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Dynamic Active Subspaces

Computational and mathematical models 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 computationalÌýmodels 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.