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，其值可能变化很大，或者可能受到数据的约束很差;因此，模型预测对参数变化的敏感性是此类计算分析中必不可少的实际考虑。 然而，穷举的、蛮力的"参数扫描"，即测试所有可能的参数，可能在计算上是昂贵的，有时根本不切实际。 所提出的研究涉及有效的数值算法计算的噪声，混沌系统的参数变化的敏感性，这些系统出现在各种不同的应用，从统计物理学到神经科学。 这项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。 这一建议结合了数值算法及其应用的研究，也是跨学科的性质，为培养未来的数学科学家提供了充分的机会，他们可以与科学家和工程师有效地合作。这一建议涉及计算非线性动力系统统计特性的算法的设计，分析和应用，这些系统是混沌的，有噪声的，并且可能是高维的。 所提出的项目的目的是（i）研究新的方差减少算法估计的期望值的观测值和它们的灵敏度噪声混沌系统;（ii）调查的效用，这种灵敏度估计在计算非线性动力学;（iii）这些一般的算法扩展到特殊类别的动力系统中的一般形式的算法可能不一定适用。 该提案包括实施，测试和分析新的数值算法的计划，以及将其应用于特定的动力系统。 由于大型，混乱和嘈杂的动力系统自然发生在各种物理和生物环境中，预计拟议的研究将产生对这些和其他领域的从业者有用的算法工具，这些类型的动力系统出现，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.