Fast gain calibration in radio astronomy using alternating direction implicit methods: Analysis and applications

Context. Modern radio astronomical arrays have (or will have) more than one order of magnitude more receivers than classical synthesis arrays, such as the VLA and the WSRT. This makes gain calibration a computationally demanding task. Several alternating direction implicit (ADI) approaches have therefore been proposed that reduce numerical complexity for this task from (P3) to (P2), where P is the number of receive paths to be calibrated Aims. We present an ADI method, show that it converges to the optimal solution, and assess its numerical, computational and statistical performance. We also discuss its suitability for application in self-calibration and report on its successful application in LOFAR standard pipelines. Methods. Convergence is proved by rigorous mathematical analysis using a contraction mapping. Its numerical, algorithmic, and statistical performance, as well as its suitability for application in self-calibration, are assessed using simulations. Results. Our simulations confirm the (P2) complexity and excellent numerical and computational properties of the algorithm. They also confirm that the algorithm performs at or close to the Cramer-Rao bound (CRB, lower bound on the variance of estimated parameters). We find that the algorithm is suitable for application in self-calibration and discuss how it can be included. We demonstrate an order-of-magnitude speed improvement in calibration over traditional methods on actual LOFAR data. Conclusions. In this paper, we demonstrate that ADI methods are a valid and computationally more efficient alternative to traditional gain calibration methods and we report on its successful application in a number of actual data reduction pipelines