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Register a new userA Frame Decomposition of the Atmospheric Tomography Operator
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Added by
Simon Hubmer
Publication date
2020
fields
Informatics Engineering
and research's language is
English
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We consider the problem of atmospheric tomography, as it appears for example in adaptive optics systems for extremely large telescopes. We derive a frame decomposition, i.e., a decomposition in terms of a frame, of the underlying atmospheric tomography operator, extending the singular-value-type decomposition results of Neubauer and Ramlau (2017) by allowing a mixture of both natural and laser guide stars, as well as arbitrary aperture shapes. Based on both analytical considerations as well as numerical illustrations, we provide insight into the properties of the derived frame decomposition and its building blocks.
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The characteristic feature of inverse problems is their instability with respect to data perturbations. In order to stabilize the inversion process, regularization methods have to be developed and applied. In this work we introduce and analyze the concept of filtered diagonal frame decomposition which extends the standard filtered singular value decomposition to the frame case. Frames as generalized singular system allows to better adapt to a given class of potential solutions. In this paper, we show that filtered diagonal frame decomposition yield a convergent regularization method. Moreover, we derive convergence rates under source type conditions and prove order optimality under the assumption that the considered frame is a Riesz-basis.
We analyze sparse frame based regularization of inverse problems by means of a diagonal frame decomposition (DFD) for the forward operator, which generalizes the SVD. The DFD allows to define a non-iterative (direct) operator-adapted frame thresholding approach which we show to provide a convergent regularization method with linear convergence rates. These results will be compared to the well-known analysis and synthesis variants of sparse $ell^1$-regularization which are usually implemented thorough iterative schemes. If the frame is a basis (non-redundant case), the thr
In this paper, we consider linear ill-posed problems in Hilbert spaces and their regularization via frame decompositions, which are generalizations of the singular-value decomposition. In particular, we prove convergence for a general class of continuous regularization methods and derive convergence rates under both a-priori and a-posteriori parameter choice rules. Furthermore, we apply our derived results to a standard tomography problem based on the Radon transform.
We consider the decomposition of bounded linear operators on Hilbert spaces in terms of functions forming frames. Similar to the singular-value decomposition, the resulting frame decompositions encode information on the structure and ill-posedness of the problem and can be used as the basis for the design and implementation of efficient numerical solution methods. In contrast to the singular-value decomposition, the presented frame decompositions can be derived explicitly for a wide class of operators, in particular for those satisfying a certain stability condition. In order to show the usefulness of this approach, we consider different examples from the field of tomography.
A systematic mathematical framework for the study of numerical algorithms would allow comparisons, facilitate conjugacy arguments, as well as enable the discovery of improved, accelerated, data-driven algorithms. Over the course of the last century, the Koopman operator has provided a mathematical framework for the study of dynamical systems, which facilitates conjugacy arguments and can provide efficient reduced descriptions. More recently, numerical approximations of the operator have enabled the analysis of a large number of deterministic and stochastic dynamical systems in a completely data-driven, essentially equation-free pipeline. Discrete or continuous time numerical algorithms (integrators, nonlinear equation solvers, optimization algorithms) are themselves dynamical systems. In this paper, we use this insight to leverage the Koopman operator framework in the data-driven study of such algorithms and discuss benefits for analysis and acceleration of numerical computation. For algorithms acting on high-dimensional spaces by quickly contracting them towards low-dimensional manifolds, we demonstrate how basis functions adapted to the data help to construct efficient reduced representations of the operator. Our illustrative examples include the gradient descent and Nesterov optimization algorithms, as well as the Newton-Raphson algorithm.
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