T. Sun

Tong Sun, David Rumsey

The new concept of numerical smoothness is applied to RKDG methods on the scalar nonlinear conservation laws. The main result is an a posteriori error estimate for the RKDG methods of arbitrary order in space and time, with optimal convergence rate. In this paper, the case of smooth solutions is the focus point. However, the error analysis framework is prepared to deal with discontinuous solutions in the future.

Tong Sun

Numerical solutions of differential equations are usually not smooth functions. However, they should resemble the smoothness of the corresponding real solutions in one way or another. In two of our recent papers, a kind of spacial smoothness indicators was defined and subsequently applied on the {\it a posteriori} error analysis. Here we prove that the boundedness of those smoothness indicators is actually a necessary condition for a piecewise polynomial function to approximate a smooth function with optimal convergence rate. This should help in validating the error analysis in the related papers. Moreover, the result of this paper provides an efficient practical method to detect the loss of convergence rate due to the lack of numerical smoothness, hence it serves as a criterion for the qualities of many numerical schemes.

Xuan Shen, Zhao Song, Yufa Zhou et al.

Transformers have emerged as the leading architecture in deep learning, proving to be versatile and highly effective across diverse domains beyond language and image processing. However, their impressive performance often incurs high computational costs due to their substantial model size. This paper focuses on compressing decoder-only transformer-based autoregressive models through structural weight pruning to improve the model efficiency while preserving performance for both language and image generation tasks. Specifically, we propose a training-free pruning method that calculates a numerical score with Newton's method for the Attention and MLP modules, respectively. Besides, we further propose another compensation algorithm to recover the pruned model for better performance. To verify the effectiveness of our method, we provide both theoretical support and extensive experiments. Our experiments show that our method achieves state-of-the-art performance with reduced memory usage and faster generation speeds on GPUs.