For large-scale system with tens of thousands of states and outputs the computation in the conventional Kalman filter becomes time-consuming such that Kalman filtering in large-scale real-time application is practically infeasible. A possible mathematical framework to lift the curse of dimensionality is to lift the problem in higher dimensions with the use of tensors and then decompose it. The tensor-train decomposition is chosen due to its computational advantages for systems with low tensor-train rank. Within this thesis two main limitations of the existing tensor Kalman filter are solved. First, a method is developed based on tensor-train rank truncation of the covariances to increase the computational speed for more general systems. Second, a MIMO tensor Kalman filter is developed for a specific class of systems. The power of the developed methods is shown on the example of adaptive optics which fits into the framework. A comparison with state-of-the-art large-scale estimation algorithms shows the computational advantage of the tensor Kalman filter at the cost of approximation errors.

The use of radial basis function (RBF) expansion as a means to represent the generalized pupil function for aberration retrieval from intensity point-spread function has been investigated in this research article. The optimal choice of RBF hyper-parameters is derived empirically to achieve an increased accuracy of approximation along with well-conditioned basis. The phase retrieval problem is solved using PhaseLift, an algorithm based on matrix rank minimization and also using a variant of the alternating projections algorithm proposed by Gerchberg-Saxton. The performance of the RBF-based method is compared in terms of accuracy and execution time with that based on the extended Nijboer-Zernike approach. Numerical results based on simulations are presented.

The visualization of objects within or beyond a turbulent medium is hampered by the aberrations the medium induces in the wavefront. The sharpness of an image is maximal when the incoming wavefront is flat, with aberrated wavefronts yielding distorted images of limited utility. In the particular case of astronomy, the flat wavefront of the light from a distant target is aberrated as it moves through the atmosphere, before reaching our telescopes. The field of Adaptive Optics (AO) is dedicated to the correction of these aberrations, with the goal of enabling the imaging of objects through turbulent media with as much detail as possible. Due to a delay inherent to the AO control loop, each control input is used to correct an aberration slightly ahead in time, which means that, for the correction to be effective, the controller must be equipped with predictive capabilities. Prediction demands a model, and fortunately, the dynamics of atmospheric turbulence can be effectively modelled using a linear system. This makes the Kalman filter, the statistically optimal state estimator for linear systems, a natural choice for prediction. However, the steady-state Kalman filter requires a solution to the computationally intensive Discrete-time Algebraic Riccati Equation (DARE), which precludes its application to large-scale systems. Furthermore, the Kalman filter assumes a precise model with particular properties is available, which often is not the case in practice. The advent of extremely large telescopes demands prediction algorithms that can handle likewise extremely large system dimensions, which means that application of the steady-state Kalman filter, in its conventional form, is unfeasible. This thesis proposes a data-driven approach to Kalman filtering that exploits the intuitive sparsity pattern of the matrices involved in the modelling of an AO system. The developed algorithm replaces the DARE and, with knowledge of the system matrices, estimates the Kalman gain from measurement data, with the exploitation of sparsity providing a substantial drop in complexity when compared to the DARE, and with its data-driven nature inherently compensating, to a certain extent, for modelling errors. This thesis further proposes a two-stage approach to reduce the burden of the online prediction operation. While the measurement and state-transition matrices of the AO system are sparse, the Kalman gain is generally dense, consequentially slowing down prediction, which should accompany the sampling period of the loop. Our proposal splits prediction into a sparse stage for prediction of local structures in the wavefront phase, and a low-dimensional dense stage for prediction of the remaining low-frequency aberrations. The two-stage predictor is inherently suboptimal, but allows for quicker prediction and identification for extremely large systems. Via tuning, the user strikes a balance between performance and execution time.