Hessian-based model reduction for large-scale data assimilation problems

Assimilation of spatially- and temporally-distributed state observations into simulations of dynamical systems stemming from discretized PDEs leads to inverse problems with high-dimensional control spaces in the form of discretized initial conditions. Solution of such inverse problems in "real-time" is often intractable. This motivates the construction of reduced-order models that can be used as surrogates of the high-fidelity simulations during inverse solution. For the surrogates to be useful, they must be able to approximate the observable quantities over a wide range of initial conditions. Construction of the reduced models entails sampling the initial condition space to generate an appropriate training set, which is an intractable proposition for high dimensional initial condition spaces unless the problem structure can be exploited. Here, we present a method that extracts the dominant spectrum of the input-output map (i.e. the Hessian of the least squares optimization problem) at low cost, and uses the principal eigenvectors as sample points. We demonstrate the efficacy of the reduction methodology on a large-scale contaminant transport problem.