Construction of local reduced spaces for Friedrichs' systems via randomized training
Article Sidebar
Published
Sep 21, 2024
Main Article Content
Abstract
This contribution extends the localized training approach, traditionally employed for multiscale problems and parameterized partial differential equations (PDEs) featuring locally heterogeneous coefficients, to the class of linear, positive symmetric operators, known as Friedrichs' operators. Considering a local subdomain with corresponding oversampling domain we prove the compactness of the transfer operator which maps boundary data to solutions on the interior domain. While a Caccioppoli-inequality quantifying the energy decay to the interior holds true for all Friedrichs' systems, showing a compactness result for the graph-spaces hosting the solution is additionally necessary. We discuss the mixed formulation of a convection-diffusion-reaction problem where the necessary compactness result is obtained by the Picard-Weck-Weber theorem. Our numerical results, focusing on a scenario involving heterogeneous diffusion fields with multiple high-conductivity channels, demonstrate the effectiveness of the proposed method.
Article Details
How to Cite
Engwer, C., Ohlberger, M., & Renelt, L.
(2024).
Construction of local reduced spaces for Friedrichs' systems via randomized training.
Proceedings Of The Conference Algoritmy, , 56 - 65.
Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/2144/1027
Section
Articles
References
[1] A. Abdulle, E. Weinan, B. Engquist, and E. Vanden-Eijnden. The heterogeneous multiscale method. Acta Numerica, 21:1–87, 2012.
[2] I. Babuska and R. Lipton. Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. Multiscale Modeling & Simulation, 9(1):373–406, 2011.
[3] P. Benner, M. Ohlberger, A. Cohen, and K. Willcox. Model Reduction and Approximation. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017.
[4] D. Broersen and R. P. Stevenson. A Petrov–Galerkin discretization with optimal test space of a mild-weak formulation of convection–diffusion equations in mixed form. IMA Journal of Numerical Analysis, 35(1):39–73, 2015.
[5] A. Buhr, C. Engwer, M. Ohlberger, and S. Rave. ArbiLoMod, a simulation technique designed for arbitrary local modifications. SIAM J. Sci. Comput., 39(4):A1435–A1465, 2017.
[6] A. Buhr, L. Iapichino, M. Ohlberger, S. Rave, F. Schindler, and K. Smetana. Localized model reduction for parameterized problems. In Handbook on Model Order Reduction. Walter De Gruyter, 2020.
[7] A. Buhr and K. Smetana. Randomized local model order reduction. SIAM journal on scientific computing, 40(4):A2120–A2151, 2018.
[8] Z. Cai, R. Lazarov, T. A. Manteuffel, and S. F. McCormick. First-order system least squares for second-order partial differential equations: Part I. SIAM Journal on Numerical Analysis, 31(6):1785–1799, 1994.
[9] Y. Efendiev, J. Galvis, and T. Y. Hou. Generalized multiscale finite element methods (GMs- FEM). Journal of computational physics, 251:116–135, 2013.
[10] A. Ern and J.-L. Guermond. Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory. SIAM journal on numerical analysis, 44(2):753–778, 2006.
[11] A. Ern, J.-L. Guermond, and G. Caplain. An Intrinsic Criterion for the Bijectivity of Hilbert Operators Related to Friedrich’Systems. Communications in partial differential equations, 32(2):317–341, 2007.
[12] K. O. Friedrichs. Symmetric positive linear differential equations. Communications on Pure and Applied Mathematics, 11(3):333–418, 1958.
[13] M. J. Gander and A. Loneland. SHEM: an optimal coarse space for RAS and its multiscale approximation. In Domain decomposition methods in science and engineering XXIII, volume 116 of Lect. Notes Comput. Sci. Eng., pages 313–321. Springer, Cham, 2017.
[14] A. Heinlein, A. Klawonn, J. Knepper, and O. Rheinbach. Multiscale coarse spaces for overlap- ping Schwarz methods based on the ACMS space in 2D. Electron. Trans. Numer. Anal., 48:156–182, 2018.
[15] T. Y. Hou and X.-H. Wu. A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of computational physics, 134(1):169–189, 1997.
[16] T. Keil, M. Ohlberger, F. Schindler, and J. Schleuß. Local training and enrichment based on a residual localization strategy. arXiv preprint arXiv:2404.16537, 2024.
[17] A. Målqvist and D. Peterseim. Localization of elliptic multiscale problems. Mathematics of Computation, 83(290):2583–2603, 2014.
[18] M. Ohlberger and F. Schindler. Error control for the localized reduced basis multiscale method with adaptive on-line enrichment. SIAM J. Sci. Comput., 37(6):A2865–A2895, 2015.
[19] R. Picard. An elementary proof for a compact imbedding result in generalized electromagnetic theory. Mathematische Zeitschrift, 187(2):151–164, 1984.
[20] F. Rellich. Ein satz über mittlere konvergenz. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1930:30–35, 1930.
[21] L. Renelt. Source code to ’Construction of local reduced spaces for Friedrichs’ systems via randomized training’. https://doi.org/10.5281/zenodo.11474448, 2024.
[22] F. Romor, D. Torlo, and G. Rozza. Friedrichs’ systems discretized with the Discontinuous Galerkin method: domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions. arXiv preprint arXiv:2308.03378, 2023.
