European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
13:30   MS29:Low-rank modelling in uncertainty quantification (Part 2)
Chair: Jan Heiland
25 mins
Energy estimates and model order reduction for bilinear systems with Lévy noise
Martin Redmann
Abstract: In this talk, we investigate a large-scale stochastic system with bilinear drift and linear diffusion term. Such high dimensional systems appear for example when discretizing a stochastic partial differential equations in space. We study a particular model order reduction technique to reduce the order of spatially-discretized systems and hence reduce computational complexity. We introduce suitable Gramians to the system and prove energy estimates that can be used to identify states which contribute only very little to the system dynamics. The reduced system is obtained by removing these states from the original system. In this talk, we present an L²-error bound for the proposed method applied to stochastic bilinear systems. This result is new even for deterministic bilinear equations. In order to achieve it, we develop a new technique which is not available in the literature so far.
25 mins
Low-rank Ensemble Kalman Filter for Nonlinear Networks: A Gas Network Example
Yue Qiu, Sara Grundel
Abstract: We consider the data assimilation problem for gas networks using the ensemble Kalman filter (EnKF). Such a network is modeled by a nonlinear differential algebraic equation (DAE). We propose a low-rank approach to approximate the states ensemble at each time step. The advantage of the low-rank EnKF are twofold. First, the number of forward model simulations at each time step is reduced from $N_{en}$ to $r_k$ compared with the standard EnKF, where $N_{en}$ is the size of ensembles, $r_k$ is the reduced rank, and $N_{en}> r_k$. Second, the low-rank EnKF further reduces the computational cost for the analysis step. Numerical experiments show that the performance of the low-rank EnKF is comparable with EnKF with a ensemble size $N_{en}$ while the computational cost is reduced dramatically.
25 mins
Low-rank parameterizations for the unsteady Navier_stokes equations in the frequency domain
Ralf Zimmermann
Abstract: For transonic flows governed by the time-accurate Navier-Stokes equations, small, approximately periodic perturbations can be calculated accurately by a transition to the frequency domain and truncating the Fourier expansion after the first harmonic. This is referred to as the linear frequency domain (LFD) method. In this talk, we discuss approaches to obtain a parametric reduced-order model for the LFD solver, where the a special focus is on the interpolation between structured system matrices. Numerical results are presented for emulating an aircraft aerodynamics in the transonic flow regime.
25 mins
Model Reduction in Micromotility Applications
Martin W Hess, Nicola Giuliani, Antonio DeSimone, Gianluigi Rozza
Abstract: We consider robotic microswimmers powered by flagella as exhibited in na- ture by bacteria, or by unicellular eukaryotic flagellataes and ciliates. Since the physical dimensions are in the micrometer range, the Reynolds number is small and the motion is governed by the Stokes equations. Using a boundary element discretization [1], we employ POD and greedy reduced basis model reduction [3] to find optimal configurations of the swimming efficiency and compare to the well-established Lighthill efficiency [2]. A number of interesting points arise from the model reduction point of view: • Although the boundary control is time-dependent, i.e., movements of the body, no time-derivative enters the equation since inertia is negligible at such low Reynolds numbers. Thus, time takes the same role as any other parameter. • The boundary element system is derived at each parameter from non-affine geometry variations. We show how to gain a computational speed-up without sacrificing geometric variability.