arXiv subjects

Chenxu Pang

Publications and source records attributed to Chenxu Pang.

4 recordsLinked to original sources

arXiv2026

An explicit scheme for stochastic Allen-Cahn equations with space-time white noise near the sharp interface limit

This article investigates time-discrete approximations of Allen-Cahn type SPDEs driven by space-time white noise near the sharp interface limit $ε\to 0$, where the small parameter $ε$ is the diffuse interface thickness. We propose an explicit and easily implementable exponential integrator with a modified nonlinearity for the considered problem. Uniform-in-time and uniform-in-$ε$ moment bounds of the scheme are established and the convergence in total variation distance of order $O(T\cdot\text{Poly}(ε^{-1})τ^γ),γ<\tfrac12$ is established, between the law of the numerical scheme and that of the SPDE over $[0,T]$. In contrast to the exponential dependence due to standard arguments, the obtained error bound depends on $ε^{-1}$ and $T$ polynomially. By incorporating carefully chosen method parameters, we only require a mild and $ε$-independent restriction on the time step-size $τ$, getting rid of the severe restriction $τ=O(ε^σ),σ\geq1$ in the literature. Also, a uniform-in-time error bound of order $O(τ^γ),γ<\tfrac12$, is obtained for a fixed $ε=1$, which improves the existing ones in the literature and matches the classical weak convergence rate in the globally Lipschitz setting. The error analysis is highly nontrivial due to the low regularity of the considered problem, the super-linear growth of the drift, the non-smooth observables inherent in the total variation metric and the presence of the small interface parameter $ε\to0$. These difficulties are addressed by introducing a new strategy of nonlinearity modification and establishing refined regularity estimates for the associated Kolmogorov equation to an auxiliary process with non-smooth test functions. Numerical experiments confirm the theoretical convergence and the ability of interface-capturing for the proposed scheme.

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arXiv2025

Projected Langevin Monte Carlo algorithms in non-convex and super-linear setting

It is of significant interest in many applications to sample from a high-dimensional target distribution $π$ with the density $π(\text{d} x) \propto e^{-U(x)} (\text{d} x) $, based on the temporal discretization of the Langevin stochastic differential equations (SDEs). In this paper, we propose an explicit projected Langevin Monte Carlo (PLMC) algorithm with non-convex potential $U$ and super-linear gradient of $U$ and investigate the non-asymptotic analysis of its sampling error in total variation distance. Equipped with time-independent regularity estimates for the associated Kolmogorov equation, we derive the non-asymptotic bounds on the total variation distance between the target distribution of the Langevin SDEs and the law induced by the PLMC scheme with order $\mathcal{O}(d^{\max\{3γ/2 , 2γ-1 \}} h |\ln h|)$, where $d$ is the dimension of the target distribution and $γ\geq 1$ characterizes the growth of the gradient of $U$. In addition, if the gradient of $U$ is globally Lipschitz continuous, an improved convergence order of $\mathcal{O}(d^{3/2} h)$ for the classical Langevin Monte Carlo (LMC) scheme is derived with a refinement of the proof based on Malliavin calculus techniques. To achieve a given precision $ε$, the smallest number of iterations of the PLMC algorithm is proved to be of order ${\mathcal{O}}\big(\tfrac{d^{\max\{3γ/2 , 2γ-1 \}}}ε \ \cdot \ln (\tfrac{d}ε) \cdot \ln (\tfrac{1}ε) \big)$. In particular, the classical Langevin Monte Carlo (LMC) scheme with the non-convex potential $U$ and the globally Lipschitz gradient of $U$ can be guaranteed by order ${\mathcal{O}}\big(\tfrac{d^{3/2}}ε \cdot \ln (\tfrac{1}ε) \big)$. Numerical experiments are provided to confirm the theoretical findings.

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arXiv2024

Antithetic multilevel Monte Carlo method for approximations of SDEs with non-globally Lipschitz continuous coefficients

In the field of computational finance, one is commonly interested in the expected value of a financial derivative whose payoff depends on the solution of stochastic differential equations (SDEs). For multi-dimensional SDEs with non-commutative diffusion coefficients in the globally Lipschitz setting, a kind of one-half order truncated Milstein-type scheme without Lévy areas was recently introduced by Giles and Szpruch (2014), which combined with the antithetic multilevel Monte Carlo (MLMC) gives the optimal overall computational cost $\mathcal{O}(ε^{-2})$ for the required target accuracy $ε$. Nevertheless, many nonlinear SDEs in applications have non-globally Lipschitz continuous coefficients and the corresponding theoretical guarantees for antithetic MLMC are absent in the literature. In the present work, we aim to fill the gap and analyze antithetic MLMC in a non-globally Lipschitz setting. First, we propose a family of modified Milstein-type schemes without Lévy areas to approximate SDEs with non-globally Lipschitz continuous coefficients. The expected one-half order of strong convergence is recovered in a non-globally Lipschitz setting, where even the diffusion coefficients are allowed to grow superlinearly. This then helps us to analyze the relevant variance of the multilevel estimator and the optimal computational cost is finally achieved for the antithetic MLMC. Since getting rid of the Lévy areas destroys the martingale properties of the scheme, the analysis of both the convergence rate and the desired variance becomes highly non-trivial in the non-globally Lipschitz setting. By introducing an auxiliary approximation process, we develop non-standard arguments to overcome the essential difficulties. Numerical experiments are provided to confirm the theoretical findings.

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arXiv2023

Linear implicit approximations of invariant measures of semi-linear SDEs with non-globally Lipschitz coefficients

This article investigates the weak approximation towards the invariant measure of semi-linear stochastic differential equations (SDEs) under non-globally Lipschitz coefficients. For this purpose, we propose a linear-theta-projected Euler (LTPE) scheme, which also admits an invariant measure, to handle the potential influence of the linear stiffness. Under certain assumptions, both the SDE and the corresponding LTPE method are shown to converge exponentially to the underlying invariant measures, respectively. Moreover, with time-independent regularity estimates for the corresponding Kolmogorov equation, the weak error between the numerical invariant measure and the original one can be guaranteed with convergence of order one. In terms of computational complexity, the proposed ergodicity preserving scheme with the nonlinearity explicitly treated has a significant advantage over the ergodicity preserving implicit Euler method in the literature. Numerical experiments are provided to verify our theoretical findings.

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