Combination technique coefficients via error splittings

We investigate a new way of choosing combination coefficients for the sparse grid combination technique. Previous work considered choosing coefficients such that the interpolation error of sufficiently smooth functions is minimised. We instead obtain an error bound using an error splitting model of approximation error and seek coefficients which minimise this. With minor modification this approach can also yield extrapolations. There are also potential applications to fault tolerance where new coefficients are required when a solution becomes unavailable due to a fault. We test the approach numerically on a scalar advection problem and compare with classical combinations from the literature. References J. Garcke. Sparse grids in a nutshell. In J. Garcke and M. Griebel editors, Sparse grids and applications, p. 57–80, Lecture Notes in Computational Science and Engineering, Vol. 88, Springer Berlin Heidelberg, 2013. doi:10.1007/978-3-642-31703-3_3 M. Griebel, M. Schneider and C. Zenger. A Combination Technique For The Solution Of Sparse Grid Problems. In P. de Groen and R. Beauwens, editors, Iterative Methods in Linear Algebra, p. 263–281. IMACS, Elsevier, North Holland, 1992. http://trove.nla.gov.au/work/34430317?q&versionId=42575630 B. Harding and M. Hegland. A robust combination technique. EMAC2012, ANZIAM J., 54:C394–C411, 2013. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/6321 B. Harding, M. Hegland, J. Larson and J. Southern. Fault tolerant computation with the sparse grid combination technique. SIAM J. Sci. Comput., 37(3):C331–C353. doi:10.1137/140964448 M. Hegland. Adaptive sparse grids. CTAC2001, ANZIAM J., 44:C335–C353, 2003. http://journal.austms.org.au/ojs/index.php/ANZIAMJ/article/view/685 M. Hegland, J. Garcke and V. Challis. The combination technique and some generalisations. Linear Algebra Appl., 420(2–3):249–275, 2007. doi:10.1016/j.laa.2006.07.014 B. Lastdrager, B. Koren and J. Verwer. The sparse-grid combination technique applied to time-dependent advection problems. Appl. Numer. Math., 38(4):377–401, 2001. doi:10.1016/S0168-9274(01)00030-7 C. Reisinger. Analysis of linear difference schemes in the sparse grid combination technique. IMA J. Numer. Anal., 33(2):544–581, 2013. doi:10.1093/imanum/drs004 C. Reisinger. Numerische Methoden fur hochdimensionale parabolische Gleichungen am Beispiel von Optionspreisaufgaben. Doctorate thesis, Universitat Heidelberg, 2004. http://www.ub.uni-heidelberg.de/archiv/495