Zexin Pan

Zexin Pan

This work analyzes the convergence of sums of the form $S_{\boldsymbolγ}(m)=\sum_{v\subseteq \mathbb{N}}γ_v m^{|v|}$ with product and order dependent (POD) weights $γ_v$. We establish that for a nonnegative sequence $\{Υ_j\mid j\in \mathbb{N}\}$, $$\sum_{v\subseteq \mathbb{N}} |v|! m^{|v|}\prod_{j\in v} Υ_j<\infty \text{ for all } m>0 \text{ if and only if } \sum_{j=1}^\infty Υ_j<\infty.$$ We further characterize the growth of $S_{\boldsymbolγ}(m)$ when $γ_v=(|v|!)^σ\prod_{j\in v}j^{-ρ}$ and prove that $\log S_{\boldsymbolγ}(m)$ is of asymptotic order $m^{1/(ρ-σ)}$ when $ρ>σ\geq 0$. We subsequently generalize both the convergence criterion and the asymptotic order of $\log S_{\boldsymbolγ}(m)$ to smoothness-driven product and order dependent (SPOD) weights, while noting that a full necessary-and-sufficient analogue remains open. Finally, we apply our theory to quasi-Monte Carlo (QMC) integration, showing that interlaced polynomial lattice rules achieve a dimension-independent convergence rate without a commonly imposed assumption in the QMC literature.

Ziyang Ye, Xiaoqun Wang, Zexin Pan

We propose a median digital-net algorithm for $L^2$-approximation of non-periodic functions over $[0,1]^s$, inspired by the recently developed median lattice algorithms for the periodic setting. The algorithm requires no smoothness or weight parameters but only a sufficiently large candidate Walsh index set $K$. It proceeds in three stages: generating multiple estimates of the Walsh coefficients in $K$ using independent randomized digital-net samples; taking the respective median of both the estimates and their absolute values; then, based on these median values, identifying the dominant coefficients and constructing a truncated Walsh series as the final approximation. We prove that if the target function has dominating mixed partial derivatives up to order $α$, all having finite Vitali variation of fractional order $λ$, then the algorithm achieves an $L^2$-error of $\mathcal{O}(M^{-α-λ+ε})$ with high probability, where $M$ is the total number of function evaluations and $ε>0$ is arbitrarily small. Furthermore, the implied constant grows at most polynomially in the dimension $s$ under suitable decay conditions on the ANOVA components of the target function. On the implementation side, we provide both parameter-dependent and -independent constructions of the index set $K$, and employ the fast Walsh--Hadamard transform and Gray code to accelerate the algorithm. Numerical experiments support the theoretical analysis and demonstrate that the proposed algorithm remains effective in high-dimensional settings.

Du Ouyang, Zexin Pan, Zhijian He

We investigate the numerical simulation of general stochastic differential equations (SDEs) using Quasi-Monte Carlo (QMC) methods. First, we provide a rigorous theoretical analysis of the QMC method applied to the Euler-Maruyama (EM) scheme, establishing that it significantly accelerates the decay of the sampling error and achieves an asymptotically superior convergence rate over the classical Monte Carlo method. Second, the traditional EM scheme exhibits a slow polynomial decay of the discretization error, which necessitates a large number of time steps and leads to a significantly high integration dimension. To address this issue, we propose a Multilevel Stochastic Time Grid (MSTG) method based on Exact Simulation techniques, and we rigorously establish its convergence rate under randomized QMC sampling, proving that it preserves the high-order convergence of the sampling error. In terms of the overall error, the truncation error of the proposed MSTG method exhibits a remarkably fast super-exponential decay. Consequently, to achieve a given accuracy level, our approach requires significantly fewer discretization steps than the EM scheme, thereby drastically reducing the actual integration dimension of the QMC method. This substantial dimensionality reduction strategy greatly enhances the practical efficiency of the QMC algorithm. Numerical experiments fully corroborate the superiority of the proposed approach.