The investigator uses tools from probability theory, partial differential equations, and numerical analysis to analyze and simulate interesting physical phenomena that occur on very long time scales or are very unlikely to be observed. Such problems arise in nearly all areas of science and often involve stochastic effects. The investigation of any stochastic system ultimately reduces to the sampling of some random variable. Most sampling applications are limited by the presence of strong spatial correlations and unlikely or long time scale events. Overcoming these obstacles is the key to understanding many of the most important scientific mysteries such as protein folding and climate change, and will be the subject of increased intellectual effort in the near future. This project results in several new methods for attacking problems of this kind as well as applications of these techniques to important problems in engineering, geophysics, and chemistry. In particular the investigator develops and applies new affine invariant ensemble sampling schemes that are relatively insensitive to spatial correlations. These samplers are applied to various statistical inverse problems. The investigator also mathematically studies the convergence of these samplers. The investigator applies new highly efficient importance sampling techniques for rare events to applications in geophysical data assimilation. For long time scale events such as chemical reactions the investigator introduces a new branching process that focuses computational effort on reactive trajectories as well as new parallel-in-time techniques to better utilize today's massively parallel supercomputers on problems involving electronic structure calculations. Finally the investigator rigorously establishes connections between various fine scale models of crystal relaxation and continuum descriptions. Continuum descriptions offer the possibility of reaching much longer time scales computational by removing strongly correlated degrees of freedom.
The investigator develops new techniques for studying phenomena that occur over long time scales or involve rare events. The techniques are applied to important problems such as exoplanet discovery, the prediction of rare transitions of the Kuroshio current running along the eastern coast of Japan, crystal surface growth, and several challenging chemical and bio-chemical problems. In general, techniques capable of reaching long time scales and simulating rare events have the potential to significantly impact many problems crucial to our national interests, e.g. drug design, global warming, hazard prediction for CO2 and nuclear waste storage, prediction of extreme geological events like earth quakes and volcanic eruptions, prediction of extreme weather events like hurricanes, prediction of extreme climate events like draughts, drug validation, deep time analysis (analysis of the geologic record), discovery of extra terrestrial life and life in extreme environments, and disease propagation. Even with the massive computing power currently available these problems are not expected to be solved by conventional "brute force" techniques any time soon. Techniques designed to directly interrogate the phenomena of interest and to make more efficient use of large-scale computing power are required. The development and analysis of these techniques form the core of this project.