A Comparative Study of Linearization Techniques With Adaptive Block Hybrid Method for Solving First‐Order Initial Value Problems

Solution of nonlinear partial differential equations from base equations

A popular and easy to implement classifier is the k-Nearest Neighbor (k-NN). However, sequentially searching for nearest neighbors in large datasets leads to inefficient classification because of the high computational cost involved. This paper presents an adaptive hybrid and cluster-based method for speeding up the k-NN classifier. The proposed method reduces the computational cost as much as possible while maintaining classification accuracy at high levels. The method is based on the wellknown k-means clustering algorithm and consists of two main parts: (i) a preprocessing algorithm that builds a two-level, cluster-based data structure, and, (ii) a hybrid classifier that classifies new items by accessing either the first or the second level of the data structure. The proposed approach was tested on seven real life datasets and the experiential measurements were statistically validated by the Wilcoxon signed ranks test. The results show that the proposed classification method can be used either to achieve high accuracy with slightly higher cost or to reduce the cost at a minimum level with slightly lower accuracy.

Linearization techniques for singularly-perturbed initial-value problems of ordinary differential equations

Multi-objective airfoil shape optimization using an adaptive hybrid evolutionary algorithm

This work develops a method for solving nonlinear evolution equations. The method, termed a bivariate spectral linear partition method (BSLPM), combines the Chebyshev spectral collocation method, bivariate Lagrange interpolation, and a linear partition technique as an underlying linearization method. It is developed for an th‐order nonlinear differential equation and then used to solve three known evolution problems. The results are compared with known exact solutions from literature. The method's applicability, reliability, and accuracy are confirmed by the congruence between the numerical and exact solutions. Tables, error graphs, and convergence graphs were generated using MATLAB (R2015a), to confirm the order of accuracy of the method and verify its convergence. The performance of the method is also observed against other methods performing well in these types of differential equations and is found to be comparable in terms of accuracy. The proposed method is also efficient as it uses minimal computation time.

This paper aims to develop a load forecasting method for short-term load forecasting, based on an adaptive two-stage hybrid network with self-organized map (SOM) and support vector machine (SVM). In the first stage, a SOM network is applied to cluster the input data set into several subsets in an unsupervised manner. Then, groups of 24 SVMs for the next day's load profile are used to fit the training data of each subset in the second stage in a supervised way. The proposed structure is robust with different data types and can deal well with the nonstationarity of load series. In particular, our method has the ability to adapt to different models automatically for the regular days and anomalous days at the same time. With the trained network, we can straightforwardly predict the next-day hourly electricity load. To confirm the effectiveness, the proposed model has been trained and tested on the data of the historical energy load from New York Independent System Operator.

This paper presents an adaptive hybrid algorithm to invert ocean acoustic field measurements for seabed geoacoustic parameters. The inversion combines a global search (simulated annealing) and a local method (downhill simplex), employing an adaptive approach to control the trade off between random variation and gradient-based information in the inversion. The result is an efficient and effective algorithm that successfully navigates challenging parameter spaces including large numbers of local minima, strongly correlated parameters, and a wide range of parameter sensitivities. The algorithm is applied to a set of benchmark test cases, which includes inversion of simulated measurements with and without noise, and cases where the model parameterization is known and where the parameterization most be determined as part of the inversion. For accurate data, the adaptive inversion often produces a model with a Bartlett mismatch lower than the numerical error of the propagation model used to compute the replica fields. For noisy synthetic data, the inversion produces a model with a mismatch that is lower than that for the true parameters. Comparison with previous inversions indicates that the adaptive hybrid method provides the best results to date for the benchmark cases.

This paper is devoted to the description of recent and prospective decorrelation techniques used in image processing. Two classes are taken into consideration : the universal or generic methods and the adaptive methods. The first class performs signal processing independently of its specificity. These methods only rely on statistical properties of the signal. They are principally based on orthogonal transforms and subband coding. In opposition, the adaptive techniques take the a posteriori contents of the images into account. Some of them adapt universal techniques to the signal content (i.e. adaptive subband coding or hybrid methods). Others are based on feature extraction processes which include more physical entities like edges and regions. For example, object approaches attempt to describe images as collections of regions characterized by their shape and their aspect.

As a typical safety-critical system, satellite has extremely stringent requirements for reliability. However, some satellite components, especially attitude sensors, are prone to suffer from performance degradation or even drastic failure in space environment, which leads to serious threats to satellite. A series of fault diagnosis methods have been investigated to diagnose these failures. However, due to the increasingly complex structure and various operating conditions of satellite, it is still a challenging task to diagnose attitude sensor failures robustly and reliably. In addition, sophisticated diagnosis methods result in heavy calculations, which may not satisfy satellite autonomy requirements. Given these issues, this article proposes an integrated fault detection, isolation, and reconstruction (FDIR) scheme for attitude sensors, such as fiber-optic gyroscopes (FOGs) and star sensor, using an adaptive hybrid method. The parity equation (PE) approach is applied to detect gyros fault. On the basis of attitude kinematics model and sensor measurement equation, a novel adaptive extended Kalman filter (EKF) is developed simultaneously. Then, the chi-square test is established using an innovation sequence for detecting sensor fault. According to fault detection results, different reconstruction strategies are presented by autonomously tuning noise covariance parameters, which will be used to reconstruct the faults of gyros and star sensor. Combined with the PE approach and chi-square test, faults in gyroscopes and star sensor can be detected reliably. Then, using this adaptive algorithm, these failures can be isolated and reconstructed accurately. Finally, the effectiveness of the presented method is verified by simulating typical faults of gyros and star sensor.

