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.