Michael Griebel, Alexander Hullmann and Oeter Oswald Optimal scaling parameters for sparse grid discretizations Numerical Linear Algebra with Applications, 22(1): 76-100 2015 http://dx.doi.org/10.1002/nla.1939
Abstract: We apply iterative subspace correction methods to elliptic PDE problems discretized by generalized sparse grid systems. The involved subspace solvers are based on the combination of all anisotropic full grid spaces that are contained in the sparse grid space. Their relative scaling is at our disposal and has significant influence on the performance of the iterative solver. In this paper, we follow three approaches to obtain close-to-optimal or even optimal scaling parameters of the subspace solvers and thus of the overall subspace correction method. We employ a Linear Program that we derive from the theory of additive subspace splittings, an algebraic transformation that produces partially negative scaling parameters which result in improved asymptotic convergence properties, and finally we use the OptiCom method as a variable non-linear preconditioner.
2014
Michael Griebel and Alexander Hullmann Dimensionality Reduction of High-Dimensional Data with a NonLinear Principal Component Aligned Generative Topographic Mapping SIAM J. Sci. Comput., 36(3): A1027-A1047 2014 http://dx.doi.org/10.1137/130931382
Michael Griebel and Alexander Hullmann A Sparse Grid Based Generative Topographic Mapping for the Dimensionality Reduction of High-Dimensional Data In Modeling, Simulation and Optimization of Complex Processes - HPSC 2012
page 51-62.
2014 http://dx.doi.org/10.1007/978-3-319-09063-4_5