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
Parameter Functions within Model Reduction for Uncertainty Quantification
Karsten Urban
Abstract: Parameter functions appear in a quite normal fashion in many parameterized problems such as optimal control, parameterized PDEs, quantum physics (variable potential), finance and others. Within the framework of MOR for UQ this would amount to include infinitely many parameters. In this talk, we show how wavelet expansions of the parameter function can be used in order to predict significant components of parameter functions. We will focus in particular on the Schrödinger equation with a variable potential. This talk is based upon joint work with Stefan Hain (Ulm).
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
Discontinuous Galerkin Model Order Reduction of Geometrically Parametrized Stokes Equation
Martin W Hess, Gianluigi Rozza, Nirav V Shah
Abstract: The present work focuses on the geometric parametrization and the reduced order modeling of the Stokes equation. We discuss the concept of a parametrized geometry and its application within a reduced order modeling technique. The full order model is based on the discontinuous Galerkin method with an interior penalty formulation. We introduce the broken Sobolev spaces as well as the weak formulation required for an affine parameter dependency. The operators are transformed from a fixed domain to a parameter dependent domain using the affine parameter dependency. The proper orthogonal decomposition is used to obtain the basis of functions of the reduced order model. By using the Galerkin projection the linear system is projected onto the reduced space. During this process, the offline-online decomposition is used to separate parameter dependent operations from parameter independent operations. Finally this technique is applied to an obstacle test problem.The numerical outcomes presented include experimental error analysis, eigenvalue decay and measurement of online simulation time.