Large Measurement Regression: Hierarchical Least Squares Multisplitting

摘要

Various measurement applications rely on indirect measurements such that the measurand is mapped through a mathematical model to the parameters of interest. Hence, measurement engineers rely on statistical models for which optimal estimators lead to sharp uncertainty bounds. Within the least squares (LS) estimation, these parameters are estimated through a regression problem. The presence of dynamics, multiple sensors and high sampling rates lead to an increased model complexity and hence high dimensional regression matrices. This paper aims to solve such massive regression problems efficiently. We revisit Renaut's Least Squares Multisplitting (LSMS) technique aimed at solving the ordinary least squares problem in parallel. The global least squares solution is replaced by an equivalent set of local smaller-sized least squares problems. At every iteration step the local solutions are recombined using an appropriate weighting scheme. Only if the scheme is convergent, it will allow a scalable and highly parallel implementation aimed at distributed systems. However, there exist regression designs for which the LSMS is divergent for each partition larger than 2. Hence, we propose a novel technique, the Hierarchical LSMS, in order to obtain the same number of parallel blocks, for which convergence is ascertained. We present its numerical properties and illustrate the technique with dedicated numerical simulations and an application within the domain of signal processing.