Simulation-based approximate solution of large-scale linear least squares problems and applications

Abstract

We consider linear least squares problems, or linear systems that can be formulated into least squares problems, of very large dimension, such as those arising for example in dynamic programming (DP) and inverse problems. We introduce an associated approximate problem, within a subspace spanned by a relatively small number of basis functions, and solution methods that use simulation, importance sampling, and low-dimensional calculations. The main components of this methodology are a regression/ regularization approach that can deal with nearly singular problems, and an importance sampling design approach that exploits existing continuity structures in the underlying models, and allows the solution of very large problems. We also investigate the use of our regression/regularization approach in temporal difference-type methods in the context of approximate DP. Finally we demonstrate the application of our methodology in a series of practical large-scale examples arising from Fredholm integral equations of the first kind.