Yinkun Wang

Yinkun Wang, Yuesheng Xu

Solutions of Fredholm integral equations of the second kind with oscillatory kernels likely exhibit oscillation. Standard numerical methods applied to solving equations of this type have poor numerical performance due to the influence of the highly rapid oscillation in the solutions. Understanding of the oscillation of the solutions is still inadequate in the literature and thus it requires further investigation. For this purpose, we introduce a notion to describe the degree of oscillation of an oscillatory function based on the dependence of its norm in a certain function space on the wavenumber. Based on this new notion, we construct structured oscillatory spaces with oscillatory structures. The structured spaces with a specific oscillatory structure can capture the oscillatory components of the solutions of Fredholm integral equations with oscillatory kernels. We then further propose oscillation preserving Galerkin methods for solving the equations by incorporating the standard approximation subspace of spline functions with a finite number of oscillatory functions which capture the oscillation of the exact solutions of the integral equations. We prove that the proposed methods have the optimal convergence order uniformly with respect to the wavenumber and they are numerically stable. A numerical example is presented to confirm the theoretical estimates.

Hui Cao, Wenlong Zou, Yinkun Wang et al.

Since the 2004 DARPA Grand Challenge, the autonomous driving technology has witnessed nearly two decades of rapid development. Particularly, in recent years, with the application of new sensors and deep learning technologies extending to the autonomous field, the development of autonomous driving technology has continued to make breakthroughs. Thus, many carmakers and high-tech giants dedicated to research and system development of autonomous driving. However, as the foundation of autonomous driving, the deep learning technology faces many new security risks. The academic community has proposed deep learning countermeasures against the adversarial examples and AI backdoor, and has introduced them into the autonomous driving field for verification. Deep learning security matters to autonomous driving system security, and then matters to personal safety, which is an issue that deserves attention and research.This paper provides an summary of the concepts, developments and recent research in deep learning security technologies in autonomous driving. Firstly, we briefly introduce the deep learning framework and pipeline in the autonomous driving system, which mainly include the deep learning technologies and algorithms commonly used in this field. Moreover, we focus on the potential security threats of the deep learning based autonomous driving system in each functional layer in turn. We reviews the development of deep learning attack technologies to autonomous driving, investigates the State-of-the-Art algorithms, and reveals the potential risks. At last, we provides an outlook on deep learning security in the autonomous driving field and proposes recommendations for building a safe and trustworthy autonomous driving system.

Yinkun Wang, Ying Li, Jianshu Luo

The moments of Bessel functions and Bessel-trigonometric functions play a basic role in many practical problems and numerical analysis. This paper presents a complete analysis for these moments based on the recursive relations of Bessel functions. To evaluate the moments of Bessel functions numerically, a fast and efficient scheme is also proposed to approximate the integral of Bessel function of the first kind and of zero order. The moments of Bessel-trigonometric functions are proved to be expressed in a closed form. In the numerical results, the accuracy and efficiency of the proposed analysis for the moments of Bessel functions is validated first and then by comparing the existing methods, a better scheme for the moments of Bessel functions is presented.

Yinkun Wang, Xiangling Chen, Ying Li et al.

A fast non-polynomial interpolation is proposed in this paper for functions with logarithmic singularities. It can be executed fast with the discrete cosine transform. Based on this interpolation, a new quadrature is proposed for a kind of logarithmically singular integrals. The interpolation and integration errors are also analyzed. Numerical examples of the interpolation and integration are shown to validate the efficiency of the proposed new interpolation and the new quadrature.