Dynamic Active Subspaces: A Data-driven Approach to Computing Time-dependent Active Subspaces in Dynamical Systems

Abstract

Computational models are aiding in the advancement of science – from biological, to engineering, to social systems. To trust the predictions of computational models, however, we must understand how the errors in the models’ inputs (i.e., through measurement error) affect the output of the systems: we must quantify the uncertainty that results from these input errors. Uncertainty quantification (UQ) becomes computationally complex when there are many parameters in the model. In such cases it is useful to reduce the dimension of the problem by identifying unimportant parameters and disregarding them for UQ studies. This makes an otherwise intractable UQ problem tractable. Active subspaces extend this idea by identifying important linear combinations of parameters, enabling more powerful and effective dimension reduction. Although active subspaces give model insight and computational tractability for scalar-valued functions, it is not enough. This analysis does not extend to time-dependent systems. In this thesis we discuss time-dependent, dynamic active subspaces. We develop a methodology by which to compute and approximate dynamic active subspaces, and introduce the analytical form of dynamic active subspaces for two cases. To highlight these methods we find dynamic active subspaces for a linear harmonic oscillator and a nonlinear enzyme kinetics system.