Hai Zhu | Searcharxiv

arXiv2026

Recursive reduction quadrature for the evaluation of Laplace layer potentials in three dimensions

A high-order quadrature scheme is constructed for the evaluation of Laplace single and double layer potentials and their normal derivatives on smooth surfaces in three dimensions. The construction begins with a harmonic approximation of the density on each patch, which allows for a natural harmonic polynomial extension in a volumetric neighborhood of the patch in the ambient space. Then by the general Stokes theorem, singular and nearly singular surface integrals are reduced to line integrals preserving the singularity of the kernel, instead of the standard origin-centered 1-forms that require expensive adaptive integration. These singularity-preserving line integrals can be semi-analytically evaluated using singularity-swap quadrature. In other words, the evaluation of singular and nearly singular surface integrals is reduced to function evaluations at the vertices on the boundary of each patch. The recursive reduction quadrature largely removes adaptive integration that is needed in most existing high-order quadratures for singular and nearly singular surface integrals, resulting in exceptional performance. The scheme achieves twelve-digit accuracy uniformly for close evaluations and offers a speedup of five times or more in constructing the sparse quadrature-correction matrix compared to previous state-of-the-art quadrature schemes.

math.NA↗

arXiv2026

SORT: A Systematically Optimized Ranking Transformer for Industrial-scale Recommenders

While Transformers have achieved remarkable success in LLMs through superior scalability, their application in industrial-scale ranking models remains nascent, hindered by the challenges of high feature sparsity and low label density. In this paper, we propose SORT (Systematically Optimized Ranking Transformer), a scalable model designed to bridge the gap between Transformers and industrial-scale ranking models. We address the high feature sparsity and low label density challenges through a series of optimizations, including request-centric sample organization, local attention, query pruning, and generative pre-training. Furthermore, we introduce a suite of refinements to the tokenization, multi-head attention (MHA), and feed-forward network (FFN) modules, which collectively stabilize the training process and enlarge the model capacity. To maximize hardware efficiency, we optimize our training system to elevate the model FLOPs utilization (MFU) to 45%. Extensive experiments demonstrate that SORT outperforms strong baselines and exhibits excellent scalability across data size, model size, and sequence length, while remaining flexible at integrating diverse features. Finally, online A/B testing in large-scale e-commerce scenarios confirms that SORT achieves significant gains in key business metrics, including orders (+7.47%), buyers (+6.67%), and GMV (+8.65%), while simultaneously cutting latency by 62% and boosting throughput nearly sevenfold (+589%). SORT has been fully deployed in production, serving all users on AliExpress.

cs.IR↗