Simulation of stochastic systems

该项目涉及复杂随机系统(涉及不确定性)的模拟和优化。随机建模和模拟是科学、工程、经济、管理和其他几个领域的主要工具。在过去的几十年里，由于计算成本的降低以及仿真方法和软件的改进，仿真变得比用真实系统进行实验更有吸引力，成本也更低。仿真往往是处理复杂系统真实模型的唯一实用工具，这些系统通常是动态、随机和非线性的。然而，设计高效、可靠的仿真方法，并利用仿真来优化这些系统中的决策/运营策略，仍然是非常困难的。随机数发生器是随机模拟的基本组成部分。例如，电脑游戏、赌博机和密码学也需要它们。它们的质量标准因应用领域的不同而不同，因此需要不同类型的设计和从不同角度进行理论研究。拟蒙特卡罗方法用比随机点更均匀地覆盖空间的点来代替模拟中使用的随机数向量，以提高模拟结果的精度。提高效率的其他方法包括自适应地将采样集中到最重要的事件上。我的研究重点是设计和分析用于各种应用的随机数发生器，(2)通过随机准蒙特卡罗进行高维数值积分，(3)提高模拟的效率(例如，通过设计方差较小的估计器)和处理罕见事件，(4)通过模拟优化随机系统。这些方法被应用于金融、风险分析、电信、管理等多个领域。包括理论方面(例如，算法的收敛分析和随机数发生器结构的数学分析)和实际方面(例如，经验实验、软件实现)。

This project concerns the simulation and optimization of complex stochastic systems (that involve uncertainty). Stochastic modeling and simulation are primary tools in science, engineering, economics, management, and several other areas. Simulation has become more attractive and less expensive than experimenting with the real systems, due to the decreasing cost of computing and the improvement in simulation methodology and software in the last decades. Simulation is often the only practical tool to deal with realistic models of complex systems, which are typically dynamic, stochastic, and nonlinear. However, designing efficient and reliable simulation methods, and using simulation to optimize decision/operation strategies in these systems, remains very difficult. Random number generators are a fundamental ingredient for stochastic simulation. They are also needed for computer games, gambling machines, and cryptology, for example. Their quality criteria differ across areas of applications, whence the need for different types of designs and for theoretical studies from different viewpoints. Quasi-Monte Carlo methods replace the vectors of random numbers used in a simulation by points that cover the space more uniformly than random points, to improve the accuracy of the simulation results. Other ways of improving efficiency include concentrating the sampling adaptively toward the most important events. My research focuses on the design and analysis of methods for: (1) random number generators for various types of applications, (2) high-dimensional numerical integration via randomized quasi-Monte Carlo, (3) improving the efficiency of simulations (e.g., by designing estimators with smaller variance) and dealing with rare events, (4) optimization of stochastic systems via simulation. These methods are applied in several areas such as finance, risk analysis, telecommunications, management, and so on. Both theoretical aspects (e.g., convergence analysis of algorithms and mathematical analysis of the structure of random number generators) and practical ones (e.g., empirical experimentation, software implementation) are covered.