arXiv subjects

Shuhuang Xiang

Publications and source records attributed to Shuhuang Xiang.

At least 19 recordsLinked to original sources

arXiv2026

Lightning Plus Polynomial Approximation: Optimal Root-Exponential Convergence for Singular Functions in Corner Domains

This paper presents a rigorous convergence analysis for the lightning plus polynomial approximation scheme, which employs rational approximations constructed with preassigned tapered, exponentially clustered poles. This pole placement strategy was originally introduced by Trefethen and his collaborators for the resolution of corner singularities. Ample numerical results indicate that this scheme achieves root-exponential convergence, and in particular attains the same optimal convergence rate as the best rational approximation to $x^α$ on $[0,1]$ established by Stahl.% which is conjectured in [SIAM J. Numer. Anal., 61:2580-2600, 2023]. In this work, we establish optimal root-exponential convergence for the class of prototype functions of the form $g(z)z^α$ or $g(z)z^α\log z$, where $g$ is analytic on a neighborhood of the sector domain. These results confirm the validity of Conjectures 3.1 and 5.3 stated in [SIAM J. Numer. Anal., 61:2580-2600, 2023], and demonstrate that the choice $σ_{\mathrm{opt}} =\frac{\sqrt{2(2 - β)}π}{\sqrtα}$ achieves the theoretically optimal convergence rate $\mathcal{O}\left(e^{-\sqrt{2(2 - β)Nα}π}\right)$. Notably, for the specific case of $β= 0$, the scheme recovers Stahl's optimal convergence rate for $x^α$. Furthermore, working within the decomposition framework for corner domains proposed by Gopal and Trefethen, this paper provides a rigorous proof of optimal root-exponential convergence for lightning plus polynomial approximation problems on corner domains, and explicitly derives the optimal pole clustering parameter.

math.NA↗

arXiv2026

Optimal asymptotic analyses on Laguerre and Hermite orthogonal approximation for functions of algebraic and logarithmic regularitiesYali

Based on the Hilb-type formula and van der Corput-type lemmas, we present optimal asymptotic estimates for the decay of the Laguerre and Hermite coefficients for functions with algebraic and logarithmic singularities, which in turn yield the convergence rates of the corresponding spectral orthogonal projections. Numerous examples are provided to verify the optimality of these asymptotic results.

math.NA↗

arXiv2025

Exact convergence rates of lightning plus polynomial approximation for branch singularities with uniform exponentially clustered poles

This paper builds rigorous analysis on the root-exponential convergence for the lightning schemes via rational functions in approximating corner (branch) singularity problems with uniform exponentially clustered poles proposed by Gopal and Trefethen. The start point is to set up the integral representations of $z^α$ and $z^α\log z$ in the slit disk and develop results akin to Paley-Wiener theorem, from which, together with the Poisson summation formula, the root-exponential convergence of the lightning plus polynomial scheme with an exact order for each clustered parameter is established in approximation of prototype functions $z^α$ or $z^α\log z$ on a sector-shaped domain, which includes $[0,1]$ as a special case. In addition, the fastest convergence rate is confirmed based upon the best choice of the clustered parameter. Furthermore, the optimal selection of the clustered parameter is employed in conformal mappings through solving Laplace problems on corner domains, building upon Lehman and Wasow's analysis of corner singularities and incorporating the domain decomposition method proposed by Gopal and Trefethen.

math.NA↗

arXiv2025

Convergence Analysis of Aggregation-Broadcast in LoRA-enabled Distributed Fine-Tuning

Federated Learning (FL) enables collaborative model training across decentralized data sources while preserving data privacy. However, the growing size of Machine Learning (ML) models poses communication and computation challenges in FL. Low-Rank Adaptation (LoRA) has recently been introduced into FL as an efficient fine-tuning method, reducing communication overhead by updating only a small number of trainable parameters. Despite its effectiveness, how to aggregate LoRA-updated local models on the server remains a critical and understudied problem. In this paper, we provide a unified convergence analysis for LoRA-based FL. We first categories the current aggregation method into two major type: Sum-Product (SP) and Product-Sum (PS). Then we formally define the Aggregation-Broadcast Operator (ABO) and derive both weak and strong convergence condition under mild assumptions. Furthermore, we present both weak and strong convergence condition that guarantee convergence of the local model and the global model respectively. These theoretical analyze offer a principled understanding of various aggregation strategies. Notably, we prove that the SP and PS aggregation methods satisfy the weak and strong convergence condition respectively, but differ in their ability to achieve the optimal convergence rate. Extensive experiments on standard benchmarks validate our theoretical findings.

