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Session Overview
MS 28b: Adaptive and high-order approximation based on Reduced Order Methods
Friday, 16/July/2021:
12:00pm - 2:00pm

Session Chair: Francesca Bonizzoni
Session Chair: Gianluigi Rozza
Virtual location: Zoom 3

Session Abstract

Reduced Order Models (ROMs) aim at providing fast, accurate and robust numerical

solution of parametrized partial differential equations (PDEs).

These techniques are important in the multi-query scenario, where the PDE has to be solved for

many different values of the parameters, as well as in the real-time scenario, where the PDE has be

to solved in few seconds.

The purpose of this minisymposium is to collect the most recent results, as well as to provide a plat-

form for exchange of new concepts and ideas on high-order and adaptive ROMs for parametrized


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12:00pm - 12:30pm

A Unified Reduced Basis Method for Elliptic and Parabolic Fractional Diffusion Problems

Tobias Danczul1, Clemens Hofreither2, Joachim Schöberl1

1TU Wien, Austria; 2Johann Radon Insitute for Computational and Applied Mathematics (RICAM), Austria

In this talk we present a unified theoretical framework to efficiently approximate the solution map to fractional diffusion problems of elliptic and parabolic type, involving the spectral fractional Laplacian and the Caputo fractional derivative. In the abstract framework of Stieltjes and complete Bernstein functions, we derive an integral representation for the desired quantities which includes solutions to parametrized integer-order PDEs. We apply a reduced basis strategy on top of a finite element method to approximate its integrand. We prove uniform convergence when using snapshots based on Zolotarev's minimal deviation problem. The latter are particularly suited for fractional diffusion as they allow for an efficient query of the solution map and do not degenerate as the fractional parameters approach zero. Numerical experiments confirm the analysis and demonstrate the efficiency of our approach.

12:30pm - 1:00pm

Error estimation in reduced basis method for systems with nonlinear boundary conditions

Laura Iapichino1, Mohammad Abbasi1, Bart Besselink2, Wil Schilders1, Nathan van de Wouw1

1Technical University of Eindhoven, The Netherlands; 2University of Groningen, The Netherlands

Many physical phenomena, such as mass transport and heat transfer, are modeled by systems of nonlinear partial differential equations. Numerical simulations of high-order discretized partial differential equations are often too computationally expensive for real-time and many-query analysis. For this reason, model complexity reduction is essential. We exploit the use of the Reduced Basis Method and in particular introduce error estimates, which play a crucial role in accelerating the offline phase of the method and also quantifying the error induced after reduction in the online phase. Burgers' equation is used as a testbed for the newly developed error estimates. A first error estimate is based on the Lur'e-type model formulation of the system obtained after the full-discretization of Burgers' equation. A second error estimate is built upon snapshots generated in the offline phase of the reduced basis method. The second error estimate is applicable to a wider range of systems compared to the first error estimate. Results reveal that when conditions for the error estimates are satisfied, the error estimates are accurate and work efficiently in terms of computational effort.

1:00pm - 1:30pm

Non-intrusive model reduction of parametric frequency response problems via minimal rational interpolation

Davide Pradovera, Fabio Nobile

CSQI, EPFL, Switzerland

Numerical methods for time-harmonic wave propagation phenomena e.g. in acoustics or elastodynamics, are often extremely computationally intensive, expecially in mid- and high-frequency regimes. As such, in many cases a direct frequency response analysis has a prohibitive computational cost, and remains a major challenge. In this framework, model order reduction (MOR) methods are very promising: starting from few expensive solves of the problem, they can provide a reliable approximation of the frequency response of the system, very cheap to evaluate in a whole range of frequencies.

In many applications, the frequency response function, which maps the frequency to (a functional of) the solution of the problem at hand, can be shown to be meromorphic, with poles at the resonances of the system. Accordingly, it is meaningful to seek a surrogate among rational functions. In this talk we consider a MOR technique in this direction, minimal rational interpolation, which builds a rational approximation of the frequency response map by solving a constrained optimization problem based only on evaluations (snapshots) of the target map. In particular, differently from other (so-called intrusive) methods, in minimal rational interpolation the original problem can be treated as a "black-box", whose underlying structure does not need to be known nor accessed.

We discuss how minimal rational interpolation can be extended to problems with multiple parameters, for instance for frequency response analyses in presence of uncertain geometries and/or materials, where the quantity of interest is multivariate, depending on both frequency and extra (geometric/material) parameters. However, particularly if the number of extra parameters is large, this method is quite inefficient, and spurious effects often arise in the resulting surrogate. To combat these issues, we describe a hybrid approach: for some fixed values of the extra parameters, one builds approximate models in frequency only, by employing minimal rational interpolation; then a joint global surrogate is obtained by combining these univariate models over the space of extra parameters. This improves the cost-effectiveness of the technique, and the spurious effects can be dealt with by post-processing the frequency surrogates before the global model is built.

Still, we show how the hybrid approach leads to new issues. Most notably, combining univariate frequency models in a sensible way is an extremely complex task, requiring an adequate theoretical understanding of how resonances can evolve as the geometry and materials change. This is also linked to the obvious problem of determining how many frequency surrogates are required to achieve reasonable accuracy, an issue which becomes critical in the high-dimensional setting.

1:30pm - 2:00pm

Registration-based model reduction of parameterized PDEs

tommaso taddei, Lei Zhang

INRIA Bordeaux South-West, France

We propose a model reduction procedure of parameterized hyperbolic systems of conservation laws with applications to hydraulics and aerodynamics. Due to the presence of parameter-dependent shock waves, hyperbolic PDEs are extremely challenging for traditional model reduction approaches based on linear approximations. The main ingredients of the proposed approach are (i) an adaptive registration procedure to align local features in a fixed reference domain, (ii) a space-time Petrov-Galerkin formulation for the computation of the mapped solution, and (iii) a hyper-reduction procedure based on empirical quadrature to speed online computations. Registration is designed to improve performance of linear approximations (e.g., POD) in the reference configuration; projection-based methods are here considered to ensure quasi-optimal reconstructions of the reduced coefficients during the online stage. We present numerical results for several model problems, to illustrate the effectiveness of our method.

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