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

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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.