arXiv subjects

Volker Schulz

Publications and source records attributed to Volker Schulz.

30 records · Page 2Linked to original sources

arXiv2020

A Hybrid Objective Function for Robustness of Artificial Neural Networks -- Estimation of Parameters in a Mechanical System

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.

cs.LG↗

arXiv2020

GivEn -- Shape Optimization for Gas Turbines in Volatile Energy Networks

This paper describes the project GivEn that develops a novel multicriteria optimization process for gas turbine blades and vanes using modern "adjoint" shape optimization algorithms. Given the many start and shut-down processes of gas power plants in volatile energy grids, besides optimizing gas turbine geometries for efficiency, the durability understood as minimization of the probability of failure is a design objective of increasing importance. We also describe the underlying coupling structure of the multiphysical simulations and use modern, gradient based multicriteria optimization procedures to enhance the exploration of Pareto-optimal solutions.

math.OC↗

arXiv2019

Exploiting multilevel Toeplitz structures in high dimensional nonlocal diffusion

We present a finite element implementation for the steady-state nonlocal Dirichlet problem with homogeneous volume constraints. Here, the nonlocal diffusion operator is defined as integral operator characterized by a certain kernel function. We assume that the domain is an arbitrary $d$-dimensional hyperrectangle and the kernel is translation invariant. Under these assumptions, we carefully analyze the structure of the stiffness matrix resulting from a continuous Galerkin method with multilinear elements and exploit this structure in order to cope with the curse of dimensionality associated to nonlocal problems. For the purpose of illustration we choose a particular kernel, which is related to space-fractional diffusion and present numerical results in 1d, 2d and for the first time also in 3d.

math.AP↗

arXiv2019

An extended model of wine fermentation including aromas and acids

The art of viticulture and the quest for making wines has a long tradition and it just started recently that mathematicians entered this field with their main contribution of modelling alcoholic fermentation. These models consist of systems of ordinary differential equations that describe the kinetics of the bio-chemical reactions occurring in the fermentation process. The aim of this paper is to present a new model of wine fermentation that accurately describes the yeast dying component, the presence of glucose transporters, and the formation of aromas and acids. Therefore the new model could become a valuable tool to predict the taste of the wine and provide the starting point for an emerging control technology that aims at improving the quality of the wine by steering a well-behaved fermentation process that is also energetically more efficient. Results of numerical simulations are presented that successfully confirm the validity of the proposed model by comparison with real data.

q-bio.MN↗

arXiv2017

A Riemannian trust-region method for low-rank tensor completion

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.

math.NA↗

arXiv2016

Quantification of airfoil geometry-induced aerodynamic uncertainties - comparison of approaches

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.

math.NA↗

arXiv2016

Efficient PDE constrained shape optimization based on Steklov-Poincaré type metrics

Recent progress in PDE constrained optimization on shape manifolds is based on the Hadamard form of shape derivatives, i.e., in the form of integrals at the boundary of the shape under investigation, as well as on intrinsic shape metrics. From a numerical point of view, domain integral forms of shape derivatives seem promising, which rather require an outer metric on the domain surrounding the shape boundary. This paper tries to harmonize both points of view by employing a Steklov-Poincaré type intrinsic metric, which is derived from an outer metric. Based on this metric, efficient shape optimization algorithms are proposed, which also reduce the analytical labor, so far involved in the derivation of shape derivatives.

math.OC↗

arXiv2016

Computational comparison of surface metrics for PDE constrained shape optimization

We compare surface metrics for shape optimization problems with constraints, consisting mainly of partial differential equations (PDE), from a computational point of view. In particular, classical Laplace-Beltrami type based metrics are compared with Steklov-Poincaré type metrics. The test problem is the minimization of energy dissipation of a body in a Stokes flow. We therefore set up a quasi-Newton method on appropriate shape manifolds together with an augmented Lagrangian framework, in order to enable a straightforward integration of geometric constraints for the shape. The comparison is focussed towards convergence behavior as well as effects on the mesh quality during shape optimization.

math.OC↗

arXiv2015

Structured inverse modeling in parabolic diffusion processess

Often, the unknown diffusivity in diffusive processes is structured by piecewise constant patches. This paper is devoted to efficient methods for the determination of such structured diffusion parameters by exploiting shape calculus. A novel shape gradient is derived in parabolic processes. Furthermore quasi-Newton techniques are used in order to accelerate shape gradient based iterations in shape space. Numerical investigations support the theoretical results.

math.OC↗

arXiv2014

Novel model for wine fermentation including the yeast dying phase

This paper presents a novel model for wine fermentation including a death phase for yeast and the influence of oxygen on the process. A model for the inclusion of the yeast dying phase is derived and compared to a model taken from the literature. The modeling ability of the several models is analyzed by comparing their simulation results.

q-bio.QM↗

arXiv2014

A Riemannian View on Shape Optimization

Shape optimization based on the shape calculus is numerically mostly performed by means of steepest descent methods. This paper provides a novel framework to analyze shape-Newton optimization methods by exploiting a Riemannian perspective. A Riemannian shape Hessian is defined yielding often sought properties like symmetry and quadratic convergence for Newton optimization methods.

math.OC↗

arXiv2012

Efficient response surface methods based on generic surrogate models

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.

math.NA↗