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

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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.