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。包含随机效应可以服务于许多不同的目的。一方面，随机性可用于补偿系统中未完全建模的元素或系统中未充分分解的尺度。本研究计划集中于分析随机性如何在大维度、扩展动力系统中传播。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来塑造随机性。随机强迫偏微分方程（SPDEs），如随机流体方程或随机强迫反应扩散方程将是一个重点提出的工作。这些系统通常在一个尺度上搅拌，理解这种搅拌在其他尺度上的影响是数学和建模的兴趣。特别地，这一提议希望阐明湍流流动中不同尺度之间的随机性传递。除了spde之外，该提案还将研究森林环境中细胞调节和养分流动中产生的大型化学网络中的相关问题。重点将放在路径动力学，新的随机数值方法，和估计问题。在我们试图理解和预测这个复杂和不断变化的世界时，对不确定性进行量化和建模变得越来越重要。从金融市场，到天气预报，到环境生成，到神经科学，到航空学，包括随机影响在内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系统的有效模型的关键。该建议特别解决了湍流水流和细胞中复杂生化途径的核心问题。更广泛地说，应用数学家是我们科技未来的前沿。由于他们受过广泛的训练，他们最有能力进入新兴的、非传统的领域，在经典学科之间架起桥梁，并将在纯数学研究中获得的理解传播到更大的科学和社会事业中。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一流的应用数学研究人员和学生。该提案包括一个本科研究部分，以鼓励感兴趣的学生选择科学相关的职业。它还包括向当地高中伸出援助之手，让学生们在更早的时候接触现代数学，以及它所带来的无数职业可能性。

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.