• Lehrstuhl IX

Posters and their abstracts

It will be possible to mount and fix the poster on the presentation walls in room E 23 starting on Monday, March 12th, around 9.00 am.

• Laura Bittner (Bergische Universität Wuppertal): Optimal Reliability for Metal Components under Cyclic Loading
Metal devices are exposed to strong forces like friction, tension and rotation which cause stress states that influence the reliability of the component significantly. Using the PDE of linear elasticity and a material theoretic approach the deterministic lifetime of the component can be calculated. But, since it is impossible to predict exactly when and where damage will happen the lifetime model becomes more realistic when a stochastic approach based on Poisson-Point-Processes is integrated. Last but not least, the shape of the component itself affects reliability. The resulting objective functional $$J(\Omega,\sigma(u_\Omega))$$ determines the failure probability depending on the shape $$\Omega$$ and the stress tensor $$\sigma(u_{\Omega})$$. We use shape calculus methods to minimize this functional in order to find shapes with optimal reliability.

• Michela Egidi (TU Dortmund): Sharp geometric condition for null-controllability of the heat equation on $$\mathbb{R}^d$$
We consider the control problem for the heat equation on $$\mathbb{R}^d$$, $$d\geq 1$$ with control set $$\omega\subset\mathbb{R}^d$$. We provide a necessary and sufficient geometric condition on $$\omega$$, called $$(\gamma, a)$$-thickness, such that the heat equation is null-controllable in any positive time. Moreover, we estimate the control cost with explicit dependency on the geometric parameters of the control set, the time $$T$$, and the dimension $$d$$, and we derive a control cost estimate for the heat equation on cubes with periodic, Dirichlet, or Neumann boundary conditions, where the control sets are again assumed to be thick. Althought these two control problems have different nature, the two control cost estimates turn out to be consistent, leading to questions on how the control problem on $$\Lambda_L$$ approximates the control problem on $$\mathbb{R}^d$$ and viceversa. Further details are available in the preprint arXiv:1711.06088.

• Robert Gruhlke (Wias Berlin): Stochastic Domain Decomposition and Application in heterogeneous material
We consider a domain decomposition approach for random PDEs with a localization of the uncertain data. As a consequence, only local problems on subdomains in local coordinates have to be treated. The aim is to obtain a (globally) significantly reduced complexity in terms of stochastic problem dimensions. However, since the local representations usually are not independent, the high-dimensional coupling along the interfaces has to be considered as in the deterministic case. In addition to sampling based techniques, we also discuss modern low-rank approximations for the interface and localized problems.

• Camilla Hahn (Bergische Universität Wuppertal): Numerical shape optimization to decrease failure probability of ceramic structures
Ceramic is a material frequently used in industry because of its favorable properties. Common approaches in shape optimization for ceramic structures aim to minimize the tensile stress acting on the component, as it is the main driver for failure. In contrast to this, we follow a more natural approach by minimizing the component's probability of failure under a given tensile load. Since the fundamental work of Weibull, the probabilistic description of the strength of ceramics is standard and has been widely applied. Here, for the first time, the resulting failure probabilities are used as objective functions in PDE constrained shape optimization. To minimize the probability of failure, we choose a gradient based method combined with a first discretize then optimize approach. For discretization finite elements are used. Using the Lagrangian formalism, the shape gradient via the adjoint equation is calculated at low computational cost. The implementation is verified by comparison of it with a finite difference method applied to a minimal 2d example. Furthermore, we construct shape flows towards an optimal / improved shape in the case of a simple beam and a bended joint.

• Sebastian Kersting (Technische Universität Darmstadt): Estimation of an improved surrogate model in uncertainty quantification by neuronal networks
Quantification of uncertainty of a technical system is often based on a surrogate model of a corresponding simulation model. In any application the simulation model will not describe the reality perfectly, and consequently the surrogate model will be imperfect. In this article we will combine observed and simulated data to construct an improved surrogate model consisting of multi-layer feedforward neural networks, and we will show that the convergence rate of the surrogate model will under suitably assumptions circumvent the curse of dimensionality. Based on this improved surrogate model we will show a convergence rate result of density estimates.

• Anoop Kodakkal (Technical University of Munich (TUM)): Multi Level Monte Carlo applied to Fluid-Structure Interaction problems
With the availability of increased computational power, computational modeling and numerical simulation is used as a method to tackle structural civil engineering problems. Unlike the conventional deterministic analysis, a stochastic analysis takes into consideration the uncertainties in the structural parameters, boundary/initial conditions, and loading to make realistic predictions. In structural wind engineering simulation, if the structure is slender, the structural responses are influenced by the coupling between structure and the fluid flow around it. This becomes a typical example of a fluid-structure interaction (FSI) problem. FSI is a multi-physics problem. The two physics (Fluid dynamics and Structural dynamics) principles are used for the respective models and also the coupling conditions are imposed on the interface. Uncertainty quantification for multi physics problems like FSI are computationally challenging as each of the deterministic solutions are expensive evaluate. Monte Carlo (MC) methods, based on sampling from the input distribution and evaluating the Quantities of Interest (QoI) at each of the sampling points are the most commonly used approaches for uncertainty quantification. MC methods have gained universal acceptance for its robustness and simplicity. However, for computationally expensive problems, like fluid-structure interaction (FSI), uncertainty quantification using Monte Carlo methods becomes computationally challenging and even infeasible in many cases. Multilevel Monte Carlo (MLMC) method is an improvement of standard MC method where the sampling is carried out from different approximations (i.e. levels) of the QoI. This approach reduces the computational cost by evaluating most samples from a low fidelity and low cost simulation. Only few samples are evaluated from high accuracy, expensive simulations. An overall reduction in computational cost is achieved by variance reduction in MLMC method. The efficacy of MLMC algorithm for expensive problems like FSI is compared in this study with classical MC. A set of benchmark test cases of FSI are used for the study. The possibility of uncertainty quantification for FSI problems with a plausible computational cost using MLMC is demonstrated.

