Hongpeng Sun

Jingzhi Li, Hongyu Liu, Zaijiu Shang et al.

We develop two inverse scattering schemes for locating multiple electromagnetic (EM) scatterers by the electric far-field measurement corresponding to a single incident/detecting plane wave. The first scheme is for locating scatterers of small size compared to the wavelength of the detecting plane wave. The multiple scatterers could be extremely general with an unknown number of components, and each scatterer component could be either an impenetrable perfectly conducting obstacle or a penetrable inhomogeneous medium with an unknown content. The second scheme is for locating multiple perfectly conducting obstacles of regular size compared to the detecting EM wavelength. The number of the obstacle components is not required to be known in advance, but the shape of each component must be from a certain known admissible class. The admissible class may consist of multiple different reference obstacles. The second scheme could also be extended to include the medium components if a certain generic condition is satisfied. Both schemes are based on some novel indicator functions whose indicating behaviors could be used to locate the scatterers. No inversion will be involved in calculating the indicator functions, and the proposed methods are every efficient and robust to noise. Rigorous mathematical justifications are provided and extensive numerical experiments are conducted to illustrate the effectiveness of the imaging schemes.

Xinhua Shen, Zaijiu Shang, Hongpeng Sun

This paper introduces a second-order convex splitting scheme for gradient flows arising in phase-field models, based on the backward differentiation formula (BDF2) for the implicit part and the Adams-Bashforth method for the nonlinear and explicit component. The method is formulated and analyzed in finite-dimensional spaces, where energy stability plays a central role in establishing rigorous convergence properties. By leveraging the Kurdyka-Łojasiewicz framework, we prove the global convergence of the discrete trajectories generated by the scheme, even in the presence of nonsmooth energy functionals, under mild assumptions on the time-step size. The Armijo line search strategy and the classical preconditioning strategies, such as symmetric Gauss-Seidel and Jacobi, are incorporated to improve its computational efficiency. Numerical experiments confirm that the proposed method achieves computational efficiency compared to existing first-order splitting approaches and other accelerated splitting algorithms, while maintaining robustness in both smooth and nonsmooth regimes.