Recursive Kronecker-based Vector Auto-Regressive identification for large-scale Adaptive Optics systems

Abstract

Adaptive Optics (AO) play an essential role in the field of astronomical seeing for ground-based telescopes, providing a corrected image from the disturbance caused by the light having to travel through the turbulent atmosphere of our Earth. To provide us with more detailed images, the sizes of these telescopes are getting increasingly larger. As a consequence to the increased size for the primary mirror comes the increase in the number of actuators necessary for the deformable mirror as well as the increase in phase point measurements by the wavefront sensor. Apart from the technical challenges of constructing such devices, there is the challenge for adaptive optics control systems to handle such a large amount of inputs and outputs. Many control methods that are used for adaptive optics rely on some prior knowledge about the atmospheric turbulence to estimate a model for use in closed loop operation. By relying on this prior knowledge, the inherently changing dynamics of the turbulence are neglected and the accuracy of the control system will suffer when the changes in atmospheric turbulence are large and frequent. Rather than relying on first principles, we derive in this thesis scalable data-driven methods to obtain better prediction performances. This thesis suggests an approach for adaptively optimizing the coefficients of a Vector Auto-Regressive (VAR) model where the coefficient matrices can be written as a sum of Kronecker products. This approach relies on the Recursive Least Squares (RLS) algorithm to continuously update the coefficients of the model to account for the changing dynamics of the atmospheric turbulence and uses the Kronecker product together with Alternating Least Squares (ALS) to reduce the computational complexity while still guaranteeing similar variance of the prediction error. The accuracy of this approach is compared to other commonly used control methods by subjecting it to generated non-stationary turbulence. Numerical experiments show that a significant increase in accuracy is achievable for this approach compared to unstructured non-recursive VAR models due to its recursive nature as well as a reduction in computational complexity due to the Kronecker structure. The algorithm is implemented by using CUDA on a Graphics Processing Unit (GPU) and is evaluated by comparing the performance to a Python implementation. Numerical experiments done in this thesis show that, irregardless of the the matrix dimensions, a speedup factor of up to 7 times can be achieved over the implementation in Python for the ALS identification segment of the algorithm. For the RLS segment it is shown that the GPU is not an optimal device for implementing recursive methods and is better implemented sequentially for the sizes considered in the experiment.