• Toni Kowalewitz (TU Chemnitz): Random Diffusion Equations with Lévy Coefficients - Numerical Simulation of a Penetrating Spheres Model
We consider an elliptic equation with a random diffusion coefficient modelling Darcy flow through a medium with random conductivity. The diffusion coefficient is a Lévy field giving rise to a random two-phase medium. We explore various quadrature schemes for estimating quantities of interest of the solution field, including product rules, Monte Carlo and the Multilevel Monte Carlo Method.

• Han Cheng Lie (Freie Universität Berlin): Strong convergence of probabilistic numerical integrators for ordinary differential equations
In the numerical solution of differential equations, multiple sources of uncertainty need to be quantified and understood. One approach to this problem is to model unknowns using random variables, and to prove contraction of the resulting probability measures to a Dirac distribution supported on the true solution of the differential equation. This contraction can be understood as a computationally low-cost, probabilistic substitute for increasing the grid resolution. One challenge in this field is to prove suitable convergence of the randomised numerical solutions to the true solution. In joint work with Andrew Stuart and Tim Sullivan, we extend the known results on the question of convergence, by proving uniform convergence results under weaker hypotheses, and by connecting the regularity of the surrogate random variables to the regularity of the random numerical solutions.

• Geoffrey Lossa (University of Mons): A Hybrid approach using Monte Carlo simulation and Polynomial Chaos Expansion for geometrical and material uncertainties propagation in the FE extraction of RLC parametres of wound inductors
In this abstract, we present a new approach that allows to propagate geometrical and material uncertainties using polynomial chaos expansion (PCE) combined to Monte Carlo (MC) simulation in order to compute the RLC parameters of wound inductors with the Finite Element (FE) method. The main contribution of the present work is twofold:

• It allows to reduce the significant computational times related to the explicit modeling of uncertainty in a FE context, by mixing two approaches for representing stochasticity in the same procedure, namely PCE and MC simulations.
• A new algorithm for representing the winding patterns geometrical uncertainties is proposed, which mimics the practical wounding patterns process turn by turn, and is compared with a standard approach.

• Manuel Marschall (WIAS Berlin): Bayesian Inversion and Adaptive Low-Rank Tensor Decomposition
The statistical Bayesian approach is a natural setting to alleviate the inherent ill-posedness of inverse problems by assigning probability densities to the considered calibration parameters. Based on a parametrized coefficient in the forward model, a sampling-free approach to Bayesian inversion with an explicit representation of the parameter densities is developed to adjust the surrogate model. The proposed sampling-free approach is discussed in the context of tensor trains, which are employed for the adaptive evaluation of the random PDE in orthogonal chaos polynomials and the subsequent high-dimensional quadrature of the log-likelihood. This modern compression technique alleviates the curse of dimensionality by hierarchical subspace approximations of the respective low-rank (solution) manifolds. All required computations can then be carried out efficiently in the low-rank format and discretization parameters are adjusted adaptively based on a posteriori error estimators or indicators. Numerical experiments, involving affine and log-normal diffusion as examples of a more general framework, demonstrate the performance and confirm the theoretical results.

• Marco Reese (Bergische Universität Wuppertal): Integrability and approximation of solutions to flow equations with conductivity given by Lévy random fields

• Laura Scarabosio (TU München): Model-based multilevel Monte Carlo for multiscale problems
This work presents theory and methods for the adaptive control of modeling error in multi-scale models of random heterogeneous media described by stochastic, elliptic boundary-value problems, and how these can be used to reduce the computational effort of Monte Carlo methods. Goal-oriented, a posteriori error estimators are used to construct lower-dimensional surrogate models for the computation of localized quantities of interest. These surrogates are then employed in a model-based multilevel Monte Carlo method that leads to significant cost savings compared to Monte Carlo.

• Ingmar Schuster (FU Berlin): Exact active subspace Metropolis-Hastings
We consider the application of active subspaces to inform a Metropolis-Hastings algorithm, thereby aggressively reducing the computational dimension of the sampling problem. We show that the original formulation, as proposed by Constantine, Kent, and Bui-Thanh (SIAM J. Sci. Comput., 38(5):A2779-A2805, 2016), possesses asymptotic bias. Using pseudo-marginal arguments, we develop an asymptotically unbiased variant. Our algorithm is applied to a synthetic multimodal target distribution as well as a Bayesian formulation of a parameter inference problem for a Lorenz-96 system.
Joint work with Paul G. Constantine, T.J. Sullivan.

• Elisa Strauch (TU Darmstadt): A multi-level Monte Carlo method for stresses along paths
We describe stress problems in the upper earth’s crust using the equations of linear elasticity. The elasticity tensor and the body force are modeled as random fields because the material parameters are often subject to measurement uncertainty. The boundary conditions remain deterministic. We analyze multi-level Monte Carlo finite elements (MLMCFE) for the approximation of the expectation of the stress along a given path. For the finite element semidiscretization, linear finite elements on a regular triangulation are used. We prove that the error of an MLMCFE approximation of the expected stress along a given path converges linearly with respect to the mesh width. The theory is illustrated by numerical results.

Contact

TU Dortmund
Fakultät für Mathematik
Lehrstuhl IX
Vogelpothsweg 87
44227 Dortmund

You find us on the 6th floor of the Math tower .

Secretary: Mrs Janine Textor
Room M 620

E-mail:
janine.textor@tu-dortmund.de

Tel. (0231) 755-3063
Fax (0231) 755-5219