10:40
MS14: Reduced Order Models for parametric PDEs: special focus on time-dependent phenomena and time-harmonic wave problems (Part 2)
Chair: Gianluigi Rozza
10:40
25 mins
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Localized model reduction for wave propagation in heterogeneous media
Andreas Buhr, Dennis Eickhorn, Mario Ohlberger
Abstract: In this contribution we study localized model order reduction [2] for wave propagation in heterogeneous media. For the approximation of high frequency wave propagation in periodic homogenization problems, the Heterogeneous Multiscale Method has recently been proposed and investigated (see e.g. [4, 6]). For high frequency scattering in non-periodic situations, the localized orthogonal decomposition method has been recently suggested in [5]. Both multi-scale approaches rely on an enrichment of a coarse scale solution space through information that reflects the macroscopic influence of the underlying fine scale structure. A very similar approach in nature is localized model order reduction, where adapted local solution spaces are constructed for parameterized heterogeneous problems. Based on ideas from [1] and [3] we present a new localized model reduction method for the Helmholtz equation that is based on local training and enrichement. In particular, we investigate randomized training with Robin-type transfer operators. We will present both theoretical results and numerical experiments.
[1] A. Buhr, C. Engwer, M. Ohlberger, S. Rave. ArbiLoMod, a Simulation Technique Designed for Arbitrary Local Modifications. SIAM J. Sci. Comput. 39(4):A1435–A1465, 2017.
[2] A. Buhr, L. Iapichino, M. Ohlberger, S. Rave, F. Schindler, K. Smetana. Localized model reduc- tion for parameterized problems. ArXiv e-prints arXiv:1902.08300 [math.NA], Preprint (Submit- ted) - 2019.
[3] A. Buhr, K. Smetana. Randomized Local Model Order Reduction. SIAM J. Sci. Comput. 40(4):A2120-A2151, 2018.
[4] P.Henning, M.Ohlberger, B.Verfürth. AnewHeterogeneousMultiscaleMethodfortime- harmonic Maxwell’s equations based on divergence-regularization. SIAM J. Numer. Anal. 54(6):3493–3522, 2016.
[5] D. Peterseim, B. Verfürth. Computational high frequency scattering from high contrast heterogeneous media. ArXiv e-prints arXiv:1902.09935[math.NA], Preprint (Submitted) - 2019.
[6] M. Ohlberger, B. Verfürth. A new Heterogeneous Multiscale Method for the Helmholtz equation with high contrast. Multiscale Model. Simul. 16(1): 385–411, 2018.
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11:05
25 mins
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A time-dependent Parametrized Background Data-Weak approach
amina benaceur
Abstract: We propose a contribution that combines model reduction with data assimilation. A dedicated Parametrized Background Data-Weak (PBDW) [2] approach has been introduced in the literature so as to combine numerical models with experimental measurements. We extend the approach to a time-dependent framework by means of a \texttt{POD-greedy} reduced basis construction. Since the construction of the basis is performed offline, the algorithm renders the time dependence of the problem we are addressing while the time stepping scheme remains unchanged.
Moreover, we devise a new inexpensive algorithm for offline basis constructions. It exploits offline state estimates in order to diminish both the dimension of the online PBDW statement and the number of required sensors collecting data.
The idea is to use \textit{in situ} observations in order to update the best-knowledge model, thereby improving the approximation capacity of the background space.
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11:30
25 mins
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Interpolatory rational model order reduction of parametric problems lacking uniform inf-sup stability
Davide Pradovera, Fabio Nobile
Abstract: In the field of model order reduction (MOR) of parametric PDEs, one sometimes encounters problems which lack the regularity necessary for the theory of usual projection-based methods (e.g. Reduced Basis) to hold. Examples of this class of PDEs include frequency-domain acoustic/electromagnetic scattering and elasticity problems with parametric wavenumber.
In such cases, the complex resonant wavenumbers of the system lead to PDEs which are not inf-sup stable over the whole parameter range. This leads to several negative consequences: for instance, the uniform quasi-optimality of the Reduced Basis method is lost. And even for uniformly inf-sup stable problem, the quasi-optimality of the technique depends quite critically on the location of the resonances just outside the parameter range. Thus, it is often important to factor the presence of ``nearby'' resonances in the MOR method of choice.
In this talk we describe a technique, named minimal rational interpolation [2, 3], aimed at the MOR of the aforementioned class of parametric problems. This method does not rely on Galerkin projection of the original system: instead, starting from snapshots of the high-fidelity (HF) solution, an approximation of the solution map (i.e. the function associating to each parameter value the corresponding solution) of the HF PDE is built by solving a small optimization problem, in a fashion somewhat reminiscent of vector fitting.
As implied by the name of our method, the approximation built through this technique is a rational function, which can be proven to interpolate the starting snapshots. In addition, under some reasonable conditions on the HF problem, we present other interesting theoretical approximation properties of minimal rational interpolants.
We remark that our technique is an explicit method, since it provides an approximation of the solution map and not a reduced version of the HF problem. As such, it is suited also for non-linear PDEs and/or in case of non-trivial parametric dependencies, thus proving easier (and, generally, quicker) to apply than usual projective-based techniques, which need to be properly adjusted to accommodate for such unfavorable circumstances.
Moreover, in this talk we show a numerical comparison between our technique and other MOR methods, namely Reduced Basis applied in an implicit Multi-Moment-Matching way [1]. A greedy-type form of minimal rational interpolation, with snapshots of the HF solution at automatically chosen points being added incrementally, is also presented and tested.
[1] P. Benner, S. Gugercin, and K. Willcox. A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems. SIAM Review, 57(4):483–531, 2015.
[2] F. Bonizzoni, F. Nobile, I. Perugia, and D. Pradovera. Fast Least-Squares Padé approximation of problems with normal operators and meromorphic structure. ArXiv e-prints, arXiv/1805.05031, 2018.
[3] F. Nobile and D. Pradovera. Interpolatory rational model order reduction of parametric problems lacking uniform inf-sup stability. In preparation, 2019.
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11:55
25 mins
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Output error estimate in reduced basis method for nonlinear systems
Mohammad Abbasi
Abstract: Burgers' equation is a nonlinear scalar partial differential equation, commonly used as a testbed for many newly developed model-order reduction techniques and error estimates. In this study, we introduce two new estimates for the error induced by projection-based model order reduction techniques applied on this nonlinear equation. The first error estimate is based on the Lur'e type coupling between the linear and the nonlinear part of the system obtained after the full-discretization of Burgers' equation. The second error estimate is built upon snapshots generated in the offline phase of the reduced basis method. The second error estimate can be applied on 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 work efficiently in terms of both computational effort and accuracy
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