cs.LG↗

arXiv2024

On the best convergence rates of lightning plus polynomial approximations

Building on introducing exponentially clustered poles, Trefethen and his collaborators introduced lightning algorithms for approximating functions of singularities. These schemes may achieve root-exponential convergence rates. In particular, based on a specific choice of the parameter of the tapered exponentially clustered poles, the lightning approximation with either a low-degree polynomial basis may achieve the optimal convergence rate simply as the best rational approximation for prototype $x^α$ on $[0,1]$, which was illustrated through delicate numerical experiments and conjectured in [SIAM J. Numer. Anal., 61:2580-2600, 2023]. By utilizing Poisson's summation formula and results akin to Paley-Wiener Theorem, we rigorously show that all these schemes with a low-degree polynomial basis achieve root-exponential convergence rates with exact orders in approximating $x^α$ for arbitrary clustered parameters theoretically, and provide the best choices of the parameter to achieve the fastest convergence rate for each type of clustered poles, from which the conjecture is confirmed as a special case. Ample numerical evidences demonstrate the optimality and sharpness of the estimates.

math.NA↗

arXiv2024

Polynomial and rational interpolation: potential, barycentric weights, and Lebesgue constants

In this paper, we focus on barycentric weights and Lebesgue constants for Lagrange interpolation of arbitrary node distributions on $[-1,1]$. The following three main works are included: estimates of upper and lower bounds on the barycentric weights are given in terms of the logarithmic potential function; for interpolation of non-equilibrium potentials, lower bounds with exponentially growing parts of Lebesgue constants are given; and for interpolation consistent with equilibrium potentials, non-exponentially growing upper bounds on their Lebesgue constants are given. Based on the work in this paper, we can discuss the behavior of the Lebesgue constant and the existence of exponential convergence in a unified manner in the framework of potential theory.

math.NA↗

arXiv2024

The root-exponential convergence of lightning plus polynomial approximation on corner domains

This paper builds further rigorous analysis on the root-exponential convergence for lightning schemes approximating corner singularity problems. By utilizing Poisson summation formula, Runge's approximation theorem and Cauchy's integral theorem, the optimal rate is obtained for efficient lightning plus polynomial schemes, newly developed by Herremans, Huybrechs and Trefethen \cite{Herremans2023}, for approximation of $g(z)z^α$ or $g(z)z^α\log z$ in a sector-shaped domain with tapered exponentially clustering poles, where $g(z)$ is analytic on the sector domain. From these results, Conjecture 5.3 in \cite{Herremans2023} on the root-exponential convergence rate is confirmed and the choice of the parameter $σ_{opt}=\frac{\sqrt{2(2-β)}π}{\sqrtα}$ may achieve the fastest convergence rate among all $σ>0$. Furthermore, based on Lehman and Wasow's study of corner singularities \cite{Lehman1954DevelopmentsIT, Wasow}, together with the decomposition of Gopal and Trefethen \cite{Gopal2019}, root-exponential rates for lightning plus polynomial schemes in corner domains $Ω$ are validated, and the best choice of lightning clustering parameter $σ$ for $Ω$ is also obtained explicitly. The thorough analysis provides a solid foundation for lightning schemes.

math.NA↗

arXiv2023

Pointwise error estimates and local superconvergence of Jacobi expansions

As one myth of polynomial interpolation and quadrature, Trefethen [30] revealed that the Chebyshev interpolation of $|x-a|$ (with $|a|<1 $) at the Clenshaw-Curtis points exhibited a much smaller error than the best polynomial approximation (in the maximum norm) in about $95\%$ range of $[-1,1]$ except for a small neighbourhood near the singular point $x=a.$ In this paper, we rigorously show that the Jacobi expansion for a more general class of $Φ$-functions also enjoys such a local convergence behaviour. Our assertion draws on the pointwise error estimate using the reproducing kernel of Jacobi polynomials and the Hilb-type formula on the asymptotic of the Bessel transforms. We also study the local superconvergence and show the gain in order and the subregions it occurs. As a by-product of this new argument, the undesired $\log n$-factor in the pointwise error estimate for the Legendre expansion recently stated in Babuuska and Hakula [5] can be removed. Finally, all these estimates are extended to the functions with boundary singularities. We provide ample numerical evidences to demonstrate the optimality and sharpness of the estimates.

math.NA↗

arXiv2022

Eigenvalue Analysis and Applications of the Legendre Dual-Petrov-Galerkin Methods for Initial Value Problems

