Radio Astronomical Image Formation using Constrained Least Squares and Krylov Subspaces

Image formation for radio astronomy can be defined as estimating the spatial power distribution of celestial sources over the sky, given an array of antennas. One of the challenges with image formation is that the problem becomes ill-posed as the number of pixels becomes large. The introduction of constraints that incorporate a-priori knowledge is crucial. In this paper we show that in addition to non-negativity, the magnitude of each pixel in an image is also bounded from above. Indeed, the classical "dirty image" is an upper bound, but a much tighter upper bound can be formed from the data using array processing techniques. This formulates image formation as a least squares optimization problem with inequality constraints. We propose to solve this constrained least squares problem using active set techniques, and the steps needed to implement it are described. It is shown that the least squares part of the problem can be efficiently implemented with Krylov subspace based techniques, where the structure of the problem allows massive parallelism and reduced storage needs. The performance of the algorithm is evaluated using simulations

Layer-oriented multigrid wavefront reconstruction algorithms for multiconjugate adaptive optics

Multi-conjugate adaptive optics (MCAO) systems with 10^4-10^5 degrees of freedom have been proposed for future giant telescopes. Using standard matrix methods to compute, optimize, and implement wavefront control algorithms for these systems is impractical, since the number of calculations required to compute and apply the reconstruction matrix scales respectively with the cube and the square of the number of AO degrees of freedom. In this paper, we develop an iterative sparse matrix implementation of minimum variance wavefront reconstruction for telescope diameters up to 32m with more than 104 actuators. The basic approach is the preconditioned conjugate gradient method, using a multigrid preconditioner incorporating a layer-oriented (block) symmetric Gauss-Seidel iterative smoothing operator. We present open-loop numerical simulation results to illustrate algorithm convergence

Computationally efficient wavefront reconstructor for simulation of multiconjugate adaptive optics on giant telescopes

Multi-conjugate adaptive optical (MCAO) systems with from 10,000 to 100,000 degrees of freedom have been proposed for future giant telescopes. Using standard matrix methods to compute, optimize, and implement wavefront reconstruction algorithms for these systems is impractical, since the number of calculations required to compute (apply) the reconstruction matrix scales as the cube (square) of the number of AO degrees of freedom. Significant improvements in computational efficiency are possible by exploiting the sparse and/or periodic structure of the deformable mirror influence matrices and the atmospheric turbulence covariance matrices appearing in these calculations. In this paper, we review recent progress in developing an iterative sparse matrix implementation of minimum variance wavefront reconstruction for MCAO. The basic method is preconditioned conjugate gradients, using a multigrid preconditioner incorporating a layer-oriented, iterative smoothing operator. We outline the key elements of this approach, including special considerations for laser guide star (LGS) MCAO systems with tilt-removed LGS wavefront measurements and auxiliary full aperture tip/tilt measurements from natural guide stars. Performance predictions for sample natural guide star (NGS) and LGS MCAO systems on 8 and 16 meter class telescopes are also presented

A robust adaptive algebraic multigrid linear solver for structural mechanics

The numerical simulation of structural mechanics applications via finite elements usually requires the solution of large-size and ill-conditioned linear systems, especially when accurate results are sought for derived variables interpolated with lower order functions, like stress or deformation fields. Such task represents the most time-consuming kernel in commercial simulators; thus, it is of significant interest the development of robust and efficient linear solvers for such applications. In this context, direct solvers, which are based on LU factorization techniques, are often used due to their robustness and easy setup; however, they can reach only superlinear complexity, in the best case, thus, have limited applicability depending on the problem size. On the other hand, iterative solvers based on algebraic multigrid (AMG) preconditioners can reach up to linear complexity for sufficiently regular problems but do not always converge and require more knowledge from the user for an efficient setup. In this work, we present an adaptive AMG method specifically designed to improve its usability and efficiency in the solution of structural problems. We show numerical results for several practical applications with millions of unknowns and compare our method with two state-of-the-art linear solvers proving its efficiency and robustness.Comment: 50 pages, 16 figures, submitted to CMAM