European Numerical Mathematics and
Advanced Applications Conference 2019
30th sep - 4th okt 2019, Egmond aan Zee, The Netherlands
10:40   MS38: Modeling of reduced order submanifolds in non-linear spaces (Part 2)
Chair: Stephan Rave
25 mins
Model Reduction of Nonlinear Hyperbolic Problems Using Low-dimensional Transport Modes
Donsub Rim, Kyle Mandli, Benjamin Peherstorfer
Abstract: Snapshot matrices formed by collecting solutions to nonlinear hyperbolic conservation laws commonly have slowly decaying singular values, but they can still have low-dimensional structure in a nonlinear sense. In this talk we will discuss a particularly simple low-dimensional structure that one can explicitly construct from the snapshots, based on some theoretical observation from relations between optimal transport problems and conservation laws. We will also discuss strategies to exploit this low-dimensional structure for model reduction.
25 mins
Model order reduction framework for problems with moving discontinuities
Harshit Bansal, Stephan Rave, Laura Iapichino, Wil Schilders, Nathan van de Wouw
Abstract: We propose a new model order reduction (MOR) approach to obtain effective reduction for transport-dominated problems or hyperbolic partial differential equations. The main ingredient is a novel decomposition of the solution into a function that tracks the evolving discontinuity and a residual part that is devoid of shock features. This decomposition ansatz is then combined with Proper Orthogonal Decomposition applied to the residual part only to develop an efficient reduced-order model representation for problems with multiple moving and possibly merging discontinuous features. Numerical case-studies show the potential of the approach in terms of computational accuracy compared with standard MOR techniques.
25 mins
A Primal-Dual Algorithm for Convex Nonsmooth Optimization on Riemannian Manifolds
Ronny Bergmann, Roland Herzog, José Vidal-Nunez, Daniel Tenbrinck
Abstract: Based on a Fenchel dual notion on Riemannian manifolds we investigate the saddle point problem related to a nonsmooth convex optimization problem. We derive a primal-dual hybrid gradient algorithm to compute the saddle point using either an exact or a linearized approach for the involved nonlinear operator. We investigate a sufficient condition for convergence of the linearized algorithm on Hadamard manifolds. Numerical examples illustrate, that on Hadamard manifolds we are on par with state of the art algorithms and on general manifolds we outperform existing approaches.
25 mins
Model Order Reduction of Combustion Processes with Complex Front Dynamics
Julius Reiss, Philipp Krah, Mario Sroka
Abstract: In this work we present a data driven method, used to improve mode-based model order reduction of transport fields with sharp fronts. We assume that the original flow field $q(\mathbf{x},t)=f(\phi(\mathbf{x},t))$ can be reconstructed by a front shape function $f$ and a level set function $\phi$. The level set function is used to generate a local coordinate, which parametrizes the distance to the front. In this way, we are able to embed the local 1D description of the front for complex 2D front dynamics with merging or splitting fronts, while seeking a low rank description of $\phi$. Here, the freedom of choosing $\phi$ far away from the front can be used to find a low rank description of $\phi$ which accelerates the convergence of $\norm{q- f(\phi_n)}$, when truncating $\phi$ after the $n$th mode. We demonstrate the ability of this new ansatz for a 2D propagating flame with a moving front.