On the convergence of the barycentric method in solving internal Dirichlet and Neumann problems in $\mathbb{R}^2$ for the Helmholtz equation

The application of the barycentric method for the numerical solution of Dirichlet and Neumann problems for the Helmholtz equation in the bounded simply connected domain $\Omega\subset\mathbb{R}^2$ is considered. The main assumption in the solution is to set the $\Omega$ boundary in a piecewise linear representation. A distinctive feature of the barycentric method is the order of formation of a global system of vector basis functions for $\Omega$ via barycentric coordinates. The existence and uniqueness of the solution of Dirichlet and Neumann problems for the Helmholtz equation by the barycentric method are established and the convergence rate estimate is determined. The features of the algorithmic implementation of the method are clarified.

A Shapelet Transform Classification over Uncertain Time Series

A shapelet is a time-series subsequence that can represent local, phase-independent similarity in shape. Time series classification with subsequences can save computing cost, improve computing speed and improve algorithm accuracy. The shapelet-based approaches for time series classification have an advantage of interpretability. Concentrating on uncertain time series, this paper tries to apply the shapelet-based method to classify uncertain time series. Due to the high dimensions of time series, the number of the generated candidate shapelets is generally huge. As a result, the calculation amount is large too. To deal with this problem, in this paper, we introduce a piecewise linear representation (PLR) method for uncertain time series based on key points so that the traditional shapelet discovery algorithm can be improved efficiently. We verify our approach with experiments. The experimental results show that the proposed shapelet algorithm can be used for uncertain time series and it can provide classification accuracy well while reducing time cost.