The unified transform method (UTM) provides a novel approach to the analysis of initial boundary value problems for linear as well as for a particular class of nonlinear partial differential equations called integrable. If the latter equations are formulated in two dimensions (either one space and one time, or two space dimensions), the UTM expresses the solution in terms of a matrix Riemann–Hilbert (RH) problem with explicit dependence on the independent variables. For nonlinear integrable evolution equations, such as the celebrated nonlinear Schrödinger (NLS) equation, the associated jump matrices are computed in terms of the initial conditions and all boundary values. The unknown boundary values are characterized in terms of the initial datum and the given boundary conditions via the analysis of the so-called global relation. In general, this analysis involves the solution of certain nonlinear equations. In certain cases, called linearizable, it is possible to bypass this nonlinear step. In these cases, the UTM solves the given initial boundary value problem with the same level of efficiency as the well-known inverse scattering transform solves the initial value problem on the infinite line. We show here that the initial boundary value problem on a finite interval with x-periodic boundary conditions (which can alternatively be viewed as the initial value problem on a circle) belongs to the linearizable class. Indeed, by employing certain transformations of the associated RH problem and by using the global relation, the relevant jump matrices can be expressed explicitly in terms of the so-called scattering data, which are computed in terms of the initial datum. Details are given for NLS, but similar considerations are valid for other well-known integrable evolution equations, including the Korteweg–de Vries (KdV) and modified KdV equations.

Scanning acoustic microscopy based on focused ultrasound waves is a promising new tool in medical imaging. In this work we apply an adaptive hybrid FEM/FDM (finite element methods/finite difference methods) method to an inverse scattering problem for the time-dependent acoustic wave equation, where one seeks to reconstruct an unknown sound velocity c(x) from a single measurement of wave-reflection data on a small part of the boundary, e.g., to detect pathological defects in bone. Typically, this corresponds to identifying an unknown object (scatterer) in a surrounding homogeneous medium. The inverse problem is formulated as an optimal control problem, where we use an adjoint method to solve the equations of optimality expressing stationarity of an associated augmented Lagrangian by a quasi-Newton method. To treat the problem of multiple minima of the objective function, the optimization procedure is first performed on a coarse grid to smooth the high frequency error, generating a starting point for optimization steps on successively refined meshes. Local refinement based on the results of previous steps will improve computational efficiency of the method. As the main result then, an a posteriori error estimate is proved for the error in the Lagrangian, and a corresponding adaptive method is formulated, where the finite element mesh is refined from residual feedback. The performance of the adaptive hybrid method and the usefulness of the a posteriori error estimator for problems with limited boundary data are illustrated in three dimensional numerical examples.

<i>Self-Similarity and Beyond: Exact Solutions of Nonlinear Problems</i>

Multi-step transversal and tangential linearization methods applied to a class of nonlinear beam equations

This chapter provides a simple Adaptive Local Variational Iteration Method (ALVIM) that can efficiently solve nonlinear differential equations and orbital problems of spacecrafts. Based on a general first-order form of nonlinear differential equations, the iteration formula is analytically derived and then discretized using Chebyshev polynomials as basis functions in the time domain. It leads to an iterative numerical algorithm that only involves the addition and multiplication of sparse matrices. Moreover, the Jacobian matrix is free from inversing. Apart from that, a straightforward adaptive scheme is proposed to refine the configuration of the algorithm, involving the length of time steps and the number of collocation nodes in a time step. With the adaptive scheme, the prescribed accuracy can be guaranteed without manually tuning the configuration of the algorithm. Since the refinement is adjusted automatically, our algorithm reduces overcalculation for smooth and slowly changing problems. Examples such as large amplitude pendulum and perturbed two-body problem are used to verify this easy-to-use adaptive method's high accuracy and efficiency.&lt;br&gt;

The two-time perturbation procedure is applied to a class of inhomogeneous initial boundary value problems for weakly nonlinear partial differential equations of wave type. An operator calculus approach is adopted to treat the problem as an initial value problem in Hilbert space. The first order equation is applied to two problems in the forced vibration of a nonlinear string. They are governed by the generalized Duffing and van der Pol equations respectively. In the case of the Duffing equation, the results show that there exists a stable, periodic vibration when small damping is present and how the system approaches this final state in the course of time. In the absence of damping, the system does not in general possess a periodic motion. For the van der Pol equation, our results generalize that of Kelley and Kogelman [1]. Here we find that there exist self excited motions in the form of many periodic motions superimposed upon an externally excited periodic motion. The dependence of the final motions on the initial conditions and the forcing function is shown explicitly. A sufficient condition is given for the existence of certain doubly periodic motions as a stable limiting solution. In the Appendix the validity of the method is proved.