Determining Degrees of Freedom in Nonlinear Complex Systems: Deterministic and Stochastic Applications

从我们的大脑功能到气候变化，我们物质世界的几乎每一个方面都是通过一个非线性复杂系统来描述的。为了从这些系统中提取具体信息，必须设计合适的数学模型，以便捕捉最相关的跨尺度相互作用和自由度。在这个项目中，PI将从两个主要角度研究探索复杂系统模型的方法的效率。首先，通过基于观测数据的物理量统计推断技术。一些应用适用于此范围，例如，通过测量稀释的被动标量（如染料）的浓度来确定背景流体速度场。第二个应用是分析某些流体力学模型的可计算正则化的长期行为。更具体地说，描述流体在粗糙表面上运动的地球物理模型，用于表示大气和海洋中的大规模过程。对这些基本问题的严格分析包含了广泛的数学工具。因此，在这里建立的技术可能允许在许多应用中开发有效的数值格式，以及相关数学理论的进步。此外，该项目还提供了教育方面的支持，包括对一名研究生的指导，以及对一名本科生的本科研究合作支持。对复杂系统的严格而完整的描述需要在无限维的水平上进行分析。在这个项目中，PI将这种无限维的方法来分析马尔科夫链蒙特卡罗（MCMC）算法的效率，如在贝叶斯方法中使用的反问题。特别重要的是MCMC算法在无限维上定义良好，这一特性允许相应的有限维近似克服维数的诅咒。本项目将解决该领域的几个开放问题，如一些无限维MCMC方法的混合率的推导及其相应的有限维近似。这里使用的技术主要依赖于Foias-Prodi型估计和某些耗散演化方程的有限数量的决定自由度的存在。此外，该项目还包括对某些水动力模型的长期行为的研究，例如弱阻尼二维欧拉方程受到轻度正则化。特别地，这将首先通过确定形式的概念来解决，结构编码系统的长期动力学基于知识的轨迹只有有限个自由度。随后，系统将在随机强迫项的作用下进行分析。这里，一个有趣的问题在于，当正则化参数消失时，如何建立不变测度的收敛性。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

Almost every aspect of our physical world is described through a nonlinear complex system, from the functioning of our brains to climatic changes. In order to extract concrete information from such systems, suitable mathematical models must be designed so as to capture the most relevant cross-scale interactions and degrees of freedom. In this project the PI will investigate the efficiency of methods for exploring complex system models under two main perspectives. First, through techniques of statistical inference of physical quantities based on observational data. Several applications fit within this scope, for example determining the background fluid velocity field from measurements of the concentration of a diluted passive scalar, such as dye. A second application is the analysis of the long-time behavior of computable regularizations of certain hydrodynamics models. More specifically, geophysical models describing the motion of fluids over a rough surface and used to represent large-scale processes in the atmosphere and ocean. A rigorous analysis of such fundamental questions encompasses a wide range of mathematical tools. As such, the techniques built here may allow for the development of efficient numerical schemes in numerous applications, as well as the advancement of the associated mathematical theory. In addition, this project presents an educational component by including support for the mentoring of one graduate student, as well as undergraduate research co-op support for one undergraduate student.A rigorous and complete description of complex systems requires an analysis at an infinite-dimensional level. In this project the PI will this infinite-dimensional approach to analyze the efficiency of Markov Chain Monte Carlo (MCMC) algorithms, as used in the Bayesian approach to inverse problems. Of particular importance are MCMC algorithms that are well-defined in infinite dimensions, a property that allows corresponding finite-dimensional approximations to beat the curse of dimensionality. This project will tackle several open questions in this field, such as the derivation of mixing rates for some infinite-dimensional MCMC methods and their corresponding finite-dimensional approximations. The techniques to be used here rely crucially on Foias-Prodi type estimates and the existence of a finite number of determining degrees of freedom for certain dissipative evolution equations. Furthermore, this project includes the study of the long-time behavior of certain hydrodynamic models, such as the weakly damped 2D Euler equations subject to a mild regularization. In particular, this will first be addressed through the concept of determining forms, structures encoding the long-time dynamics of the system based on knowledge on the trajectories of only a finite number of degrees of freedom. Later, the system will be analyzed under the action of a stochastic forcing term. Here, an interesting question consists in establishing the convergence of invariant measures as the regularization parameter vanishes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.