# Local training and enrichment based on a residual localization strategy

## Main Article Content

## Abstract

To efficiently tackle parametrized multi and/or large scale problems, we propose an adaptive localized model order reduction framework combining both local offline training and local online enrichment with localized error control. For the latter, we adapt the residual localization strategy introduced in [Buhr, Engwer, Ohlberger, Rave, SIAM J. Sci. Comput., 2017] which allows to derive a localized a posteriori error estimator that can be employed to adaptively enrich the reduced solution space locally where needed. Numerical experiments demonstrate the potential of the proposed approach.

## Article Details

How to Cite

Keil, T., Ohlberger, M., Schindler, F., & Schleuß, J.
(2024).
Local training and enrichment based on a residual localization strategy.

*Proceedings Of The Conference Algoritmy,*, 76 - 84. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/2157/1029
Section

Articles

## References

[1] F. Albrecht, B. Haasdonk, S. Kaulmann, and M. Ohlberger, The localized reduced basis multiscale method, in Proceedings of Algoritmy 2012, Conference on Scientific Computing, Vysoke Tatry, Podbanske, 2012, Slovak University of Technology in Bratislava, Publishing House of STU, 2012, pp. 393–403.

[2] I. Babuška and R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Model. Simul., 9 (2011), pp. 373–406.

[3] P. Bastian, M. Blatt, A. Dedner, N.A. Dreier, C. Engwer, R. Fritze, C. Grüninger, D. Kempf, R. Klöfkorn, M. Ohlberger, O. Sander, The DUNE framework: basic concepts and recent developments, Comput. Math. Appl., 81 (2021), pp. 75–112.

[4] P. Benner, A. Cohen, M. Ohlberger, and K. Willcox, eds., Model reduction and approximation, vol. 15 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2017. Theory and algorithms.

[5] J. Blechta, J. Málek, and M. Vohralı́k, Localization of the W-1, q norm for local a poste- riori efficiency, IMA J. Numer. Anal., 40 (2020), pp. 914–950.

[6] A. Buhr, C. Engwer, M. Ohlberger, and S. Rave, ArbiLoMod, a simulation technique designed for arbitrary local modifications, SIAM J. Sci. Comput., 39 (2017), pp. A1435–A1465.

[7] A. Buhr, L. Iapichino, M. Ohlberger, S. Rave, F. Schindler, and K. Smetana., Localized model reduction for parameterized problems, in P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Sileira (eds.). Model Order Reduction, Volume 2, Snapshot-Based Methods and Algorithms., Walter De Gruyter GmbH, Berlin, 2020.

[8] A. Buhr and K. Smetana, Randomized Local Model Order Reduction, SIAM J. Sci. Comput., 40 (2018), pp. A2120–A2151.

[9] P. Ciarlet and M. Vohralı́k, Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients, ESAIM Math. Model. Numer. Anal., 52 (2018), pp. 2037–2064.

[10] N. Halko, P. G. Martinsson, and J. A. Tropp, Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53 (2011), pp. 217–288.

[11] M. Kartmann, T. Keil, M. Ohlberger, S. Volkwein, B. Kaltenbacher, Adaptive Reduced Basis Trust Region Methods for Parameter Identification Problems, arXiv preprint, arXiv:2309.07627 (2023).

[12] T. Keil, L. Mechelli, M. Ohlberger, F. Schindler, S. Volkwein, A non-conforming dual approach for adaptive trust-region reduced basis approximation of PDE-constrained parameter optimization, ESAIM Math. Model. Numer. Anal. 55 (2021), pp. 1239–1269.

[13] T. Keil and M. Ohlberger, A relaxed localized trust-region reduced basis approach for optimization of multiscale problems, ESAIM Math. Model. Numer. Anal. 58 (2024), pp. 79–105.

[14] T. Keil, M. Ohlberger, and F. Schindler, Adaptive Localized Reduced Basis Methods for Large Scale Parameterized Systems, arXiv preprint, arXiv:2303.03074 (2023). Accepted for publication in Large-Scale Scientific Computing, 14th International Conference, LSSC 2023, Sozopol, Bulgaria, June 5–9, 2023, edited by I. Lirkov, S. Margenov.

[15] C. Ma, R. Scheichl, and T. Dodwell, Novel design and analysis of generalized finite element methods based on locally optimal spectral approximations, SIAM J. Numer. Anal., 60 (2022), pp. 244–273.

[16] R. Milk, S. Rave, and F. Schindler, pyMOR – generic algorithms and interfaces for model order reduction, SIAM J. Sci. Comput., 38 (2016), pp. S194–S216.

[17] M. Ohlberger and F. Schindler, Error control for the localized reduced basis multiscale method with adaptive on-line enrichment, SIAM J. Sci. Comput., 37 (2015), pp. A2865– A2895.

[18] J. Schleuß and K. Smetana, Optimal local approximation spaces for parabolic problems, Multiscale Model. Simul., 20 (2022), pp. 551–582.

[19] J. Schleuß, K. Smetana, and L. ter Maat, Randomized quasi-optimal local approximation spaces in time, SIAM J. Sci. Comput., 45 (2023), pp. A1066–A1096.

