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In this paper, a new approach based on least squares support vector machines (LS-SVMs) is proposed for solving linear and nonlinear ordinary differential equations (ODEs). The approximate solution is presented in closed form by means of LS-SVMs, whose parameters are adjusted to minimize an appropriate error function. For the linear and nonlinear cases, these parameters are obtained by solving a system of linear and nonlinear equations, respectively. The method is well suited to solving mildly stiff, nonstiff, and singular ODEs with initial and boundary conditions. Numerical results demonstrate the efficiency of the proposed method over existing methods.

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This paper proposes a multiclass semisupervised learning algorithm by using kernel spectral clustering (KSC) as a core model. A regularized KSC is formulated to estimate the class memberships of data points in a semisupervised setting using the one-versus-all strategy while both labeled and unlabeled data points are present in the learning process. The propagation of the labels to a large amount of unlabeled data points is achieved by adding the regularization terms to the cost function of the KSC formulation. In other words, imposing the regularization term enforces certain desired memberships. The model is then obtained by solving a linear system in the dual. Furthermore, the optimal embedding dimension is designed for semisupervised clustering. This plays a key role when one deals with a large number of clusters.

Structural health monitoring refers to the process of measuring damagesensitive variables to assess the functionality of a structure. In principle, vibration data can capture the dynamics of the structure and reveal possible failures, but environmental and operational variability can mask this information. Thus, an effective outlier detection algorithm can be applied only after having performed data normalization (i.e. filtering) to eliminate external influences. Instead, in this article we propose a technique which unifies the data normalization and damage detection steps. The proposed algorithm, called adaptive kernel spectral clustering (AKSC), is initialized and calibrated in a phase when the structure is undamaged. The calibration process is crucial to ensure detection of early damage and minimize the number of false alarms. After the calibration, the method can automatically identify new regimes which may be associated with possible faults. These regimes are discovered by means of two complementary damage (i.e. outlier) indicators. The proposed strategy is validated with a simulated example and with real-life natural frequency data from the Z24 pre-stressed concrete bridge, which was progressively damaged at the end of a one-year monitoring period.

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