Computational Nonlinear Dynamics: Variance Reduction Methods and Numerical Studies of Large, Chaotic, and Noisy Systems

科学家和工程师越来越依赖于数学模型的计算分析来理解、预测、设计和控制物理和生物系统中的动态过程。模型通常包含数值参数，这些参数的值可能变化很大，或者可能不受数据的约束；因此，在这种计算分析中，模型预测对参数变化的敏感性是一个重要的实际考虑因素。然而，穷举的、暴力的“参数扫描”，即测试所有可能的参数，在计算上可能是昂贵的，有时甚至是不切实际的。提出的研究涉及计算噪声、混沌系统对参数变化敏感性的有效数值算法；这些系统出现在各种不同的应用中，从统计物理到神经科学。提出的研究可能有助于这些领域的研究人员更有效地进行数学模型的计算分析。该提案结合了数值算法及其应用的研究，本质上也是跨学科的，并为培训未来能够与科学家和工程师有效合作的数学科学家提供了充足的机会。本提案涉及设计、分析和应用算法来计算混沌、噪声和潜在高维的非线性动力系统的统计特性。提出的项目旨在(i)研究新的方差减少算法，用于估计嘈杂混沌系统的可观测值期望值及其灵敏度；（ii）研究这种灵敏度估计器在计算非线性动力学中的效用；（iii）将这些一般算法扩展到可能不一定适用所提出算法的一般形式的特殊类型的动力系统。该提案包括实施，测试和分析新的数值算法的计划，以及将它们应用于特定的动力系统。由于大的、混沌的、嘈杂的动力系统在各种物理和生物环境中自然发生，拟议的研究有望产生对这些和其他领域的从业者有用的算法工具，这些领域中出现了这些类型的动力系统，并将直接适用于PI、他的学生和合作者感兴趣的一系列问题。

Scientists and engineers increasingly depend on computational analyses of mathematical models to understand, predict, design, and control dynamic processes in physical and biological systems. Models often contain numerical parameters whose values may vary widely, or may be poorly constrained by data; the sensitivity of model predictions to parameter variations is thus an essential practical consideration in such computational analyses. However, exhaustive, brute-force "parameter sweeps," in which one tests all possible parameters, can be computationally expensive and is sometimes simply impractical. The proposed research concerns efficient numerical algorithms for computing sensitivities of noisy, chaotic systems to parameter variations; these systems arise in a variety of different applications, ranging from statistical physics to neuroscience. The proposed research can potentially help researchers in these fields perform computational analyses of mathematical models more efficiently. The proposal, combining as it does the study of numerical algorithms and their applications, is also interdisciplinary in nature and provides ample opportunities for the training of future mathematical scientists who can collaborate effectively with scientists and engineers.This proposal concerns the design, analysis, and application of algorithms for computing the statistical properties of nonlinear dynamical systems that are chaotic, noisy, and potentially high-dimensional. The proposed projects aim to (i) study novel variance reduction algorithms for estimating expectation values of observables and their sensitivities for noisy chaotic systems; (ii) investigate the utility of such sensitivity estimators in computational nonlinear dynamics; (iii) extend these general algorithms to special classes of dynamical systems where the general form of the proposed algorithms may not necessarily apply. The proposal includes plans for implementing, testing, and analyzing novel numerical algorithms, as well as applying them to specific dynamical systems. As large, chaotic, and noisy dynamical systems occur naturally in a variety of physical and biological contexts, the proposed research is expected to produce algorithmic tools useful to practitioners in these and other fields where these types of dynamical systems arise, and will be directly applicable to a range of problems of interest to the PI, his students, and collaborators.