Complex Stochastic Systems and the Effect of Discretization

复杂随机系统和离散化的影响

• 批准号: 1855788
• 负责人: Arnab Ganguly
• 金额: $ 17.39万
• 依托单位: Louisiana State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-07-01 至 2023-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1855788&HistoricalAwards=false
• 关键词: Complex; Stochastic; Systems; Effect; Discretization

Ubiquitous presence of randomness in behaviors of various physical and biological systems is universally acknowledged. For example, cellular processes, fluctuations in stock prices, weather patterns, movement of microscopic particles exhibit different types of random behaviors. Furthermore, randomness is a vital ingredient in most modern algorithms that are popular in data-science. Hence, it is of fundamental importance that various types of randomness are properly characterized for detailed understanding of system properties. Mathematical models incorporating such randomness are typically expressed in terms of stochastic equations, which often have complex dynamics. An important component of mathematical studies of these equations includes computer simulations of their temporal evolution, which are necessary for understanding their behavioral patterns over time. Accurate statistical estimation of some key parameters of these stochastic equations from observed data is also crucial, as it leads to the "most appropriate mathematical model" underlying these data points. This in turn leads to better predictive power of such models. The research supported by this award will undertake proper mathematical analyses of various numerical schemes that are used for these purposes. More specifically, research in this direction will involve precise calculations of probabilities of getting accurate results from these schemes and will describe conditions under which these probabilities are very high. There is a critical need for such theoretical results, since they will inevitably lead to design of faster and more efficient algorithms, which in turn will be beneficial to society. The project will involve undergraduate and graduate students and will help them to gain valuable analytical and computational skills. The results of the project will be published in well-known scientific journals and will also be presented at domestic and international conferences. The research will focus on the limiting behaviors of complex stochastic systems discretized by properly scaled step sizes. Discretization is at the heart of various numerical schemes, but its effect on the desired convergence properties is not always clearly understood. For example, it is well known that approximating stationary distribution of an ergodic stochastic differential equation (SDE) by time averages of sample paths obtained from an Euler-Maruyama type discretization scheme (using fixed step-size) could be problematic. In particular, such an estimator will have a bias, which may or may not be quantifiable. These are infinite-time horizon problems, and there is a deficit of proper asymptotic results in this direction even for regular Ito-diffusions. Exploring proper scaling techniques to get desired convergence along with convergence rates for such discretization-based schemes is the central goal of this project. Stochastic models that will be considered covers switching jump-diffusion models, multiscale systems and systems of interacting SDEs with possible jumps. The last class is particularly important for understanding particle-based methods for approximating nonlinear integro-differential equations including Boltzmann-type equations. These equations are connected to appropriate systems of interacting SDEs with jumps through McKean-Vlasov type limits. The stochastic equations of interest will also include Langevin SDEs, which model dynamics of molecular systems in presence of particle interaction potential, damping and random forces, and which also play a pivotal role in certain Markov Chain Monte Carlo algorithms. The research puts special emphasis on large deviation asymptotics of error probabilities, which, importantly, because of presence of both upper and lower bounds, give the optimal exponential decay rate. This project is jointly funded by Probability program and the Established Program to Stimulate Competitive Research (EPSCoR).This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.

众所周知，各种物理和生物系统的行为普遍存在随机性。例如，细胞过程、股票价格波动、天气模式、微观粒子的运动表现出不同类型的随机行为。此外，随机性是数据科学中流行的大多数现代算法的重要组成部分。 因此，正确表征各种类型的随机性对于详细理解系统属性至关重要。包含这种随机性的数学模型通常用随机方程来表示，这些方程通常具有复杂的动力学。 这些方程的数学研究的一个重要组成部分包括对其时间演化的计算机模拟，这对于理解它们随时间的行为模式是必要的。根据观测数据对这些随机方程的一些关键参数进行准确的统计估计也至关重要，因为它可以得出这些数据点背后的“最合适的数学模型”。这反过来又导致此类模型具有更好的预测能力。该奖项支持的研究将对用于这些目的的各种数值方案进行适当的数学分析。更具体地说，这个方向的研究将涉及从这些方案中获得准确结果的概率的精确计算，并将描述这些概率非常高的条件。对这样的理论结果的需求非常迫切，因为它们将不可避免地导致更快、更有效的算法的设计，而这反过来又将造福于社会。该项目将涉及本科生和研究生，并将帮助他们获得宝贵的分析和计算技能。该项目的成果将发表在知名科学期刊上，并将在国内和国际会议上发表。 该研究将重点关注通过适当缩放的步长离散化的复杂随机系统的限制行为。离散化是各种数值方案的核心，但它对所需收敛特性的影响并不总是清楚地理解。例如，众所周知，通过从 Euler-Maruyama 型离散化方案（使用固定步长）获得的样本路径的时间平均值来近似遍历随机微分方程 (SDE) 的平稳分布可能会出现问题。特别是，这样的估计器会存在偏差，该偏差可能是可量化的，也可能是不可量化的。这些是无限时间范围的问题，即使对于常规的伊藤扩散，在这个方向上也缺乏适当的渐近结果。探索适当的缩放技术以获得所需的收敛以及此类基于离散化的方案的收敛速率是该项目的中心目标。将考虑的随机模型涵盖切换跳跃扩散模型、多尺度系统以及具有可能跳跃的相互作用 SDE 的系统。最后一类对于理解基于粒子的方法来逼近非线性积分微分方程（包括玻尔兹曼型方程）特别重要。这些方程连接到适当的相互作用 SDE 系统，并跨越 McKean-Vlasov 类型限制。感兴趣的随机方程还包括 Langevin SDE，它对存在粒子相互作用势、阻尼和随机力的分子系统动力学进行建模，并且在某些马尔可夫链蒙特卡罗算法中也发挥着关键作用。 该研究特别强调误差概率的大偏差渐近，重要的是，由于存在上限和下限，给出了最佳指数衰减率。 该项目由概率计划和刺激竞争研究既定计划 (EPSCoR) 联合资助。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响审查标准进行评估，被认为值得支持。

项目成果

Inhomogeneous functionals and approximations of invariant distributions of ergodic diffusions: Central limit theorem and moderate deviation asymptotics

• DOI: 10.1016/j.spa.2020.10.009
• 发表时间: 2021-03-01
• 期刊: STOCHASTIC PROCESSES AND THEIR APPLICATIONS
• 影响因子: 1.4
• 作者: Ganguly, Arnab;Sundar, P.
• 通讯作者: Sundar, P.

Moment stability of stochastic processes with applications to control systems

随机过程的力矩稳定性及其在控制系统中的应用

• DOI: 10.3934/mcrf.2023008
• 发表时间: 2023
• 期刊: Mathematical Control and Related Fields
• 影响因子: 1.2
• 作者: Ganguly, Arnab;Chatterjee, Debasish
• 通讯作者: Chatterjee, Debasish

Infinite-dimensional optimization and Bayesian nonparametric learning of stochastic differential equations

随机微分方程的无限维优化和贝叶斯非参数学习

• 发表时间: 2023
• 期刊: Journal of machine learning research
• 影响因子: 6
• 作者: Ganguly, Arnab;Mitra, Riten;Zhou, Jinpu
• 通讯作者: Zhou, Jinpu