Reduced-Complexity Algorithms for Data Assimilation of Large-Scale Systems.

Abstract

Data assimilation is the use of measurement data to improve estimates of the state of dynamical systems using mathematical models. Estimates from models alone are inherently imperfect due to the presence of unknown inputs that affect dynamical systems and model uncertainties. Thus, data assimilation is used in many applications: from satellite tracking to biological systems monitoring. As the complexity of the underlying model increases, so does the complexity of the data assimilation technique. This dissertation considers reduced-complexity algorithms for data assimilation of large-scale systems. For linear discrete-time systems, an estimator that injects data into only a specified subset of the state estimates is considered. Bounds on the performance of the new filter are obtained, and conditions that guarantee the asymptotic stability of the new filter for linear time-invariant systems are derived. We then derive a reduced-order estimator that uses a reduced-order model to propagate the estimator state using a finite-horizon cost, and hence solutions of algebraic Riccati and Lyapunov equations are not required. Finally, a reduced-rank square-root filter that propagates only a few columns of the square root of the state-error covariance is developed. Specifically, the columns are chosen from the Cholesky factor of the state-error covariance. Next, data assimilation algorithms for nonlinear systems is considered. We first compare the performance of two suboptimal estimation algorithms, the extended Kalman filter and unscented Kalman filter. To reduce the computational requirements, variations of the unscented Kalman filter with reduced ensemble are suggested. Specifically, a reduced-rank unscented Kalman filter is introduced whose ensemble members are chosen according to the Cholesky decomposition of the square root of the pseudo-error covariance. Finally, a reduced-order model is used to propagate the pseudo-error covariance, while the full-order model is used to propagate the estimator state. To compensate for the neglected correlations, a complementary static estimator gain based on the full-order steady-state correlations is also used. We use these variations of the unscented Kalman filter for data assimilation of one-dimensional compressible flow and two-dimensional magnetohydrodynamic flow.