Volker Schulz

Gennadij Heidel, Volker Schulz

The goal of tensor completion is to fill in missing entries of a partially known tensor (possibly including some noise) under a low-rank constraint. This may be formulated as a least-squares problem. The set of tensors of a given multilinear rank is known to admit a Riemannian manifold structure, thus methods of Riemannian optimization are applicable. In our work, we derive the Riemannian Hessian of an objective function on the low-rank tensor manifolds using the Weingarten map, a concept from differential geometry. We discuss the convergence properties of Riemannian trust-region methods based on the exact Hessian and standard approximations, both theoretically and numerically. We compare our approach to Riemannian tensor completion methods from recent literature, both in terms of convergence behaviour and computational complexity. Our examples include the completion of randomly generated data with and without noise and recovery of multilinear data from survey statistics.

Dishi Liu, Alexander Litvinenko, Claudia Schillings et al.

Uncertainty quantification in aerodynamic simulations calls for efficient numerical methods since it is computationally expensive, especially for the uncertainties caused by random geometry variations which involve a large number of variables. This paper compares five methods, including quasi-Monte Carlo quadrature, polynomial chaos with coefficients determined by sparse quadrature and gradient-enhanced version of Kriging, radial basis functions and point collocation polynomial chaos, in their efficiency in estimating statistics of aerodynamic performance upon random perturbation to the airfoil geometry which is parameterized by 9 independent Gaussian variables. The results show that gradient-enhanced surrogate methods achieve better accuracy than direct integration methods with the same computational cost.

Benjamin Rosenbaum, Volker Schulz

Surrogate models are used for global approximation of responses generated by expensive computer experiments like CFD applications. In this paper, we make use of structural similarities which are shared by a class of related problems. We identify these structures by applying statistical shape models. They are used to build a generic surrogate model approximation to sample data of a new problem of the same class. In a variable fidelity framework the generic surrogate model is combined with the sample data to generate an efficient and globally accurate interpolation model, which requires less costly sample evaluations than ordinary response surface methods. We demonstrate our method with an aerodynamic test case and show that it significantly improves the approximation quality.

Jan Sokolowski, Volker Schulz, Udo Schröder et al.

In several studies, hybrid neural networks have proven to be more robust against noisy input data compared to plain data driven neural networks. We consider the task of estimating parameters of a mechanical vehicle model based on acceleration profiles. We introduce a convolutional neural network architecture that is capable to predict the parameters for a family of vehicle models that differ in the unknown parameters. We introduce a convolutional neural network architecture that given sequential data predicts the parameters of the underlying data's dynamics. This network is trained with two objective functions. The first one constitutes a more naive approach that assumes that the true parameters are known. The second objective incorporates the knowledge of the underlying dynamics and is therefore considered as hybrid approach. We show that in terms of robustness, the latter outperforms the first objective on noisy input data.