arXiv subjects

Jiarui Du

Publications and source records attributed to Jiarui Du.

6 recordsLinked to original sources

arXiv2026

A deterministic multiple-shift lattice algorithm for function approximation in Korobov and half-period Cosine spaces

Approximating multivariate periodic functions in weighted Korobov spaces via rank-1 lattices is fundamentally limited by frequency aliasing. Existing optimal-rate methods rely on randomized constructions or large pre-computations. We propose a fully deterministic multiple-shift lattice algorithm without pre-computation. First, we develop a simplified multiple shift framework for aliased frequency fibers that reduces sampling costs. Second, leveraging the Chinese Remainder Theorem and the Weil bound, we introduce an adaptive hybrid construction that algebraically guarantees the full rank and bounded condition number of the reconstruction matrix. We rigorously prove that this deterministic method maintains the optimal convergence rate in the worst-case setting. Furthermore, we extend this framework to non-periodic, half-period cosine spaces via the tent transformation. By establishing a strict projection equivalence, we prove that the algorithm attains optimal $L_2$ and $L_\infty$ approximation orders in the half-period cosine space, successfully resolving an open theoretical problem posed by Suryanarayana et al. (2016). This mathematically also validates the proposed algorithm as a generic meshless spectral solver for high-dimensional boundary value problems, such as the Poisson equation with Neumann conditions. Numerical experiments corroborate the theoretical bounds, demonstrating an order-of-magnitude reduction in sampling complexity over probabilistic baselines while ensuring absolute deterministic stability.

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arXiv2026

The inverse of the star discrepancy of a union of randomly shifted Korobov rank-1 lattice point sets depends polynomially on the dimension

The inverse of the star discrepancy, $N(\epsilon, s)$, defined as the minimum number of points required to achieve a star discrepancy of at most $\epsilon$ in dimension $s$, is known to depend linearly on $s$. However, explicit constructions achieving this optimal linear dependence remain elusive. Recently, Dick and Pillichshammer (2025) made significant progress by showing that a multiset union of randomly digitally shifted Korobov polynomial lattice point sets almost achieve the optimal dimension dependence with high probability. In this paper, we investigate the analog of this result in the setting of classical integer arithmetic using Fourier analysis. We analyze point sets constructed as multiset unions of Korobov rank-1 lattice point sets modulo a prime $N$. We provide a comprehensive analysis covering four distinct construction scenarios, combining either random or fixed integer generators with either continuous torus shifts or discrete grid shifts. We prove that in all four cases, the star discrepancy is bounded by a term of order $O(s \log(N_{tot}) / \sqrt{N_{tot}})$ with high probability, where $N_{tot}$ is the total number of points. This implies that the inverse of the star discrepancy for these structured sets depends quadratically on the dimension $s$. While the proofs are probabilistic, our results significantly reduce the search space for optimal point sets from a continuum to a finite set of candidates parameterized by integer generators and random shifts.

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arXiv2025

The $L_p$-error rate for randomized quasi-Monte Carlo self-normalized importance sampling of unbounded integrands

Self-normalized importance sampling (SNIS) is a fundamental tool in Bayesian inference when the posterior distribution involves an unknown normalizing constant. In many applications, both the test function of interest and the underlying state space are unbounded, making direct $L_1$-error (mean absolute error) and $L_2$-error (root mean square error) estimates challenging for SNIS under randomized quasi-Monte Carlo (RQMC) sampling. In this work, we derive the $L_p$-error rate $(p\ge1)$ for RQMC-based SNIS (RQMC-SNIS) estimators with unbounded integrands on unbounded domains. A key step in our analysis is to first establish the $L_p$-error rate for plain RQMC integration. Our results allow for a broader class of transport maps used to generate samples from RQMC points. Under mild function boundary growth conditions, we further establish the $L_p$-error rate of order $\mathcal{O}(N^{-\beta + \epsilon})$ for RQMC-SNIS estimators, where $\epsilon>0$ is arbitrarily small, $N$ is the sample size, and $\beta \in (0,1]$ depends on the boundary growth rate of the resulting integrand. Numerical experiments validate the theoretical results.

