Least Squares Problems

Abstract

In the past decade, the developments of efficient multilevel preconditioning techniques have revived the interest in least squares discretizations of boundary value problems, see e.g. [BG, BLP1, Sta]. They appear to be an interesting direction of research, although not without several opposing perspectives. On one hand, they are particularly tempting because, in principle, they allow to turn a variety of problems into a positive definite symmetric variational formulation. This may impose less stringent compatibility constraints on the underlying discretizations and also allows for the employment of the many available iterative methods for symmetric positive definite problems. Since, on the other hand, the approach inherently squares the problem, this may enhance ill-conditioning. This in turn might be avoided by suitable preconditioning techniques combined with a judicious choice of the least squares functional. However, these least squares functionals often turn out to be difficult to deal with in practice since they may require to evaluate norms for noninteger or even negative function spaces. Of course, the employment of new tools and concepts may well enhance the one or other advantage.