Simulation of stochastic systems

这个项目涉及复杂随机系统（包含不确定性）的模拟和优化。随机建模和模拟是科学、工程、经济、管理和其他一些领域的主要工具。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，仿真已经变得比用真实系统进行实验更有吸引力和更便宜。模拟通常是处理复杂系统的现实模型的唯一实用工具，这些模型通常是动态的、随机的和非线性的。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的决策/操作策略，仍然是非常困难的。随机数生成器是随机模拟的基本组成部分。例如，计算机游戏、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。拟蒙特卡罗方法将模拟中使用的随机数向量替换为比随机点更均匀地覆盖空间的点，以提高模拟结果的准确性。其他提高效率的方法包括将采样自适应地集中在最重要的事件上。我的研究重点是设计和分析方法：(1)各种类型应用的随机数生成器，(2)通过随机准蒙特卡罗进行高维数值积分，(3)提高模拟效率（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。这些方法应用于金融、风险分析、电信、管理等多个领域。理论方面（例如，算法的收敛分析和随机数生成器结构的数学分析）和实践方面（例如，经验实验，软件实现）都包括在内。

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.