arXiv subjects

Xiannan Hu

Publications and source records attributed to Xiannan Hu.

3 recordsLinked to original sources

arXiv2026

From Pólya's Conditions to a Complete Characterization of the $L^2$ Convergence of Hyperinterpolation

It has remained open to identify the necessary and sufficient conditions for the $L^2$ convergence of hyperinterpolation since it was introduced by Sloan in 1995. We show that the $L^1$-$L^2$ Marcinkiewicz-Zygmund (MZ) condition, together with the asymptotic functional approximation property for polynomials, is the answer. We further prove that the optimal $L^1$-$L^2$ MZ constant coincides with the operator norm of the hyperinterpolation operator, and it admits a natural Banach space duality interpretation. With an explicit construction, we also show that Pólya's classical conditions for quadrature convergence are not sufficient for the $L^2$ convergence of hyperinterpolation. This reveals a fundamental distinction between the convergence of linear functionals (quadrature formulas) and that of linear operators (hyperinterpolation operators). We establish a strict logical hierarchy for the stability and accuracy conditions governing the convergence of quadrature and hyperinterpolation.

math.NA↗

arXiv2026

An Optimization Approach to Weight Collocation for Scattered Spherical Data

We introduce an optimization approach for constructing spherical quadrature rules on arbitrarily scattered data. Rather than designing node placements, the new approach focuses on optimally computing the weights for fixed configurations. Motivated by Pólya's necessary and sufficient conditions for quadrature convergence in 1933, we argue that pursuing weight positivity and high algebraic exactness for scattered data approximation is not necessary. To align the quadrature design with the underlying theory of approximation, we construct convex optimization models with suitable objective functionals by examining the accuracy of numerical integration with reproducing kernels of Sobolev spaces and the performance of hyperinterpolation with Marcinkiewicz-Zygmund (MZ) inequalities. The resulting optimization models encode the spatial distribution of the scattered sites and the analytic properties of the target function spaces. The proposed approach enables the derivation of rigorous theoretical stability bounds, and the resulting quadrature weights are efficiently computable by modern convex optimization techniques. Numerical results are reported to demonstrate the performance of the optimization approach for fundamental approximation tasks such as numerical integration and hyperinterpolation for scattered spherical data.

math.NA↗

arXiv2025

BestServe: Serving Strategies with Optimal Goodput in Collocation and Disaggregation Architectures

Serving large language models (LLMs) to millions of users requires efficient resource allocation and parallelism strategies. It is a labor intensive trial-and-error process to find such a strategy. We present BestServe, a novel framework for ranking serving strategies by estimating goodput under various operating scenarios. Supporting both collocated and disaggregated architectures, BestServe leverages an inference simulator built on an adapted roofline model and CPU-GPU dispatch dynamics. Our framework determines the optimal strategy in minutes on a single standard CPU, eliminating the need for costly benchmarking, while achieving predictions within a $20\%$ error margin. It appeals to be practical for rapid deployment planning because of its lightweight design and strong extensibility.

cs.LG↗