A Finite Element--Wavelet Hybrid Algorithm for Atmospheric Tomography

Mykhaylo Yudytskiy*, Tapio Helin and Ronny Ramlau

The problem of atmospheric tomography arises in ground--based telescope imaging with adaptive optics, where one aims to compensate in real--time for the rapidly changing optical distortions in the atmosphere. The mathematical formulation of the problem resembles limited angle tomography. With the growing dimension of adaptive optics systems, current state of the art direct solvers for atmospheric reconstruction become computationally heavy. The matrix--free formulation of the underlying operators make iterative methods a viable alternative. \newline In this talk we present a novel iterative method for atmospheric reconstruction. The method combines multi--scale wavelet--based techniques, efficient operator representation with finite elements and conjugate gradient schemes. The problem of atmospheric tomography is formulated in the Bayesian framework and regularization utilizes statistical information. In consequence, the hybrid finite element--wavelet approach is robust even in the presence of high measurement noise. \newline In this talk we give a short introduction to the topic, discuss the theoretical results, and present a few numerical examples.

Mathematics Subject Classification: 65R32 45Q05 65T60 85A35

Keywords: atmospheric tomography; adaptive optics; wavelets; Bayesian inference

Minisymposion: Iterative Methods for Ill-Posed Problems