Arbitrarily Oriented Phase Randomization of Design Ground Motions by Continuous Wavelets

For dynamic analysis in seismic design, selection of input ground motions is of huge importance. In the presented scheme, complex Continuous Wavelet Transform (CWT) is utilized to simulate stochastic ground motions from historical records of earthquakes with phase disturbance arbitrarily localized in time-frequency domain. The complex arguments of wavelet coefficients are determined as phase spectrum and an innovative formulation is constructed to improve computational efficiency of inverse wavelet transform with a pair of random complex arguments introduced and make more candidate wavelets available in the article. The proposed methodology is evaluated by numerical simulations on a two-degree-of-freedom system including spectral analysis and dynamic analysis with Shannon wavelet basis and Gabor wavelet basis. The result shows that the presented scheme enables time-frequency range of disturbance in time-frequency domain arbitrarily oriented and complex Shannon wavelet basis is verified as the optimal candidate mother wavelet for the procedure in case of frequency information maintenance with phase perturbation.

Natural Graph Wavelet Packet Dictionaries

We introduce a set of novel multiscale basis transforms for signals on graphs that utilize their “dual” domains by incorporating the “natural” distances between graph Laplacian eigenvectors, rather than simply using the eigenvalue ordering. These basis dictionaries can be seen as generalizations of the classical Shannon wavelet packet dictionary to arbitrary graphs, and do not rely on the frequency interpretation of Laplacian eigenvalues. We describe the algorithms (involving either vector rotations or orthogonalizations) to construct these basis dictionaries, use them to efficiently approximate graph signals through the best basis search, and demonstrate the strengths of these basis dictionaries for graph signals measured on sunflower graphs and street networks.

Model Predictive Control for First-Order Hyperbolic System Based on Quasi-Shannon Wavelet Basis

In this paper, we consider a class of first-order hyperbolic distributed parameter systems. Our focus is on the design of a new class of model predictive control schemes using a quasi-Shannon wavelet basis. First, the first-order hyperbolic distributed parameter system is transformed into an equivalent system using collocation techniques for the approximation of spatial derivatives and Euler forward difference method for the approximation of the time component. Then, a model reduction method is applied to obtain a reduced-order system on which a nonlinear model predictive controller is designed through solving a nonlinear quadratic programming problem with input constraints. For illustration, the temperature control of a flow-control long-duct heating system is considered to be an example. A comparative simulation study is conducted to demonstrate the effectiveness of the proposed method.

Point-Symmetric Extension-Based Interval Shannon-Cosine Spectral Method for Fractional PDEs

The approximation accuracy of the wavelet spectral method for the fractional PDEs is sensitive to the order of the fractional derivative and the boundary condition of the PDEs. In order to overcome the shortcoming, an interval Shannon-Cosine wavelet based on the point-symmetric extension is constructed, and the corresponding spectral method on the fractional PDEs is proposed. In the research, a power function of cosine function is introduced to modulate Shannon function, which takes full advantage of the waveform of the Shannon function to ensure that many excellent properties can be satisfied such as the partition of unity, smoothness, and compact support. And the interpolative property of Shannon wavelet is held at the same time. Then, based on the point-symmetric extension and the general variational theory, an interval Shannon-Cosine wavelet is constructed. It is proved that the first derivative of the approximated function with this interval wavelet function is continuous. At last, the wavelet spectral method for the fractional PDEs is given by means of the interval Shannon-Cosine wavelet. By means of it, the condition number of the discrete matrix can be suppressed effectively. Compared with Shannon and Shannon-Gabor wavelet quasi-spectral methods, the novel scheme has stronger applicability to the shockwave appeared in the solution besides the higher numerical accuracy and efficiency.

A Shannon Wavelet Method for Pricing American Options under Two-Factor Stochastic Volatilities and Stochastic Interest Rate

In the paper, the pricing of the American put options under the double Heston model with Cox–Ingersoll–Ross (CIR) interest rate process is studied. The characteristic function of the log asset price is derived, and thereby Bermuda options are well evaluated by means of a state-of-the-art Shannon wavelet inverse Fourier technique (SWIFT), which is a robust and highly efficient pricing method. Based on the SWIFT method, the price of American option can be approximated by using Richardson extrapolation schemes on a series of Bermudan options. Numerical experiments show that the proposed pricing method is efficient, especially for short-term American put options.

A fast filtering method based on adaptive impulsive wavelet for the gear fault diagnosis

The useful fault features applied for the fault diagnosis are usually overwhelmed by noise and other interference factors in rotation machinery. The impulses masked in vibration signals can represent the faults of gears or bearings in a gearbox. The key to finding impulsive components is to identify the modeling parameters (such as damping ratio, central frequency) of a transient (Morlet wavelet, Laplace wavelet), which can be used as an adaptive filter to denoise the vibration signal. However, its engineering application is limited by the time-consuming computation. In order to tackle this issue, a fast algorithm based on an adaptive impulsive wavelet is proposed to filter the fault signal so that the fault characteristic frequency can be identified. Firstly, a correlation coefficient maximum criterion is employed to find one of the optimal parameters of the impulsive wavelet. Then, the other parameter is optimized by the minimum Shannon wavelet entropy criterion. Finally, the impulsive wavelet filter with optimal parameters is applied to extract the fault characteristic frequency. Simulation signals are applied to verify the efficiency of the proposed approach, and comparison analysis is conducted as well. Further, the proposed method is applied to detect the gear fault of a gearbox. The experimental results show that the proposed method is effective with high efficiency.

Improving Classification with CNNs using Wavelet Pooling with Nesterov-Accelerated Adam

Wavelet pooling methods can improve the classification accuracy of Convolutional Neural Networks (CNNs). Combining wavelet pooling with the Nesterov-accelerated Adam (NAdam) gradient calculation method can improve both the accuracy of the CNN. We have implemented wavelet pooling with NAdam in this work using both a Haar wavelet (WavPool-NH) and a Shannon wavelet (WavPool-NS). The WavPool-NH and WavPool- NS methods are most accurate of the methods we considered for the MNIST and LIDC- IDRI lung tumor data-sets. The WavPool-NH and WavPool-NS implementations have an accuracy of 95.92% and 95.52%, respectively, on the LIDC-IDRI data-set. This is an improvement from the 92.93% accuracy obtained on this data-set with the max pooling method. The WavPool methods also avoid overfitting which is a concern with max pool- ing. We also found WavPool performed fairly well on the CIFAR-10 data-set, however, overfitting was an issue with all the methods we considered. Wavelet pooling, especially when combined with an adaptive gradient and wavelets chosen specifically for the data, has the potential to outperform current methods.