[20] K. Smetana and A. T. Patera, Optimal local approximation spaces for component-based static condensation procedures, SIAM J. Sci. Comput., 38 (2016), pp. A3318–A3356.

[21] K. Smetana and T. Taddei, Localized model reduction for nonlinear elliptic partial differential equations: localized training, partition of unity, and adaptive enrichment, SIAM J. Sci. Comput., 45 (2022), pp. A1300–A1331.

[2] I. Babuška and R. Lipton, Optimal local approximation spaces for generalized finite element methods with application to multiscale problems, Multiscale Model. Simul., 9 (2011), pp. 373–406.

[3] P. Bastian, M. Blatt, A. Dedner, N.A. Dreier, C. Engwer, R. Fritze, C. Grüninger, D. Kempf, R. Klöfkorn, M. Ohlberger, O. Sander, The DUNE framework: basic concepts and recent developments, Comput. Math. Appl., 81 (2021), pp. 75–112.

[4] P. Benner, A. Cohen, M. Ohlberger, and K. Willcox, eds., Model reduction and approximation, vol. 15 of Computational Science & Engineering, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2017. Theory and algorithms.

[5] J. Blechta, J. Málek, and M. Vohralı́k, Localization of the W-1, q norm for local a poste- riori efficiency, IMA J. Numer. Anal., 40 (2020), pp. 914–950.

[6] A. Buhr, C. Engwer, M. Ohlberger, and S. Rave, ArbiLoMod, a simulation technique designed for arbitrary local modifications, SIAM J. Sci. Comput., 39 (2017), pp. A1435–A1465.

[7] A. Buhr, L. Iapichino, M. Ohlberger, S. Rave, F. Schindler, and K. Smetana., Localized model reduction for parameterized problems, in P. Benner, S. Grivet-Talocia, A. Quarteroni, G. Rozza, W.H.A. Schilders, L.M. Sileira (eds.). Model Order Reduction, Volume 2, Snapshot-Based Methods and Algorithms., Walter De Gruyter GmbH, Berlin, 2020.

[8] A. Buhr and K. Smetana, Randomized Local Model Order Reduction, SIAM J. Sci. Comput., 40 (2018), pp. A2120–A2151.

[9] P. Ciarlet and M. Vohralı́k, Localization of global norms and robust a posteriori error control for transmission problems with sign-changing coefficients, ESAIM Math. Model. Numer. Anal., 52 (2018), pp. 2037–2064.

[10] N. Halko, P. G. Martinsson, and J. A. Tropp, Finding structure with randomness: probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53 (2011), pp. 217–288.

[11] M. Kartmann, T. Keil, M. Ohlberger, S. Volkwein, B. Kaltenbacher, Adaptive Reduced Basis Trust Region Methods for Parameter Identification Problems, arXiv preprint, arXiv:2309.07627 (2023).

[12] T. Keil, L. Mechelli, M. Ohlberger, F. Schindler, S. Volkwein, A non-conforming dual approach for adaptive trust-region reduced basis approximation of PDE-constrained parameter optimization, ESAIM Math. Model. Numer. Anal. 55 (2021), pp. 1239–1269.

[13] T. Keil and M. Ohlberger, A relaxed localized trust-region reduced basis approach for optimization of multiscale problems, ESAIM Math. Model. Numer. Anal. 58 (2024), pp. 79–105.

[14] T. Keil, M. Ohlberger, and F. Schindler, Adaptive Localized Reduced Basis Methods for Large Scale Parameterized Systems, arXiv preprint, arXiv:2303.03074 (2023). Accepted for publication in Large-Scale Scientific Computing, 14th International Conference, LSSC 2023, Sozopol, Bulgaria, June 5–9, 2023, edited by I. Lirkov, S. Margenov.

[15] C. Ma, R. Scheichl, and T. Dodwell, Novel design and analysis of generalized finite element methods based on locally optimal spectral approximations, SIAM J. Numer. Anal., 60 (2022), pp. 244–273.

[16] R. Milk, S. Rave, and F. Schindler, pyMOR – generic algorithms and interfaces for model order reduction, SIAM J. Sci. Comput., 38 (2016), pp. S194–S216.

[17] M. Ohlberger and F. Schindler, Error control for the localized reduced basis multiscale method with adaptive on-line enrichment, SIAM J. Sci. Comput., 37 (2015), pp. A2865– A2895.

[18] J. Schleuß and K. Smetana, Optimal local approximation spaces for parabolic problems, Multiscale Model. Simul., 20 (2022), pp. 551–582.

[19] J. Schleuß, K. Smetana, and L. ter Maat, Randomized quasi-optimal local approximation spaces in time, SIAM J. Sci. Comput., 45 (2023), pp. A1066–A1096.

[20] K. Smetana and A. T. Patera, Optimal local approximation spaces for component-based static condensation procedures, SIAM J. Sci. Comput., 38 (2016), pp. A3318–A3356.

[21] K. Smetana and T. Taddei, Localized model reduction for nonlinear elliptic partial differential equations: localized training, partition of unity, and adaptive enrichment, SIAM J. Sci. Comput., 45 (2022), pp. A1300–A1331.