Opticom and the Iterative Combination Technique for Convex Minimisation

Abstract

Since “A combination technique for the solution of sparse grid problems” Griebel et al. (1992), the sparse grid combination technique has been successfully employed to approximate sparse grid solutions of multi-dimensional problems. In this paper we study the technique for a minimisation problem coming from statistics. Our methods can be applied to other convex minimisation problems. We improve the combination technique by adapting the “Opticom” method developed in Hegland et al. (Linear Algebra Appl 420:249–275, 2007). We also suggest how the Opticom method can be extended to other numerical problems. Furthermore, we develop a new technique of using the combination technique iteratively. We prove this method yields the true sparse grid solution rather than an approximation. We also present numerical results which illustrate our theory.KeywordsCoarse GridSparse GridConvex MinimisationNewton StepCombination TechniqueThese keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.