Asymptotic Problems of Partial Differential Equations with Random Coefficients: Homogenization and Beyond

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This research is devoted to the analysis of systems with parameters that contain uncertainties and vary on multiple scales, and the goal is to understand their effects on the dynamics of the systems. Examples of such systems are ubiquitous in nature: climate and weather developments, oceanography, and seismology, just to name a few. Similarly, multiscale processes with built-in uncertainty are common in many large-scale societal phenomena, from stock market behavior to propagation of information through social media. The mathematical subject is partial differential equations (PDE) with random and highly oscillatory coefficients, and the project is aimed at deriving effective models that incorporate the impacts from the randomness and the separation of scales. The project is expected to result in connections to multi-scale algorithms, inverse problems, imaging, and uncertainty quantification, as well as to have applications in subjects such as climate prediction, geophysics, and materials science. The project involves three investigations, with a focus on questions related to stochastic homogenization, wave in random media and stochastic PDE: (i) study the quantitative aspects, including convergence rates and statistical fluctuations, of stochastic homogenization and macroscopic models of wave propagation in random media; (ii) analyze the effects of different correlation properties of the randomness on both qualitative and quantitative aspects; and (iii) investigate the transition from deterministic to stochastic models, and study the convergence to stochastic PDE beyond the homogenization regime.