Model order reduction is an effective strategy to address problems with (possibly random) parameters. The idea is to project the underlying equations onto a small finite-dimensional subspace spanned by few cleverly constructed deterministic modes thus leading to a reduced size problem on which UQ or parametric analysis can be cheaply performed by e.g. sampling or quadrature techniques. However, for time-dependent problems with complex dynamics, the optimal subspace on which to approximate the solution at each time instant can considerably change over time. We address in this minisymposium recent dynamical techniques to construct time-varying reduced subspaces. These include, for instance, local-adaptive-transformed reduced basis methods as well as dynamical low-rank tensor approximations.
14:00
Tensor Methods for Model Reduction of Dynamical Systems
Antonio Falcó Montesinos | Universidad CEU Cardenal Herrera | Spain
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Antonio Falcó Montesinos | Universidad CEU Cardenal Herrera | Spain
The aim of this talk is to ask the question as whether it is possible, for a given dynamical system defined by a vector field over an infinite dimensional space, to construct a reduced-order model over a infinite dimensional manifold. We prove that if the manifold under consideration is an immersed submanifold of the vector space, considered as ambient manifold, then it is possible to construct explicitly a reduced-order vector field over this submanifold.
14:30
Time discretization and stability estimates for dynamical low rank approximations of random parabolic equations
Eva Vidlicková | Ecole Polytechnique Fédérale de Lausanne / CSQI-MATH | Switzerland
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Eva Vidlicková | Ecole Polytechnique Fédérale de Lausanne / CSQI-MATH | Switzerland
Yoshihito Kazashi | Ecole Polytechnique Fédérale de Lausanne / CSQI-MATH | Switzerland
Fabio Nobile | Ecole Polytechnique Fédérale de Lausanne / CSQI-MATH | Switzerland
The numerical quantification of uncertainties can be particularly challenging for problems requiring long time integration as the structure of the random solution might considerably change over time. In this respect, Dynamical Low Rank (DLR) methods are very appealing. They can be seen as reduced basis methods, thus solvable at a relatively low computational cost, in which the solution is expanded as a linear combination of few well chosen deterministic functions with random coefficients. The distinctive feature of the DLR method is that the spatial basis is computed on the fly and is free to evolve in time, thus adjusting at each time to the current structure of the random solution. In this talk we will discuss possible time discretizations for DLR approximation of random parabolic equations and state their stability properties.
15:00
Reduced Order Modeling for Hyperbolic PDEs with Shock Collisions
Gerrit Welper | University of Central Florida | United States
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Gerrit Welper | University of Central Florida | United States
Because of shock discontinuities the efficient simulation of parametric or stochastic hyperbolic PDEs still poses significant challenges. To this end, in recent years some non-linear reconstruction techniques, such as shifted POD, TSMOR, displacement interpolation or transformed snapshot interpolation, have been developed. These methods approximate the PDE's solution by time or parameter dependent subspaces constructed from shifts, transport or transformation of snapshots in the physical variables. The aim of this extra step is to move the shocks of the snapshots to the correct locations for the time/parameters where one wants to reconstruct the solution.
In the recent literature, it has been demonstrated that this approach is efficient for for individual, spatially separated or linearly separable jumps, but the matching of shocks is still problematic if the shock structure changes, e.g. for for emerging or colliding shocks. To this end, we introduce another adaption mechanism, loosely related to a WENO interpolation, which locally selects snapshots with the targeted jump-set structure. To this end, we discuss several challenges, including that the transforms for alignment become singular at collision points and selection mechanisms for the active snapshots.
15:30
Model Reduction for Nonlinear Parameterized Transport Equations
Philipp Schulze | Technische Universität Berlin | Germany
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Philipp Schulze | Technische Universität Berlin | Germany
Model order reduction (MOR) techniques aim for reducing the computational effort of numerical simulations, especially, in the context of multi-query tasks as occurring, e.g., in uncertainty quantification and optimization. In various applications, standard MOR schemes have been successfully deployed, but they are known to perform poorly for systems exhibiting traveling waves with sharp fronts. In the past years, new methods have been developed to overcome these shortcomings. One common idea is to account for the transport of the waves by appropriate time-dependent coordinate transforms, for instance, by approximating the FOM solution by a linear combination of transformed modes.
In this talk, we present a method for creating reduced-order models (ROMs) in order to compute the time-dependent mode amplitudes as well as the dynamic parameters, which determine the coordinate transforms, in the online phase. In the case that the FOM is linear, an efficient offline/online decomposition can be obtained. In order to achieve an efficient offline/online decomposition even in the case that the FOM is nonlinear, we introduce an additional approximation of the nonlinearity. This new approach is based on an extension of the empirical interpolation method which allows for an effective application to transport-dominated problems. The potency of this new approach is illustrated by means of numerical examples.