Long Time Scales and Unlikely Events: Sampling and Coarse Graining Strategies

WeareDMS-1109731研究人员使用概率论、偏微分方程组和数值分析的工具来分析和模拟在很长时间尺度上发生的有趣的物理现象或非常不可能观察到的物理现象。这类问题几乎出现在科学的所有领域，而且往往涉及随机效应。对任意随机系统的研究最终归结为对某个随机变量的抽样。大多数采样应用都受到强空间相关性和不太可能的或长时间尺度事件的存在的限制。克服这些障碍是理解许多最重要的科学谜团的关键，例如蛋白质折叠和气候变化，并将在不久的将来成为更多智力努力的主题。这个项目导致了几种新的方法来解决这类问题，并将这些技术应用于工程、地球物理和化学中的重要问题。特别是，研究人员开发和应用了对空间相关性相对不敏感的新的仿射不变集成采样方案。这些采样器被应用于各种统计反问题。研究人员还从数学上研究了这些采样器的收敛问题。研究人员将针对罕见事件的新的高效重要抽样技术应用于地球物理数据同化。对于长时间尺度的事件，如化学反应，研究人员引入了一种新的分支过程，将计算工作集中在反应轨迹上，以及新的时间并行技术，以更好地利用今天的大规模并行超级计算机来处理涉及电子结构计算的问题。最后，研究人员严格地建立了各种精细的晶体松弛模型和连续介质描述之间的联系。连续体描述提供了通过去除强相关自由度来达到计算的更长时间尺度的可能性。研究人员开发了研究长时间尺度上发生的现象或涉及罕见事件的新技术。这些技术被应用于重要的问题，如系外行星的发现，日本东海岸黑潮洋流罕见过渡的预测，晶体表面的生长，以及几个具有挑战性的化学和生物化学问题。一般而言，能够达到长时间尺度和模拟罕见事件的技术可能会对许多对我们国家利益至关重要的问题产生重大影响，例如药物设计、全球变暖、二氧化碳和核废料储存的危险预测、地震和火山喷发等极端地质事件的预测、飓风等极端天气事件的预测、药物验证、深度时间分析(地质记录分析)、发现地外生命和极端环境中的生命以及疾病传播。即使目前拥有巨大的计算能力，这些问题也不可能很快通过传统的“蛮力”技术得到解决。需要设计技术来直接询问感兴趣的现象，并更有效地利用大规模计算能力。这些技术的开发和分析构成了本项目的核心。

WeareDMS-1109731 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.