Computational Methods in Applied Nonlinear Dynamical Systems

本文提出了计算数学与应用动力系统的交叉研究，包括提高我们对混沌动力学在计算机模拟中的作用的理解，以及从时间序列数据推断动力结构信息的新计算方法。模拟工作是对确定性建模（包括复杂系统的计算模型）中出现的大统计误差的持续研究。第二个主要领域是研究者在解释实验多元时间序列的基本问题上的工作，包括来自物理和生物实验的数据。将特别注意由网络动力学产生的物理、化学、工程和生物/医学设置的复杂确定性时间序列。我们提出了新的方法来解决识别问题，推断网络结构和动态，并在结构已知的情况下使用数据同化方法跟踪和减少轨迹中的噪声。该项目侧重于开发新的方法来研究计算机模拟的有效性，以及对实验数据的解释。计算机模拟是现代科学的一个重要组成部分。依赖于微分方程的解的模拟受到很小的建模和截断误差的影响。该建议为分析误差对物理相关模型长期模拟结果可能产生的影响奠定了基础。该项目的第二个重点是在只有部分数学模型可用的情况下，解释从实验系统和自然界收集的数据。我们将研究一系列技术，这些技术取决于可用模型的完整性，可用于尝试重建过程的模型和动态行为，具有预测或控制过程的潜力。该项目的两个部分都涉及科学和工程的许多领域，并与生物系统特别相关。

Investigations are proposed at the overlap of computational mathematics and applied dynamical systems, including improving our understanding of the role of chaotic dynamics in computer simulations, and new computational methods for inferring information about dynamical structure from time series data. The work on simulation is an ongoing study of large statistical errors occurring in deterministic modeling, including computational models of complex systems. The second major area is work by the investigator on fundamental questions in the interpretation of experimental multivariate time series, including data from physical and biological experiments. Particular attention will be paid to complex deterministic time series from physical, chemical, engineering and biological/medical settings that are produced by network dynamics. We propose new methods to solve discrimination problems, infer network structure and dynamics, and to track and reduce noise from trajectories using data assimilation methods in cases where the structure is known. The project focuses on the development of new approaches to study computer simulation validity, and to the interpretation of experimental data. Computer simulation is a critical ingredient of modern science. Simulations that depend on the solution of differential equations are subject to small modeling and truncation errors. The proposal builds a foundation for analyzing the possible effect of the errors on long-term 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, 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. Both parts of the project have implications in many areas of sciences and engineering, and special relevance to biological systems.