Fork/Join is a simple but effective technique for exploiting the parallelism. When developing a parallel program using Fork/Join, one of the main things is how a large task is decomposed into subtasks whose results can be combined as a final result. In this paper we show how to develop Fork/Join parallel programs through refinement and decomposition. We take Fork/Join style task decomposition as a refinement which we call Fork/Join refinement. Proof obligations of refinement can ensure the correctness of decomposition. For practical application, we provide a refinement pattern for the Fork/Join refinement and extend an atomicity decomposition diagram to illustrate it. Our approach provides a good framework for modeling Fork/Join parallel programs and showing proof obligations of correctness for such programs. We illustrate the approach by applying it on a small case.
In this study, a multivariate time series forecasting model applicable to high‐dimensional short‐term forecasting is proposed. The Reservoir Computing Network (RCN) state reduction method is investigated to address the dimensionality arising from high‐dimensional data. To enable the states to express very distant dependencies in time so that the RCN satisfies the echo state property, a bidirectional RCN model is investigated to capture the dependencies of the data forward and backward in time. To solve the short‐term data prediction, the spatiotemporal transformation theory is combined with the RCN to exploit the conjugate property between delayed and nondelayed embeddings of dynamical systems, and the correlation between variables in high‐dimensional systems is used to compensate for the shortcomings of short‐term series, design the method of the RCN training, and complete the prediction of future states. In the experiments, four existing models based on two different data sets were used for comparison, and the results showed that our method has better prediction performance.
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