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arXiv2025

Randomized Quasi-Monte Carlo and Importance Sampling for Super-Fast Growing Functions with Applications to Finance

Many problems can be formulated as high-dimensional integrals of discontinuous functions that exhibit significant boundary growth, challenging the error analysis and applications of randomized quasi-Monte Carlo (RQMC) methods. This paper studies RQMC methods for super-fast growing functions satisfying generalized exponential growth conditions, with a special focus on financial derivative pricing. The main contribution of this paper is threefold. First, by combining RQMC with importance sampling (IS), we derive a new error bound for a class of integrands, whose values and derivatives are bounded by the critical growth function $e^{A|\boldsymbol{x}|^2}$ with $A = 1/2$. This result extends the existing results in the literature, which are limited to the case $A < 1/2$. We demonstrate that by imposing a light-tailed condition on the proposal distribution of IS, RQMC can achieve an error rate of $O(n^{-1 + \epsilon})$ with a sample size n and an arbitrarily small $\epsilon >0$. Second, we verify that the Gaussian proposals used in Optimal Drift Importance Sampling (ODIS) satisfy the required light-tailed condition, providing a rigorous theoretical guarantees for RQMC-ODIS in critical growth scenarios. Third, for discontinuous integrands from finance, we prove that the integrands after preintegration satisfy the exponential growth condition. This ensures that the preintegrated functions can be seamlessly incorporated into our RQMC-IS framework. Numerical experiments on financial derivative pricing validate our theory, showing that the RQMC-IS with preintegration is effective in handling problems with discontinuous payoffs, successfully achieving the expected convergence rates.

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arXiv2025

Enhanced convergence rates of Adaptive Importance Sampling with recycling schemes via quasi-Monte Carlo methods

This article investigates the integration of quasi-Monte Carlo (QMC) methods using the Adaptive Multiple Importance Sampling (AMIS). Traditional Importance Sampling (IS) often suffers from poor performance since it heavily relies on the choice of the proposal distributions. The AMIS and the Modified version of AMIS (MAMIS) address this by iteratively refining proposal distributions and reusing all past samples through a recycling strategy. We introduce the RQMC methods into the MAMIS, achieving higher convergence rates compared to the Monte Carlo (MC) methods. Our main contributions include a detailed convergence analysis of the MAMIS estimator under randomized QMC (RQMC) sampling. Specifically, we establish the $L^q$ $(q \geq 2)$ error bound for the RQMC-based estimator using a smoothed projection method, which enables us to apply the H\"older's inequality in the error analysis of the RQMC-based MAMIS estimator. As a result, we prove that the root mean square error of the RQMC-based MAMIS estimator converges at a rate of $\mathcal{O}(\bar{N}_T^{-1+\epsilon})$, where $\bar{N}_T$ is the average number of samples used in each step over $T$ iterations, and $\epsilon > 0$ is arbitrarily small. Numerical experiments validate the effectiveness of our method, including mixtures of Gaussians, a banana-shaped model, and Bayesian Logistic regression.

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arXiv2024

Unbiased Markov chain quasi-Monte Carlo for Gibbs samplers

In statistical analysis, Monte Carlo (MC) stands as a classical numerical integration method. When encountering challenging sample problem, Markov chain Monte Carlo (MCMC) is a commonly employed method. However, the MCMC estimator is biased after a fixed number of iterations. Unbiased MCMC, an advancement achieved through coupling techniques, addresses this bias issue in MCMC. It allows us to run many short chains in parallel. Quasi-Monte Carlo (QMC), known for its high order of convergence, is an alternative of MC. By incorporating the idea of QMC into MCMC, Markov chain quasi-Monte Carlo (MCQMC) effectively reduces the variance of MCMC, especially in Gibbs samplers. This work presents a novel approach that integrates unbiased MCMC with MCQMC, called as an unbiased MCQMC method. This method renders unbiased estimators while improving the rate of convergence significantly. Numerical experiments demonstrate that for Gibbs sampling, unbiased MCQMC with a sample size of $N$ yields a faster root mean square error (RMSE) rate than the $O(N^{-1/2})$ rate of unbiased MCMC, toward an RMSE rate of $O(N^{-1})$ for low-dimensional problems. Surprisingly, in a challenging problem of 1049-dimensional P\'olya Gamma Gibbs sampler, the RMSE can still be reduced by several times for moderate sample sizes. In the setting of parallelization, unbiased MCQMC also performs better than unbiased MCMC, even running with short chains.

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