Intrinsic Complexity of Random Fields and Its Connections to Random Matrices and Stochastic Differential Equations

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Scientific computing together with effective use of data plays a more and more important role in many science and engineering applications. Modeling and understanding uncertainty or randomness inherited in these application is of utmost importance. Random fields are commonly used for modeling of space (or time) dependent stochastic processes in science and engineering problems, such as image and signal processing, Bayesian inference, data analysis, uncertainty quantification, and many other applications. It has the flexibility and generality to model randomness with spatial structures or vice versa. This project will characterize the intrinsic complexity of a random field.For the purpose of analysis as well as modeling and computation in practice, a separable representation or approximation of a random field in the form of separating deterministic and stochastic variables is very useful. This project will characterize the intrinsic complexity of a random field by providing accurate and computable lower bounds on the number of terms needed in a separable approximation of a random field for a given accuracy. This characterization can be related to the well-known notion of Kolmogorov n-width in information theory. It can reveal the intrinsic degrees of freedom (or richness) of a random field. It is also useful for an estimation of the intrinsic complexity of a system that is modeled upon a random field in real applications. For example, the investigator will study the intrinsic complexity of the solution space for partial differential equations that involve random material properties and develop efficient numerical methods that can explore low dimensional structures in these systems. By regarding a set of random vectors as the discrete sampling of a random field and vice versa, the investigator will also study the question of random vector embedding and explore its connections to random matrix theories.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.