A variational method and approximations of a Cauchy problem for elliptic equations

A Cauchy problem for general elliptic second-order linear partial differential equations in which the Dirichlet data in H½(?1 ? ?3) is assumed available on a larger part of the boundary ? of the bounded domain O than the boundary portion ?1 on which the Neumann data is prescribed, is investigated using a conjugate gradient method. We obtain an approximation to the solution of the Cauchy problem by minimizing a certain discrete functional and interpolating using the finite diference or boundary element method. The minimization involves solving equations obtained by discretising mixed boundary value problems for the same operator and its adjoint. It is proved that the solution of the discretised optimization problem converges to the continuous one, as the mesh size tends to zero. Numerical results are presented and discussed.

A procedure for the reconstruction of a stochastic stationary temperature field

An iterative procedure is proposed for the reconstruction of a temperature field from a linear stationary heat equation with stochastic coefficients, and stochastic Cauchy data given on a part of the boundary of a bounded domain. In each step, a series of mixed well-posed boundary-value problems are solved for the stochastic heat operator and its adjoint. Well-posedness of these problems is shown to hold and convergence in the mean of the procedure is proved. A discretized version of this procedure, based on a Monte Carlo Galerkin finite-element method, suitable for numerical implementation is discussed. It is demonstrated that the solution to the discretized problem converges to the continuous as the mesh size tends to zero.

Application of IEEE 802.16 mesh networks as the backhaul of multihop cellular networks

Cellular networks have been widely used to support many new audio-and video-based multimedia applications. The demand for higher data rate and diverse services has driven the research on multihop cellular networks (MCNs). With its ad hoc network features, an MCN can offer many additional advantages, such as increased network throughput, scalability and coverage. However, providing ad hoc capability to MCNs is challenging as it may require proper wireless interfaces. In this article, the architecture of IEEE 802.16 network interface to provide ad hoc capability for MCNs is investigated, with its focus on the IEEE 802.16 mesh networking and scheduling. Several distributed routing algorithms based on network entry mechanism are studied and compared with a centralized routing algorithm. It is observed from the simulation results that 802.16 mesh networks have limitations on providing sufficient bandwidth for the traffic from the cellular base stations when a cellular network size is relatively large. © 2007 IEEE.