Yi Zhang

Xiaobing Feng, Yukun Li, Yi Zhang

This paper develops and analyzes a fully discrete finite element method for a class of semilinear stochastic partial differential equations (SPDEs) with multiplicative noise. The nonlinearity in the diffusion term of the SPDEs is assumed to be globally Lipschitz and the nonlinearity in the drift term is only assumed to satisfy a one-side Lipschitz condition. The semilinear SPDEs considered in this paper is a direct generalization of the SODEs considered in [13]. There are several difficulties which need to be overcome for this generalization. First, obviously the spatial discretization, which does not appear in the SODE case, adds an extra layer of difficulty. It turns out a special discretization must be designed to guarantee certain properties for the numerical scheme and its stiffness matrix. In this paper we use a finite element interpolation technique to discretize the nonlinear drift term. Second, in order to prove the strong convergence of the proposed fully discrete finite element method, stability estimates for higher order moments of the $H^1$-seminorm of the numerical solution must be established, which are difficult and delicate. A judicious combination of the properties of the drift and diffusion terms and a nontrivial technique borrowed from [16] is used in this paper to achieve the goal. Finally, stability estimates for the second and higher order moments of the $L^2$-norm of the numerical solution is also difficult to obtain due to the fact that the mass matrix may not be diagonally dominant. This is done by utilizing the interpolation theory and the higher moment estimates for the $H^1$-seminorm of the numerical solution. After overcoming these difficulties, it is proved that the proposed fully discrete finite element method is convergent in strong norms with nearly optimal rates of convergence.

Nobuki Takayama, Takaharu Yaguchi, Yi Zhang

Definite integrals with parameters of holonomic functions satisfy holonomic systems of linear partial differential equations. When we restrict parameters to a one dimensional curve, the system becomes a linear ordinary differential equation (ODE) with respect to a curve in the parameter space. We can evaluate the integral by solving the linear ODE numerically. This approach to evaluate numerically definite integrals is called the holonomic gradient method (HGM) and it is useful to evaluate several normalizing constants in statistics. We will discuss and compare methods to solve linear ODE's to evaluate normalizing constants.

Shengqian Zhu, Chengrong Yu, Qiang Wang et al.

Class incremental medical image segmentation (CIMIS) aims to preserve knowledge of previously learned classes while learning new ones without relying on old-class labels. However, existing methods 1) either adopt one-size-fits-all strategies that treat all spatial regions and feature channels equally, which may hinder the preservation of accurate old knowledge, 2) or focus solely on aligning local prototypes with global ones for old classes while overlooking their local representations in new data, leading to knowledge degradation. To mitigate the above issues, we propose Prototype-Guided Calibration Distillation (PGCD) and Dual-Aligned Prototype Distillation (DAPD) for CIMIS in this paper. Specifically, PGCD exploits prototype-to-feature similarity to calibrate class-specific distillation intensity in different spatial regions, effectively reinforcing reliable old knowledge and suppressing misleading information from old classes. Complementarily, DAPD aligns the local prototypes of old classes extracted from the current model with both global prototypes and local prototypes, further enhancing segmentation performance on old categories. Comprehensive evaluations on two widely used multi-organ segmentation benchmarks demonstrate that our method outperforms state-of-the-art methods, highlighting its robustness and generalization capabilities.