European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
13:30   MS26:New challenges and opportunities for model order reduction (Part 2)
Chair: Katrin Smetana
25 mins
Gradient based methods for nonlinear dimensionality reduction
Olivier Zahm, Daniele Bigoni, Clémentine Prieur, Youssef Marzouk
Abstract: Model order reduction techniques such as polynomial chaos expansion or tensor methods suffer when the number of input parameters is large. Identifying the parametric directions where the input-to-output function does not vary significantly is a key preprocessing step to reduce the complexity of the approximation algorithms. We propose a gradient-based method that permits to detect such a low-dimensional structure of a function using evaluation of its gradients. The methodology consists in minimizing an upper-bound of the approximation error obtained using Poincaré-type inequalities. We then explain how this methodology naturally extends to nonlinear dimension reduction, e.g. when the function is not constant along a subspace but along a low-dimensional manifold.
25 mins
Reduced Modeling of Riemannian Manifolds. Application to 1D Conservative PDE's
Olga Mula
Abstract: Model reduction of transport-dominated PDEs requires the development of nonlinear approximation methods. We present a novel strategy in this direction which involves tangent spaces of Riemanian manifolds. We will show some theoretical and numerical results illustrating its efficiency on several one-dimensional conservative PDEs: pure transport problems, inviscous and viscous Burger's equations, Camassa-Holm and Korteveg de Vries. This is a joint work with V. Ehrlacher, D. Lombardi and F.X. Vialard.
25 mins
Structure-Preserving Model Reduction for Advection-Dominated Phenomena
Philipp Schulze
Abstract: Model order reduction (MOR) has been proven to be a powerful tool for reducing the computational effort in many applications, especially, where large-scale simulations have to be performed in a multi-query setting. However, when it comes to systems which experience advection of high-gradient structures, as for instance, shocks, standard MOR methods are known to perform poorly. Recently, various methods emerged which address this problem, for instance, by applying coordinate transformations in order to account for the advective transport. Even though these methods are capable of efficiently reducing advection-dominated systems by preserving the advective behavior of the original model, they do in general not preserve other important system properties such as stability and passivity. In this talk we focus on the shifted proper orthogonal decomposition (shifted POD) which is an extension of the classical POD using coordinate transformations to account for the advective transport. Especially, we demonstrate how to project the original model onto the span of the shifted POD modes and how to preserve the port-Hamiltonian structure of the original model in order to ensure the stability and passivity of the reduced-order model. Moreover, we present numerical experiments to illustrate the effectiveness of the new approach.
25 mins
Projection-based reduced order models for a cut finite element method in parametrized domains
Efthymios N. Karatzas, Francesco Ballarin, Gianluigi Rozza
Abstract: The talk will present a reduced order modelling technique built on a high fidelity embedded mesh finite element method, discussed in our preprint [1]. Embedded methods, and in particular the CutFEM method, are attractive in the generation of projection-based reduced order models thanks to their capabilities to seamlessly handle large deformations of parametrized domains. The combination of embedded methods and reduced order models allows us to obtain fast evaluation of parametrized problems, avoiding remeshing as well as the reference domain formulation, often used in the reduced order modelling for boundary fitted finite element formulations. The resulting novel methodology is presented on linear elliptic and Stokes problems, together with several test cases to assess its capability. The role of a proper extension and, especially, transport of embedded solutions to a common background is discussed. The methodological connection with a wider class of transport dominated problems is higlighted, together with an outline of ongoing works (in particular, variational multiscale modelling, fluid-structure interaction problems) on this topic. [1] E. N. Karatzas, F. Ballarin, and G. Rozza, "Projection-based reduced order models for a cut finite element method in parametrized domains", submitted, 2019. Arxiv preprint 1901.03846.