Statistical Inference for Stochastic Processes, Analysis of MCMC Algorithms and Applications to Climate Science

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This proposal consists of three complementary themes with a particular research focus on statistical inference for data arising from dynamical systems, partial differential equations (often representing physical phenomena) and diffusions. The research questions in the proposal are motivated by the genuine need for novel statistical inference in these areas where the data is naturally high dimensional. A highlight of this proposal is the interdisciplinary nature of the problems which requires the integration of techniques from a wide spectrum of fields in applied mathematics, probability and statistics. The main themes are 1) stability of Markov Chain Monte Carlo algorithms in high dimensions, 2) statistical inference for inverse problems from diffusions and dynamical systems, and 3) applications to climate science and temperature prediction. The first two themes aim at developing methods and improving the theoretical understanding of statistical inference procedures whilst the third will directly implement the insights gained from the first two to answer a few concrete relevant and open problems in climate science. The main thread connecting the above three themes of the research is the development and theoretical analysis of novel and efficient Markov Chain Monte Carlo techniques.Advances in technology and computing power have made many historically intractable problems in statistics amenable to routine implementation using certain probabilistic algorithms. Despite two decades of intense research, our theoretical understanding of the behavior of these complex algorithms in high dimensions is still primitive. The PI proposes to study these algorithms and quantify their behavior in high dimensions, and apply them to solve concrete problems in climate science.