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 that result in improved asymptotic convergence properties, and finally, we use the OptiCom method as a variable nonlinear preconditioner

Most high-dimensional real-life data exhibit some dependencies such that data points do not populate the whole data space but lie approximately on a lower-dimensional manifold. A major problem in many data mining applications is the detection of such a manifold and the expression of the given data in terms of a moderate number of latent variables. We present a method which is derived from the generative topographic mapping (GTM) and can be seen as a nonlinear generalization of the principal component analysis (PCA). It can detect certain nonlinearities in the data but does not suffer from the curse of dimensionality with respect to the latent space dimension as the original GTM and thus allows for higher embedding dimensions. We provide experiments that show that our approach leads to an improved data reconstruction compared to the purely linear PCA and that it can furthermore be used for classification.

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