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Xiaoda Xu

Publications and source records attributed to Xiaoda Xu.

13 recordsLinked to original sources
arXiv2026

The Partition Principle Revisited: Non-Equal Volume Designs Achieve Minimal Expected Star Discrepancy

We study the expected star discrepancy under a newly designed class of non-equal volume partitions. The main contributions are twofold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed non-equal volume partitions yield stratified sampling point sets with lower expected star discrepancy than classical jittered sampling. Specifically, we prove that $\mathbb{E}(D^{*}_{N}(Z)) < \mathbb{E}(D^{*}_{N}(Y))$, where $Y$ and $Z$ represent jittered sampling and our non-equal volume partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our non-equal volume partition models, which improve upon existing bounds for jittered sampling. Our results provide a theoretical foundation for using non-equal volume partitions in high-dimensional numerical integration.

stat.ML↗
arXiv2026

Expected star discrepancy based on stratified sampling

We present two main contributions to the expected star discrepancy theory. First, we derive a sharper expected upper bound for jittered sampling, improving the leading constants and logarithmic terms compared to the state-of-the-art [Doerr, 2022]. Second, we prove the strong partition principle for star discrepancy, showing that any equal-measure stratified sampling yields a strictly smaller expected discrepancy than simple random sampling, thereby resolving an open question in [Kiderlen and Pausinger, 2022]. Numerical simulations confirm our theoretical advances and illustrate the superiority of stratified sampling in low to moderate dimensions.

math.ST↗
arXiv2026

Weighted $L_p$-Discrepancy Bounds for Parametric Stratified Sampling and Applications to High-Dimensional Integration

This paper studies the expected $L_p$-discrepancy ($2 \leq p < \infty$) for stratified sampling schemes under importance sampling. We introduce a parametric family of equivolume partitions $Ω_{θ,\sim}$ and leverage recent exact formulas for the expected $L_2$-discrepancy \cite{xian2025improved}. Our main contribution is a weighted discrepancy reduction lemma that relates weighted $L_p$-discrepancy to standard $L_p$-discrepancy with explicit constants depending on the weight function. For $p=2$, we obtain explicit bounds using the exact discrepancy formulas. For $p>2$, we derive probabilistic bounds via dyadic chaining techniques. The results yield uniform error estimates for multivariate integration in Sobolev spaces $\mathcal{H}^1(K)$ and $F^*_{d,q}$, demonstrating improved performance over classical jittered sampling in importance sampling scenarios. Numerical experiments validate our theoretical findings and illustrate the practical advantages of parametric stratified sampling.

math.NA↗
arXiv2026

Sharp Non-Asymptotic Bounds for the Star Discrepancy of Double-Infinite Random Matrices via Optimal Covering Numbers

We establish sharp non-asymptotic probabilistic bounds for the star discrepancy of double-infinite random matrices -- a canonical model for sequences of random point sets in high dimensions. By integrating the recently proved \textbf{optimal covering numbers for axis-parallel boxes} (Gnewuch, 2024) into the dyadic chaining framework, we achieve \textbf{explicitly computable constants} that improve upon all previously known bounds. For dimension $d \ge 3$, we prove that with high probability, \[ D_N^d \le \sqrt{αA_d + βB \frac{\ln \log_2 N}{d}} \sqrt{\frac{d}{N}}, \] where $A_d$ is given by an explicit series and satisfies $A_3 \le 745$, a \textbf{14\% improvement} over the previous best constant of 868 (Fiedler et al., 2023). For $d=2$, we obtain the currently smallest known constant $A_2 \le 915$. Our analysis reveals a \textbf{precise trade-off} between the dimensional dependence and the logarithmic factor in $N$, highlighting how optimal covering estimates directly translate to tighter discrepancy bounds. These results immediately yield improved error guarantees for \textbf{quasi-Monte Carlo integration, uncertainty quantification, and high-dimensional sampling}, and provide a new benchmark for the probabilistic analysis of geometric discrepancy. \textbf{Keywords:} Star discrepancy, double-infinite random matrices, covering numbers, dyadic chaining, high-dimensional integration, quasi-Monte Carlo, probabilistic bounds.

math.ST↗
arXiv2026

On a Class of Partitions with Lower Expected Star Discrepancy and Its Upper Bound than Jittered Sampling

We investigate the expected star discrepancy under a newly designed class of convex equivolume partition models. The main contributions are two-fold. First, we establish a strong partition principle for the star discrepancy, showing that our newly designed partitions yield stratified sampling point sets with lower expected star discrepancy than both classical jittered sampling and simple random sampling. Specifically, we prove that $\mathbb{E}(D^{*}_{N}(Z))\leq\mathbb{E}(D^{*}_{N}(Y))<\mathbb{E}(D^{*}_{N}(X))$, where $X$, $Y$, and $Z$ represent simple random sampling, jittered sampling, and our new partition sampling, respectively. Second, we derive explicit upper bounds for the expected star discrepancy under our partition models, which improve upon existing bounds for jittered sampling. Our results resolve Open Question 2 posed in Kiderlen and Pausinger (2021) regarding the strong partition principle for star discrepancy.

