CAREER: Multiscale Stochastic Processes, Monte Carlo Methods and Irreversibility

Sponsor: National Science Foundation (NSF)

Award Number: 1550918

PI: Konstantinos Spiliopoulos

One of the challenges facing today applied mathematics and probability is to obtain accurate and provably efficient methods to approximate and simulate a range of complex systems using probabilistic models. Probabilistic models are commonly used to represent physical, biological and financial phenomena that are often however too complex to solve, approximate or even simulate. The primary purpose of this research is to rigorously investigate problems related to multiscale systems, rare events and related Monte Carlo simulation methods. We are interested in studying stochastic dynamical systems that may have different time scales and understudying and quantifying related events that may be rare in a given time scale, but can have important consequences for the system itself. Exact solutions are typically impossible, so one has to rely on simulation and for this reason rigorous development of provably-efficient approximation methods and simulation Monte Carlo methods is in the core of our analysis. The problems of interest in this project are motivated both by fundamental mathematical questions and by a broad array of questions in other branches of science. Examples range from estimation of rare event probabilities in chemical physics, genetic switch models in biology, tracking loop problems in engineering to network failure and large portfolio losses in complex financial systems, where multiscale features and rare events are core issues. In addition, the research project is integrated with an educational program that is designed to help in the training of undergraduate and graduate students in applied mathematics, physics, engineering and chemistry in the exploration of rare events, multiscale processes and their analysis, calibration and simulation.

This research project leads to the development of a rigorous mathematical framework that allows to design provably efficient algorithms associated to modeling and estimation of rare events as well as statistical calibration methods. We are interested in both the large deviations regime (tail or rare events) as well as in the moderate deviations regime (the gap between the typical center and the tail of the distribution). We develop provably-efficient Monte Carlo methods such as importance sampling and statistical calibration methods. In a closely related direction we rigorously investigate the consequences of violation of time-reversibility in the design of Monte Carlo methods such as steady state simulation and algorithms for acceleration of convergence to equilibrium. The research project attempts to crystallize concepts and methods that are not well understood such as the effect of metastability on Monte Carlo methods and of multiple scales on large and moderate deviations, on Monte Carlo methods and on statistical estimation methods. The goal is to provide a useful and reliable methodology for simulation and statistical procedures for complex stochastic multiscale dynamical systems that can be used by a wide collection of disciplines and also open frontiers for research. Metastability behavior and multiscale phenomena are of central interest.

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