Computational Methods and Data Assimilation in Nonlinear Dynamics

计算机模拟是现代科学的重要组成部分。依赖于微分方程解的模拟会出现小的建模、截断和定点算术误差。一套相对较新的算法，称为面向集合的方法，已被开发用于计算动力学中的结构分析。我们建议建立一个基础，分析这些技术和计算动力学中使用的其他错误的可能影响。该项目的第二个主要重点是当只有部分数学模型可用时，对从实验系统和自然中收集的数据进行解释。我们将研究一系列的技术，这取决于可用模型的完整性，可用于尝试重建模型和过程的动力学行为，具有预测或控制过程的潜力。由网络动力学产生的物理和生物/医学环境的复杂确定性时间序列将是特别的应用。该项目集中在两个主要领域：研究计算机模拟有效性的新方法的发展，以及非线性过程实验数据的解释。计算机模拟是现代科学的重要组成部分。模拟在建模和计算方面都受到误差的影响。该建议建立了一个基础，分析可能的影响，面向集合的仿真结果的错误，物理相关的模型。该项目的第二个主要重点是当只有部分数学模型可用时，对从实验系统和自然中收集的数据进行解释。我们将研究一系列可用于尝试重建模型和动力学行为的技术，这些技术具有预测或控制过程的潜力。

Computer simulation is a critical ingredient of modern science. Simulations that depend on the solution of differential equations are subject to small modeling, truncation, and floating point arithmetic errors. A relatively new set of algorithms, called set-oriented methods, have been developed for the analysis of structures in computational dynamics. We propose to build a foundation for analyzing the possible effect of the errors on these techniques and others used in computational dynamics. The second major focus of this project is the interpretation of data collected from experimental systems and nature when only a partial mathematical model is available. We willinvestigate a range of techniques which, depending on the completeness of the available model, can be used to attempt to reconstruct the model and dynamical behavior of the process, with potential to predict or control the process. Complex deterministic time series from physical, and biological/medical settings that are produced by network dynamics will be particular applications.The project focuses on two major areas: the development of new approaches to study computer simulation validity, and the interpretation of experimental data from nonlinear processes. Computer simulation is a critical ingredient of modern science. Simulations are subject to errors both in modeling and computation. The proposal builds a foundation for analyzing the possible effect of the errors on set-oriented simulation results, for physically relevant models. The second major focus of this project is the interpretation of data collected from experimental systems and nature when only a partial mathematical model is available. We will investigate a range of techniques which can be used to attempt to reconstruct the model and dynamical behavior, with potential to predict or control the process.