CAREER: Stochastic analysis and numerics in partial differential equations and extended dynamical systems

随机建模在所有工程和科学学科中的重要性日益增加。包含随机效应可以用于许多不同的目的。一方面，随机性可以用于补偿系统中未完全建模的元素或系统中分辨率不足的尺度。这项研究计划集中在随机性如何通过大规模的扩展动力系统传播的分析。 其中一个核心问题是如何通过许多非线性相互作用或尺度形状的随机性。随机受迫偏微分方程（SPDE），如随机流体方程或随机受迫反应扩散方程将是拟议工作的重点。这些系统经常在一个尺度上被搅动，理解这种搅动在其他尺度上的效果具有数学和建模的意义。特别是，这一建议希望阐明湍流流体流动中不同尺度之间的随机性转移。除了SPDE外，该提案还将研究森林环境中细胞调节和营养流动所产生的大型化学网络中的相关问题。重点将放在路径动态，新的随机数值方法，估计problem.Quantifying和建模的不确定性是越来越重要的，在我们试图理解和预测我们的复杂和不断变化的世界。 从金融市场到天气预测，到环境生成，到神经科学，到航空，包括随机影响的模型已经成为物理科学，社会科学和工程学的核心。 特别重要的是理解纯粹的随机性是如何形成我们观察到的结构的，特别是当随机性在不同的空间尺度上转移时。这种理解是产生预测复杂真实的系统的有效模型的关键。这项提案特别针对湍流水流和细胞中复杂的生化途径的核心问题。更广泛地说，应用数学家是我们科学和技术未来的前沿。 由于他们广泛的培训，他们是最好的装备进入新的，非传统的领域，因为他们出现，建立经典学科之间的桥梁，并传播在更纯粹的数学研究获得更大的科学和社会事业的理解。美国在科学和技术方面的领导地位要求我们继续培养世界级的应用数学研究人员和学生。该提案包括本科研究部分，以鼓励感兴趣的学生选择与科学相关的职业。它包括推广到当地高中，让学生在更早的年龄接触现代数学和它所提供的无数职业机会。

The importance of stochastic modeling is increasing in all engineering and scientific disciplines. The inclusion of random effects can serve a number of distinct purposes. On one hand, randomness can be used to compensate for elements of a system not fully modeled or for insufficiently resolved scales in the system. This research proposal concentrates on the analysis of how randomness spreads through large dimensional, extended dynamical systems. One of the central questions is how passing through many nonlinear interactions or scales shapes the randomness. Stochastically forced partial differential equations (SPDEs) such as stochastic fluid equations or stochastically forced reaction diffusion equations will be a focus of the proposed work. These systems are often agitated at one scale and it is of both mathematical and modeling interest to understand the effect of this agitation at other scales. In particular, this proposal hopes to shed light on the transfer of randomness between different scales in turbulent fluid flow. In addition to SPDEs, the proposal will study related questions in large chemical networks arising in cell regulation and nutrient flow in forest environments. The emphasis will be on the pathwise dynamics, novel stochastic numerical methods, and estimation problems.Quantifying and modeling uncertainty is increasingly important in our attempts to understand and predict our complex and changing world. From the financial markets, to weather prediction, to environmental engendering, to neuroscience, to aeronautics, models including random influences have become central to the physical sciences, to the social sciences and to engineering. Of particular significance is understanding how pure randomness is shaped into the structures we observe, especially when the randomness is transferred across disparate spatial scales. This understanding is the key to producing effective models with which to predict complex real systems. This proposal specifically addresses questions central to turbulent water flow and complicated biochemical pathways in cells. More broadly, applied mathematicians are on the leading edge ofour scientific and technological future. Because of their broad training, they are the best equipped to move into new, non-traditional fields as they appear, build the bridges between classical disciplines, and disseminate the understanding gained in purer mathematical studies to larger scientific and social endeavors. America's leadership in science and technology requires usto continue producing world-class researchers and students in applied mathematics. This proposal includes an undergraduate research component to encourage interested students to choose a science-related career. It includes outreach to local high schools to expose students at an earlier age to modern mathematics and the myriad of career possibilities it presents.