Collaborative Research: Optimal Monte Carlo Estimation via Randomized Multilevel Methods

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This research project will investigate a comprehensive set of tools to enable efficient and unbiased Monte Carlo methods in a wide range of settings such as: steady-state computations and stochastic differential equations (SDEs). The PIs extend the applicability and power of a recently introduced technique called multilevel Monte Carlo (MLMC), which has rapidly grown in popularity and has shown to be highly successful, particularly in the context of numerical solutions to SDEs. The PIs strategy rests on two basic ingredients. First, they abstract the main ideas of MLMC. This abstraction makes it clear that MLMC can be applied to many problem settings (beyond the SDE context), for example in problems such as: estimating steady-state expectations of Markov random fields, and solving distributional fixed point equations. Second, the PIs introduce a simple, yet powerful, extra randomization step. This randomization step will permit to not only completely delete the bias, which so far is present in every single application of the multilevel method, but it will also permit to more easily optimize parameters (often user-defined) that arise in classical multilevel applications. At the core of our abstraction of the MLMC method lies the construction of a suitable sequence of strong (almost sure) approximations under some metric. The freedom that is implicit in constructing such approximations yields a rich research program that touches upon many of the elements of modern probability, including random matrices, Markov random fields, mean field fixed point equations and Lyapunov stability. The PIs will investigate a methodology that enables high-performance computing in the context of simulation of stochastic systems. The PIs methodology will substantially extend a recently developed approach, called Multilevel Monte Carlo (MLMC), which has typically been applied only to compute numerical solutions of stochastic differential equations (SDEs). More generally, this research project addresses a wide range of problems that lie at the center of modern scientific computing, beyond the important setting of SDEs which arise in virtually all areas of modeling in engineering and science. For example, the PIs will generalize the MLMC approach to accurately perform so-called steady-state simulation for Markov chains indexed by trees. These computational problems arise very often in statistical inference applications, ranging from imaging to classification problems. The PIs research also improves upon the classical MLMC technique by optimizing its design and allowing the study of, for example, steady-state analysis of SDEs (i.e. combining traditional areas of study with new methodological applications). The PIs will in particular apply these optimized computational techniques to solve problems in service and manufacturing engineering. The PIs plan to develop a new jointly designed course, on the topic of this proposal, and the course material will be made available online to increase the dissemination and the potential applicability of the project's findings. The PIs will attempt to recruit high-quality personnel from under-represented groups and will disseminate the scientific output of the research via open access sites, in addition to the standard vehicles such as conferences and journal publications.