Collaborative Proposal: Strong Stochastic Simulation of Stochastic Processes Theory and Applications

连续随机结构的高性能计算在大量的科学和工程研究中出现。例如，这些结构被用于不同地理区域的洪水环境模型，这些模型受到随机测量误差的影响。它们还用于潜在灾害的预测和减灾规划。然而，由于这些随机结构的连续性，它们不可能在不产生偏差的情况下在计算机中捕获。本课题研究了一种连续随机结构数值分析的新框架。与目前最先进的方法相比，它实现了更强的误差控制，而计算成本基本相同。如果成功，所研究的框架和算法将促进对广大科学家和工程师感兴趣的基本随机结构的分析和性能评估。为了加强更广泛的影响，首席研究员将通过研究培养研究生，并将研究结果整合到科学计算的新研究生课程中。本项目研究连续随机结构（如微分方程和随机场）的一种新的蒙特卡罗框架。该框架的主要创新特征是能够通过完全可模拟的（通常是分段常量）对象来近似连续随机对象，该对象在路径空间中具有100%确定性的统一误差界。错误界限由用户指定，可以按顺序进行细化。研究项目包括为感兴趣的基本随机结构开发模拟算法。这些包括：高斯随机场，列维过程，分数布朗运动，最大稳定场等。该算法具有可扩展性，通过应用连续映射原理和可量化的误差分析，可以很容易地扩展到更复杂的模型。该方法的一个重要方面是在随机分析的背景下建立了蒙特卡罗模拟和粗糙路径理论之间的联系。

High performance computing of continuous random structures arises in a large body of scientific and engineering investigations. For example, these structures are used in environmental models for floods in different geographical areas, which are subject to random measurement errors. They are also used in the prediction and mitigation planning of potential disasters. However, these random structures are impossible to capture in a computer without incurring bias, due to their continuous nature. This research project investigates a new framework for the numerical analysis of continuous random structures. It achieves stronger error control, compared to current state-of-the-art methods, at basically the same computational cost. If successful, the framework and algorithms to be investigated will facilitate analysis and performance evaluation of fundamental random structures of interests to a broad community of scientists and engineers. To enhance the broader impact, the Principal Investigators will train graduate students through research and integrate the results from this research into new graduate courses in scientific computing. This project investigates a new Monte Carlo framework for continuous stochastic structures (such as differential equations and random fields). The main innovative feature of the framework is the ability to approximate a continuous random object by a fully simulatable (typically piece-wise constant) object with a uniform error bound in the path space with 100% certainty. The error bound is user-specified and can be sequentially refined. Research projects involve developing simulation algorithms for fundamental random structures of interests. These include: Gaussian random fields, Levy processes, fractional Brownian motion, max-stable fields, etc. The algorithms are scalable in the sense of being easily extendable to more complex models by applying the continuous mapping principle with quantifiable error analysis. An important aspect of the methodology is the connection established between Monte Carlo simulation and the theory of rough paths in the setting of stochastic analysis.