math.PR↗
arXiv2025

Random uniform approximation under weighted importance sampling of a class of stratified input

We consider random discrepancy under weighted importance sampling of a class of stratified input. We give the expected $L_p-$discrepancy($2\leq p<\infty$) upper bound in weighted form under a class of stratified sampling. This result contributes to the error estimate of the upper bound of the integral approximation under weighted importance sampling, and and our sampling pattern is a stratified input.

math.PR↗
arXiv2024

Mathematical Analysis and Numerical Computation of String Vibration Equations with Elastic Supports for Bridge Cable Force Evaluation

This study focuses on a critical aspect of bridge engineering -- the evaluation of cable forces, paying particular attention to the cables that are internally constrained by elastic supports. Detecting these cable forces is important for the safety and stability of bridges. The practical problem introduces a novel mathematical challenge: how to effectively address string vibration equations with one or multiple internal elastic supports,~which remains a theoretical issue not fully solved in engineering. To tackle this, it is necessary to firstly establish an appropriate mathematical model and accurately define initial-boundary value problems. We then formulate the well-posedness of the solution using both classical and weak solution approaches, supplementing the existing numerical results available in engineering. Meanwhile, we attempt to use PINNs (Physics-Informed Neural Networks) instead of traditional FEM (Finite Element Method) in engineering. Consequently, in contrast to the classical solution method, we demonstrate that for a string with finite elastic supports, the weak solution method not only improves mathematical modeling efficiency but also simplifies the process of explaining the well-posedness of the solution.

math.AP↗
arXiv2023

Star discrepancy bounds based on Hilbert space filling curve stratified sampling and its applications

In this paper, we consider the upper bound of the probabilistic star discrepancy based on Hilbert space filling curve sampling. This problem originates from the multivariate integral approximation, but the main result removes the strict conditions on the sampling number of the classical grid-based jittered sampling. The main content has three parts. First, we inherit the advantages of this new sampling and achieve a better upper bound of the random star discrepancy than the use of Monte Carlo sampling. In addition, the convergence order of the upper bound is improved from $O(N^{-\frac{1}{2}})$ to $O(N^{-\frac{1}{2}-\frac{1}{2d}})$. Second, a better uniform integral approximation error bound of the function in the weighted space is obtained. Third, other applications will be given. Such as the sampling theorem in Hilbert spaces and the improvement of the classical Koksma-Hlawka inequality. Finally, the idea can also be applied to the proof of the strong partition principle of the star discrepancy version.

math.ST↗
arXiv2022

Expected $L_2-$discrepancy bound for a class of new stratified sampling models

We introduce a class of convex equivolume partitions. Expected $L_2-$discrepancy are discussed under these partitions. There are two main results. First, under this kind of partitions, we generate random point sets with smaller expected $L_2-$discrepancy than classical jittered sampling for the same sampling number. Second, an explicit expected $L_2-$discrepancy upper bound under this kind of partitions is also given. Further, among these new partitions, there is optimal expected $L_2-$discrepancy upper bound.

math.ST↗
arXiv2022

Star discrepancy for new stratified random sampling I: optimal expected star discrepancy

We introduce a class of convex equivolume partitions. Expected star discrepancy results are compared for stratified samples under these partitions, including simple random samples. There are four main parts of our results. First, among these newly designed partitions, there is one that minimizes the expected star discrepancy, thus we partly answer an open question in [F. Pausinger, S. Steinerberger, J. Complex. 2016]. Second, there are an infinite number of such class of partitions, which generate point sets with smaller expected discrepancy than classical jittered sampling for large sampling number, leading to an open question in [M. Kiderlen, F. Pausinger, Monatsh. Math. 2021] being solved. Third, we prove a strong partition principle and generalize the expected star discrepancy under these partition models from $L_2-$discrepancy to star discrepancy, hence an open question in [M. Kiderlen, F. Pausinger, J. Complex. 2021] is answered. In the end, optimal expected star discrepancy upper bound under this class of partitions is given, which is better than using jittered sampling.

math.ST↗
arXiv2021

Expected uniform integration approximation under general equal measure partition

In this paper, we study bounds of expected $L_2-$discrepancy to give mean square error of uniform integration approximation for functions in Sobolev space $\mathcal{H}^{\mathbf{1}}(K)$, where $\mathcal{H}$ is a reproducing Hilbert space with kernel $K$. Better order $O(N^{-1-\frac{1}{d}})$ of approximation error is obtained, comparing with previously known rate $O(N^{-1})$ using crude Monte Carlo method. Secondly, we use expected $L_{p}-$discrepancy bound($p\ge 1$) of stratified samples to give several upper bounds of $p$-moment of integral approximation error in general Sobolev space $F_{d,q}^{*}$.

math.NA↗