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| Determining Degrees of Freedom in Nonlinear Complex Systems: Deterministic and Stochastic Applications | |
| 项目编号 | 2009859 |
| Cecilia Mondaini | |
| 项目主持机构 | Drexel University |
| 开始日期 | 2020-08-01 |
| 结束日期 | 07/31/2023 |
| 英文摘要 | 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. |
| 资助机构 | US-NSF |
| 项目经费 | $127,085.00 |
| 项目类型 | Continuing Grant |
| 国家 | US |
| 语种 | 英语 |
| 文献类型 | 项目 |
| 条目标识符 | http://gcip.llas.ac.cn/handle/2XKMVOVA/212600 |
| 推荐引用方式 GB/T 7714 | Cecilia Mondaini.Determining Degrees of Freedom in Nonlinear Complex Systems: Deterministic and Stochastic Applications.2020. |
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