In this paper, we show that the eigenvalues and eigenvectors of the spectral discretisation matrices resulted from the Legendre dual-Petrov-Galerkin (LDPG) method for the $m$th-order initial value problem (IVP): $u^{(m)}(t)=σu(t),\, t\in (-1,1)$ with constant $σ\not=0$ and usual initial conditions at $t=-1,$ are associated with the generalised Bessel polynomials (GBPs). The essential idea of the analysis is to properly construct the basis functions for the solution and its dual spaces so that the matrix of the $m$th derivative is an identity matrix, and the mass matrix is then identical or approximately equals to the Jacobi matrix of the three-term recurrence of GBPs with specific integer parameters. This allows us to characterise the eigenvalue distributions and identify the eigenvectors. As a by-product, we are able to answer some open questions related to the very limited known results on the collocation method at Legendre points (studied in 1980s) for the first-order IVP, by reformulating it into a Petrov-Galerkin formulation. Moreover, we present two stable algorithms for computing zeros of the GBPs, and develop a general space-time spectral method for evolutionary PDEs using either the matrix diagonalisation, which is restricted to a small number of unknowns in time due to the ill-conditioning but is fully parallel, or the QZ decomposition which is numerically stable for a large number of unknowns in time but involves sequential computations. We provide ample numerical results to demonstrate the high accuracy and robustness of the space-time spectral methods for some interesting examples of linear and nonlinear wave problems.

math.NA↗

arXiv2021

Sparse Solutions of a Class of Constrained Optimization Problems

In this paper, we consider a well-known sparse optimization problem that aims to find a sparse solution of a possibly noisy underdetermined system of linear equations. Mathematically, it can be modeled in a unified manner by minimizing $\|\bf{x}\|_p^p$ subject to $\|A\bf{x}-\bf{b}\|_q\leqσ$ for given $A \in \mathbb{R}^{m \times n}$, $\bf{b}\in\mathbb{R}^m$, $σ\geq0$, $0\leq p\leq 1$ and $q \geq 1$. We then study various properties of the optimal solutions of this problem. Specifically, without any condition on the matrix $A$, we provide upper bounds in cardinality and infinity norm for the optimal solutions, and show that all optimal solutions must be on the boundary of the feasible set when $0<p<1$. Moreover, for $q \in \{1,\infty\}$, we show that the problem with $0<p<1$ has a finite number of optimal solutions and prove that there exists $0<p^*<1$ such that the solution set of the problem with any $0<p<p^*$ is contained in the solution set of the problem with $p=0$ and there further exists $0<\bar{p}<p^*$ such that the solution set of the problem with any $0<p\leq\bar{p}$ remains unchanged. An estimation of such $p^*$ is also provided. In addition, to solve the constrained nonconvex non-Lipschitz $L_p$-$L_1$ problem ($0<p<1$ and $q=1$), we propose a smoothing penalty method and show that, under some mild conditions, any cluster point of the sequence generated is a KKT point of our problem. Some numerical examples are given to implicitly illustrate the theoretical results and show the efficiency of the proposed algorithm for the constrained $L_p$-$L_1$ problem under different noises.

math.OC↗

arXiv2021

Fast linear barycentric rational interpolation for singular functions via scaled transformations

In this paper, applied strictly monotonic increasing scaled maps, a kind of well-conditioned linear barycentric rational interpolations are proposed to approximate functions of singularities at the origin, such as $x^α$ for $α\in (0,1)$ and $\log(x)$. It just takes $O(N)$ flops and can achieve fast convergence rates with the choice the scaled parameter, where $N$ is the maximum degree of the denominator and numerator. The construction of the rational interpolant couples rational polynomials in the barycentric form of second kind with the transformed Jacobi-Gauss-Lobatto points. Numerical experiments are considered which illustrate the accuracy and efficiency of the algorithms. The convergence of the rational interpolation is also considered.

math.NA↗

arXiv2020

Convergence rates of spectral orthogonal projection approximation for functions of algebraic and logarithmatic regularities

Based on the Hilb type formula between Jacobi polynomials and Bessel functions, optimal decay rates on Jacobi expansion coefficients are derived, by applying van der Corput type lemmas, for functions of logarithmatic singularities, which leads to the optimal convergence rates on the Jacobi, Gegenbauer and Chebyshev orthogonal projections. It is interesting to see that for boundary singularities, one may get faster convergence rate on the Jacobi or Gegenbauer projection as $(α,β)$ and $λ$ increases. The larger values of parameter, the higher convergence rates can be achieved. In particular, the Legendre projection has one half order higher than Chebyshev. Moreover, if $\min\{α,β\}>0$ and $λ>\frac{1}{2}$, the Jacobi and Gegenbauer orthogonal projections have higher convergence orders compared with Legendre. While for interior singularity, the convergence order is independent of $(α,β)$ and $λ$.

