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.
10:30
Hierarchical Partitioning Format for adaptive dynamical low rank approximations
Damiano Lombardi | Inria Paris and Sorbonne Université / LJLL | France
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Authors:
Damiano Lombardi | Inria Paris and Sorbonne Université / LJLL | France
Laura Grigori | Inria Paris and Sorbonne Université / LJLL | France
Virginie Ehrlacher | Inria Paris and Ecole Nationale des Ponts Paristech / CERMICS, | France
In this work, a tensor method is proposed to deal with the approximation of parametric time-dependent solutions. The key feature of the method is that, instead of considering a global tensor approximation, we build an adaptive, dynamical, piece-wise tensor approximation. Neither the partition nor the rank of each sub-tensor approximation are fixed a priori. Instead, they are computed to fulfil an error criterion and optimise the storage to some extent. Some examples and numerical experiments are shown on a parametric advection-diffusion system and on the Boltzmann equation.
11:00
Dynamical low-rank approximation method for linear conservation laws
Marie Billaud Friess | Centrale Nantes / LMJL | France
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Authors:
Marie Billaud Friess | Centrale Nantes / LMJL | France
Thomas Heuzé | Centrale Nantes / GeM | France
This talk concerns the approximation of discontinuous solutions of parameter-dependent linear conservation laws, especially arising in the context of uncertainty quantification.
Here, we propose a dynamical low-rank approximation method that relies on a projection based strategy. To that goal a time dependent reduced basis is constructed from transformed snapshots of the true solution (with respect to the parameters) using the characteristics. At the numerical level, these snapshots are obtained from conservative finite volume solver. Moreover, the snapshot transformation is performed with a Reconstruct, Transform then Average (RTA) procedure. Then, at each time step, the low-rank approximation is computed as the best-approximation of the solution that would be predicted by the finite volume scheme.
Preliminary numerical results for transport equation and linear elastodynamics equations will illustrate the behavior of the proposed approach.
This work has been partially funded by the CNRS Energy unit (Cellule Energie) through the project DROME.
11:30
Manifold Approximation via Transported Spaces (MATS)
Donsub Rim | NYU / Courant Institute of Mathematical Sciences | United States
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Authors:
Donsub Rim | NYU / Courant Institute of Mathematical Sciences | United States
Benjamin Peherstorfer | NYU / Courant Institute of Mathematical Sciences | United States
Kyle T. Mandli | Columbia University / Applied Physics and Applied Mathematics | United States
We introduce a model reduction approach for time-dependent nonlinear scalar conservation laws. Our approach, Manifold Approximation via Transported Spaces (MATS), exploits structure via a nonlinear approximation that is obtained by transporting reduced spaces along the characteristic curves. The notion of Kolmogorov N-width is extended to account for this new nonlinear approximation and the connection to deep neural networks is established. We also present an online efficient time-stepping algorithm based on MATS with costs independent of the dimension of the full model. Numerical results with stiff source terms demonstrate that reduced models based on MATS achieve orders of magnitude speedups compared to full models and traditional (linear) reduced models.
12:00
Reduced Basis Approximation of Problems with Moving Discontinuities via Nonlinear State Space Transformation
Stephan Rave | University of Münster | Germany
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Authors:
Stephan Rave | University of Münster | Germany
Harshit Bansal | TU Eindhoven | Netherlands
Christoph Lehrenfeld | University of Göttigen | Germany
Mario Ohlberger | University of Münster | Germany
A well-known issue for reduced order modeling of evolution problems with moving shock discontinuities, problems with moving internal interfaces or moving boundaries is the slow decay of the Kolmogorov n-widths of their corresponding solution manifolds, which means that there cannot exist linear low-order approximation spaces which accurately approximate the solution at all times or parameters of interest. Thus, successful reduced order modeling of these problems necessarily has to employ some form of nonlinear approximation.
In this talk we present two such approaches, both based on the idea of transforming the problem's underlying spatial domain: In the first approach, reduced basis methods are combined with the so-called 'method of freezing' to derive efficient reduced order models for equivariant hyperbolic problems with moving shocks. For free boundary problems, the second approach employs an ALE formulation to transform the problem to a reference domain with fixed boundary.