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:
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.