, the numerical experiments show the efficiency of our methods.
Abstract: The least mean square (LMS) algorithmis one of the
most well-known algorithms for mobile communication systems
due to its implementation simplicity. However, the main limitation
is its relatively slow convergence rate. In this paper, a booster
using the concept of Markov chains is proposed to speed up the
convergence rate of LMS algorithms. The nature of Markov
chains makes it possible to exploit the past information in the
updating process. Moreover, since the transition matrix has a
smaller variance than that of the weight itself by the central limit
theorem, the weight transition matrix converges faster than the
weight itself. Accordingly, the proposed Markov-chain based
booster thus has the ability to track variations in signal
characteristics, and meanwhile, it can accelerate the rate of
convergence for LMS algorithms. Simulation results show that the
LMS algorithm can effectively increase the convergence rate and
meantime further approach the Wiener solution, if the
Markov-chain based booster is applied. The mean square error is
also remarkably reduced, while the convergence rate is improved.