Zaijiu Shang

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.

Xiaohua Ding, Hongyu Liu, Zaijiu Shang et al.

Based on reasonable testing model problems, we study the preservation by symplectic Runge-Kutta method (SRK) and symplectic partitioned Runge-Kutta method (SPRK) of structures for fixed points of linear Hamiltonian systems. The structure-preservation region provides a practical criterion for choosing step-size in symplectic computation. Examples are given to justify the investigation.

Geng Sun, Siqing Gan, Hongyu Liu et al.

Symmetric method and symplectic method are classical notions in the theory of Runge-Kutta methods. They can generate numerical flows that respectively preserve the symmetry and symplecticity of the continuous flows in the phase space. Adjoint method is an important way of constructing a new Runge-Kutta method via the symmetrisation of another Runge-Kutta method. In this paper, we introduce a new notion, called symplectic-adjoint Runge-Kutta method. We prove some interesting properties of the symmetric-adjoint and symplectic-adjoint methods. These properties reveal some intrinsic connections among several classical classes of Runge-Kutta methods. In particular, the newly introduced notion and the corresponding properties enable us to develop a novel and practical approach of constructing high-order explicit Runge-Kutta methods, which is a challenging and longly overlooked topic in the theory of Runge-Kutta methods.

Zhaodong Ding, Zaijiu Shang

In this paper, we study the persistence of invariant tori of integrable Hamiltonian systems satisfying Rüssmann's non-degeneracy condition when symplectic integrators are applied to them. Meanwhile, we give an estimate of the measure of the set occupied by the invariant tori in the phase space. On an invariant torus, the one-step map of the scheme is conjugate to a one parameter family of linear rotations with a step size dependent frequency vector in terms of iteration. These results are a generalization of Shang's theorems (1999, 2000), where the non-degeneracy condition is assumed in the sense of Kolmogorov. In comparison, Rüssmann's condition is the weakest non-degeneracy condition for the persistence of invariant tori in Hamiltonian systems. These results provide new insight into the nonlinear stability of symplectic integrators.

Zhaodong Ding, Zaijiu Shang, Bo Xie

We prove a Nekhoroshev-type theorem for nearly integrable symplectic map. As an application of the theorem, we obtain the exponential stability symplectic algorithms. Meanwhile, we can get the bounds for the perturbation, the variation of the action variables, and the exponential time respectively. These results provide a new insight into the nonlinear stability analysis of symplectic algorithms. Combined with our previous results on the numerical KAM theorem for symplectic algorithms (2018), we give a more complete characterization on the complex nonlinear dynamical behavior of symplectic algorithms.

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.

Xinran Ma, Zhaoqi Zhou, Chuan Zhou et al.

Graph-based approaches to approximate nearest neighbor search (ANNS) enable fast, high-recall retrieval on billion-scale vector datasets. Among them, the Sparse Neighborhood Graph (SNG) is widely used due to its strong search performance. However, the lack of theoretical understanding of SNG leads to expensive tuning of the truncation parameter that controls graph sparsification. In this work, we present OPT-SNG, a principled framework for analyzing and optimizing SNG construction. We introduce a martingale-based model of the pruning process that characterizes the stochastic evolution of candidate sets during graph construction. Using this framework, we prove that SNG has a maximum out-degree of \(O(n^{2/3+ε})\), where \(ε>0\) is an arbitrarily small constant, and an expected search path length of \(O(\log n)\). Building on these insights, we derive a closed-form rule for selecting the optimal truncation parameter \(R\), thereby eliminating the need for costly parameter sweeping. Extensive experiments on real-world datasets demonstrate that OPT-SNG achieves an average \(5.9\times\) speedup in index construction time, with peak improvements reaching \(15.4\times\), while consistently maintaining or improving search performance.