Jul 27 – 31, 2020
Virtual event
Europe/Berlin timezone

Adaptive Interpolatory MOR by Learning the Error Estimator in the Parameter Domain

Jul 29, 2020, 7:30 PM
1h
Poster Posters 2

Speaker

Mr Sridhar Chellappa (Max Planck Institute for Dynamics of Complex Technical Systems)

Description

Interpolatory methods offer a powerful framework for generating reduced‑order models for non‑parametric or parametric systems with time‑varying inputs. Choosing the interpolation points adaptively remains an area of active interest. A greedy framework has been introduced in [1, 2] to choose interpolation points automatically using a posteriori error estimators. Nevertheless, when the parameter range is large or if the parameter space dimension is larger than two, the greedy algorithm may take considerable time, since the training set needs to include a considerable number of parameters.

In this work, we introduce an adaptive training technique by learning an efficient a posteriori error estimator over the parameter domain. A fast learning process is created by interpolating the error estimator using radial basis functions over a fine parameter training set, representing the whole parameter domain. The error estimator is evaluated only on a coarse training set consisting of only a few parameter samples. The algorithm is an extension of the work in [3] to interpolatory model order reduction in the frequency domain. Possibilities exist to use other sophisticated machine‑learning techniques like artificial neural networks, etc. to learn the error estimator, based on data at a few parameter samples. However, we do not pursue this in the present work. Selected numerical examples demonstrate the efficiency of the proposed approach.

References

[1] Feng, L., Antoulas, A.C., Benner, P.: Some a posteriori error bounds for reduced‑order modelling of (non‑)parametrized linear systems. ESAIM: Math. Model. Numer. Anal. 51(6), 2127–2158 (2017).

[2] Feng, L., Benner, P.: A new error estimator for reduced‑order modeling of linear parametric systems. IEEE Trans. Microw. Theory Techn. 67(12), 4848–4859 (2019).

[3] Chellappa, S., Feng, L., Benner, P.: An adaptive sampling approach for the reduced basis method. e‑prints 1910.00298, arXiv (2019). URL https://arxiv.org/abs/1910.00298. Math.NA.

[4] Chellappa, S., Feng, L., de la Rubia, V., Benner, P.: Adaptive Interpolatory MOR by Learning the Error Estimator in the Parameter Domain. e‑prints 2003.02569, arXiv (2020).
URL https://arxiv.org/abs/2003.02569. Math.NA.

Primary authors

Mr Sridhar Chellappa (Max Planck Institute for Dynamics of Complex Technical Systems) Dr Lihong Feng (Max Planck Institute for Dynamics of Complex Technical Systems) Dr Valentin de la Rubia (Universidad Politécnica de Madrid) Prof. Peter Benner (Max Planck Institute for Dynamics of Complex Technical Systems)

Presentation materials