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Dynamic Active Subspaces

Computational and mathematical 91ÃÛÌÒ¸ó of physical systems are essential tools in modern engineering design, epidemiologicalÌýanalysis, and scientific exploration. Sensitivity analysis of such systems becomes computationally complex when there areÌýmany parameters in the model.ÌýActive subspacesÌý[Constantine, 2016] identify the most important linear combinationsÌýof parameters. Such analysis gives model insight and computational tractability for scalar-valued functions.

This analysis is not enough! It does not extend to time-dependent systems.ÌýExtending active subspaces to time-dependent systems will enable uncertainty quantification, sensitivity analysis, and parameter estimation for computationalÌý91ÃÛÌÒ¸ó that have explicit dependence on time.Ìý

The state-of-the-art method for identifying time-dependent active subspaces is to compute them at individual time steps. UsingÌýthis approach we identify active subspaces in various engineering and biological dynamical systems. This approach isÌýcomputationally expensive, however: it requires resampling, computing, and decomposing at every time step. In rapidÌýtransients, necessarily small time steps lead to many more computations.

To reduce computational cost we implementÌýDynamic Mode DecompositionÌý(DMD) [Kutz et al., 2016] andÌýSparse Identification for Nonlinear Dynamical SystemsÌý(SINDy) [Brunton et al., 2016] to reconstruct and predictÌýfuture active subspaces. We also derive analytical forms of time-dependent active subspaces for time-dependent outputs of twoÌýlinear parameterized dynamical systems. This analysis and computation inform visualization and insight of parameterÌýdependence in various dynamical systems.