math.NA↗

arXiv2019

Levin methods for highly oscillatory integrals with singularities

In this paper, new Levin methods are presented for calculating oscillatory integrals with algebraic and/or logarithmic singularities. To avoid singularity, the technique of singularity separation is applied and then the singular ODE occurring in classic Levin methods is converted into two kinds of non-singular ODEs. The solutions of one can be obtained explicitly, while those of the other can be solved efficiently by collocation methods. The proposed methods can attach arbitrarily high asymptotic orders and also enjoy superalgebraic convergence with respect to the number of collocation points. Several numerical experiments are presented to validate the efficiency of the proposed methods.

math.NA↗

arXiv2019

A Levin method for logarithmically singular oscillatory integrals

We propose a new stable Levin method to compute oscillatory integrals with logarithmic singularities and without stationary points. To avoid the singularity, we apply the technique of singularity separation and transform the singular ODE into two non-singular ODEs, which can be solved efficiently by the collocation method. Applying the equivalency of the new Levin method for the singular oscillatory integrals and the Filon method when the oscillator is linear, we consider the convergence of the new Levin method. This new method shares the proposition that less error for higher oscillation. Several numerical experiments are presented to validate the efficiency of the proposed method.

math.NA↗

arXiv2016

Computation of highly oscillatory Bessel transforms with algebraic singularities

In this paper, we consider the Clenshaw-Curtis-Filon method for the highly oscillatory Bessel transform $\int_0^1x^α(1-x)^βf(x) J_ν(ωx)dx$, where $f$ is a smooth function on $[0, 1]$, and $ν\geq0.$ The method is based on Fast Fourier Transform (FFT) and fast computation of the modified moments. We give a recurrence relation for the modified moments and present an efficient method for the evaluation of modified moments by using recurrence relation. Moreover, the corresponding error bound in inverse powers of $N$ for this method for the integral is presented. Numerical examples are provided to support our analysis and show the efficiency and accuracy of the method.

math.NA↗

arXiv2015

On Interpolation Approximation: Convergence rates for polynomial interpolation for functions of limited regularity

The convergence rates on polynomial interpolation in most cases are estimated by Lebesgue constants. These estimates may be overestimated for some special points of sets for functions of limited regularities. In this paper, by applying the Peano kernel theorem and Wainerman's lemma, new formulas on the convergence rates are considered. Based upon these new estimates, it shows that the interpolation at strongly normal pointsystems can achieve the optimal convergence rate, the same as the best polynomial approximation. Furthermore, by using the asymptotics on Jacobi polynomials, the convergence rates are established for Gauss-Jacobi, Jacobi-Gauss-Lobatto or Jacobi-Gauss-Radau pointsystems. From these results, we see that the interpolations at the Gauss-Legendre, Legendre-Gauss-Lobatto pointsystem, or at strongly normal pointsystems, has essentially the same approximation accuracy compared with those at the two Chebyshev piontsystems, which also illustrates the equally accuracy of the Gauss and Clenshaw-Curtis quadrature. In addition, numerical examples illustrate the perfect coincidence with the estimates, which means the convergence rates are optimal.

math.NA↗

arXiv2014

On the Optimal Rates of Convergence for Quadratures Derived from Chebyshev Points

In this paper, we study the optimal general convergence rates for quadratures derived from Chebyshev points. By building on the aliasing errors on integration of Chebyshev polynomials, together with the asymptotic formulae on the coefficients of Chebyshev expansions, new and optimal convergence rates for $n$-point Clenshaw-Curtis, Fejér's first and second quadrature rules are established for Jacobi weights or Jacobi weights multiplied by $\ln((x+1)/2)$. The convergence orders are attainable for some functions of finite regularities. In addition, by using refined estimates on aliasing errors on integration of Chebyshev polynomials by Gauss-Legendre quadrature, an improved convergence rate for Gauss-Legendre is given too.

math.NA↗

arXiv2014

On Fast Implementation of Higher Order Hermite-Fejer Interpolation

The problem of barycentric Hermite interpolation is highly susceptible to overflows or underflows. In this paper, based on Sturm-Liouville equations for Jacobi orthogonal polynomials, we consider the fast implementation on the second barycentric formula for higher order Hermite-Fejér interpolation at Gauss-Jacobi or Jacobi-Gauss-Lobatto pointsystems, where the barycentric weights can be efficiently evaluated and cost linear operations corresponding to the number of grids totally. Furthermore, due to the division of the second barycentric form, the exponentially increasing common factor in the barycentric weights can be canceled, which yields a superiorly stable method for computing the simplified barycentric weights, and leads to a fast implementation of the higher order Hermite-Fejér interpolation with linear operations on the number of grids. In addition, the convergence rates are derived for Hermite-Fejér interpolation at Gauss-Jacobi pointsystems.

math.NA↗