Guanghan Li

Zhicheng Hu, Guanghan Li, Chunwu Wang et al.

A high-order Newton multigrid method is proposed for steady-state shallow water flows in open channels with regular and irregular geometries. The method integrates a finite volume discretization with third-order weighted essentially non-oscillatory (WENO) reconstruction and a Newton multigrid framework with an efficient approximation of the Jacobian matrix for solving the resulting discrete system. In high-order schemes, the computational cost of Jacobian construction becomes dominant due to the wide stencil. Meanwhile, only a small fraction of the non-zero Jacobian entries exhibit large magnitudes. Based on this observation, a simplified Jacobian approximation is introduced using reduced stencils, in which selected off-stencil contributions are neglected, thereby achieving a substantial reduction in computational cost. The proposed approach is verified numerically to show significant efficiency improvement while maintaining comparable convergence behavior to that obtained with the full Jacobian approach. To further enhance performance, a geometric multigrid method incorporating a successive over-relaxation iteration as the smoother is applied to solve the linear systems arising in each Newton step. A variety of numerical experiments, including a one-dimensional smooth subcritical flow, flows over a hump, and a two-dimensional hydraulic jump over a wedge, are carried out to illustrate the third-order accuracy, efficiency, and robustness of the proposed method.

Guanghan Li, Xun Zhang, Yufei Zhang et al.

Large language models (LLMs), endowed with exceptional reasoning capabilities, are adept at discerning profound user interests from historical behaviors, thereby presenting a promising avenue for the advancement of recommendation systems. However, a notable discrepancy persists between the sparse collaborative semantics typically found in recommendation systems and the dense token representations within LLMs. In our study, we propose a novel framework that harmoniously merges traditional recommendation models with the prowess of LLMs. We initiate this integration by transforming ItemIDs into sequences that align semantically with the LLMs space, through the proposed Alignment Tokenization module. Additionally, we design a series of specialized supervised learning tasks aimed at aligning collaborative signals with the subtleties of natural language semantics. To ensure practical applicability, we optimize online inference by pre-caching the top-K results for each user, reducing latency and improving effciency. Extensive experimental evidence indicates that our model markedly improves recall metrics and displays remarkable scalability of recommendation systems.