Computational Comparison of Continuous and Discontinuous Galerkin Time-Stepping Methods for Nonlinear Initial Value Problems

This article centers on the computational performance of the continuous and discontinuous Galerkin time stepping schemes for general first-order initial value problems in R n , with continuous nonlinearities. We briefly review a recent existence result for discrete solutions from [6], and provide a numerical comparison of the two time discretization methods.

On ANOVA Decompositions of Kernels and Gaussian Random Field Paths

The FANOVA (or “Sobol’-Hoeffding”) decomposition of multivariate functions has been used for high-dimensional model representation and global sensitivity analysis. When the objective function f has no simple analytic form and is costly to evaluate, computing FANOVA terms may be unaffordable due to numerical integration costs. Several approximate approaches relying on Gaussian random field (GRF) models have been proposed to alleviate these costs, where f is substituted by a (kriging) predictor or by conditional simulations. Here we focus on FANOVA decompositions of GRF sample paths, and we notably introduce an associated kernel decomposition into 4 d 4d terms called KANOVA. An interpretation in terms of tensor product projections is obtained, and it is shown that projected kernels control both the sparsity of GRF sample paths and the dependence structure between FANOVA effects. Applications on simulated data show the relevance of the approach for designing new classes of covariance kernels dedicated to high-dimensional kriging.