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An automatic L1 regularized parameter is proposed with monotonicity property. We consider the L1 regularized iterative forward backward splitting (IFBS) algorithm for image restoration. The main aim of the paper is to develop a fast and adaptive method with an automatic selection of regularization parameter. The regularization parameter is updated in each iteration without requiring the initial value of the parameter. We establish analytically monotonicity results of the adaptive parameter. Such an algorithm corresponds to solving an adaptive iterative thresholding process. Experimental results demonstrate that the adaptive parameter method is efficient and provide competitive performance.

Synchronization in classes of continuous-time dynamical unweighted networks with different topologies is investigated. A synchronization-optimal network model based on rewiring of links is proposed. Compared with other networks, it exhibits a stronger synchronizability. We presented link density and investigated the correlation of synchronizability, link density and heterogeneity in degree distribution. In this work, it will be shown that synchronizability of Type I networks is independent of heterogeneity in the degree distribution when the link density is smaller than 0.02. Synchronizability and link density are proportional. When the link density is larger than 0.025, heterogeneity decides the curve slope. The synchronizability of Type II networks is drastically enhanced by enhancing the link density when the link density is smaller than 0.025. However synchronizability increases weakly by enhancing the link density when the link density is larger than 0.025.

Synchronization in classes of continuous-time dynamical networks with different topologies is investigated. A synchronization-optimal algorithm based on the synchronization criterion is proposed so as to get the appropriate topology such that complex network has the optimal synchronizability. It has been argued that heterogeneity suppresses synchronization in unweighted networks. However, it is shown in this work that synchronizability of Type I networks is independent of heterogeneity in the degree distribution, while synchronizability of Type II networks is dependent of both heterogeneity and scale of networks. It is presented the more heterogeneous and larger scale, the poorer synchronizability of Type II networks.