[23] J. Schleuß and K. Smetana. Optimal local approximation spaces for parabolic problems. Multiscale Modeling & Simulation, 20(1):551–582, 2022.
[24] K. Smetana and A. T. Patera. Optimal local approximation spaces for component-based static condensation procedures. SIAM J. Sci. Comput., 38(5):A3318–A3356, 2016.
[25] C. Weber and P. Werner. A local compactness theorem for Maxwell’s equations. Mathematical Methods in the Applied Sciences, 2(1):12–25, 1980.
[26] N. Weck. Maxwell’s boundary value problem on Riemannian manifolds with nonsmooth boundaries. Journal of Mathematical Analysis and Applications, 46(2):410–437, 1974.
[2] I. Babuska and R. Lipton. Optimal local approximation spaces for generalized finite element methods with application to multiscale problems. Multiscale Modeling & Simulation, 9(1):373–406, 2011.
[3] P. Benner, M. Ohlberger, A. Cohen, and K. Willcox. Model Reduction and Approximation. Society for Industrial and Applied Mathematics, Philadelphia, PA, 2017.
[4] D. Broersen and R. P. Stevenson. A Petrov–Galerkin discretization with optimal test space of a mild-weak formulation of convection–diffusion equations in mixed form. IMA Journal of Numerical Analysis, 35(1):39–73, 2015.
[5] A. Buhr, C. Engwer, M. Ohlberger, and S. Rave. ArbiLoMod, a simulation technique designed for arbitrary local modifications. SIAM J. Sci. Comput., 39(4):A1435–A1465, 2017.
[6] A. Buhr, L. Iapichino, M. Ohlberger, S. Rave, F. Schindler, and K. Smetana. Localized model reduction for parameterized problems. In Handbook on Model Order Reduction. Walter De Gruyter, 2020.
[7] A. Buhr and K. Smetana. Randomized local model order reduction. SIAM journal on scientific computing, 40(4):A2120–A2151, 2018.
[8] Z. Cai, R. Lazarov, T. A. Manteuffel, and S. F. McCormick. First-order system least squares for second-order partial differential equations: Part I. SIAM Journal on Numerical Analysis, 31(6):1785–1799, 1994.
[9] Y. Efendiev, J. Galvis, and T. Y. Hou. Generalized multiscale finite element methods (GMs- FEM). Journal of computational physics, 251:116–135, 2013.
[10] A. Ern and J.-L. Guermond. Discontinuous Galerkin methods for Friedrichs’ systems. I. General theory. SIAM journal on numerical analysis, 44(2):753–778, 2006.
[11] A. Ern, J.-L. Guermond, and G. Caplain. An Intrinsic Criterion for the Bijectivity of Hilbert Operators Related to Friedrich’Systems. Communications in partial differential equations, 32(2):317–341, 2007.
[12] K. O. Friedrichs. Symmetric positive linear differential equations. Communications on Pure and Applied Mathematics, 11(3):333–418, 1958.
[13] M. J. Gander and A. Loneland. SHEM: an optimal coarse space for RAS and its multiscale approximation. In Domain decomposition methods in science and engineering XXIII, volume 116 of Lect. Notes Comput. Sci. Eng., pages 313–321. Springer, Cham, 2017.
[14] A. Heinlein, A. Klawonn, J. Knepper, and O. Rheinbach. Multiscale coarse spaces for overlap- ping Schwarz methods based on the ACMS space in 2D. Electron. Trans. Numer. Anal., 48:156–182, 2018.
[15] T. Y. Hou and X.-H. Wu. A multiscale finite element method for elliptic problems in composite materials and porous media. Journal of computational physics, 134(1):169–189, 1997.
[16] T. Keil, M. Ohlberger, F. Schindler, and J. Schleuß. Local training and enrichment based on a residual localization strategy. arXiv preprint arXiv:2404.16537, 2024.
[17] A. Målqvist and D. Peterseim. Localization of elliptic multiscale problems. Mathematics of Computation, 83(290):2583–2603, 2014.
[18] M. Ohlberger and F. Schindler. Error control for the localized reduced basis multiscale method with adaptive on-line enrichment. SIAM J. Sci. Comput., 37(6):A2865–A2895, 2015.
[19] R. Picard. An elementary proof for a compact imbedding result in generalized electromagnetic theory. Mathematische Zeitschrift, 187(2):151–164, 1984.
[20] F. Rellich. Ein satz über mittlere konvergenz. Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse, 1930:30–35, 1930.
[21] L. Renelt. Source code to ’Construction of local reduced spaces for Friedrichs’ systems via randomized training’. https://doi.org/10.5281/zenodo.11474448, 2024.
[22] F. Romor, D. Torlo, and G. Rozza. Friedrichs’ systems discretized with the Discontinuous Galerkin method: domain decomposable model order reduction and Graph Neural Networks approximating vanishing viscosity solutions. arXiv preprint arXiv:2308.03378, 2023.
[23] J. Schleuß and K. Smetana. Optimal local approximation spaces for parabolic problems. Multiscale Modeling & Simulation, 20(1):551–582, 2022.
[24] K. Smetana and A. T. Patera. Optimal local approximation spaces for component-based static condensation procedures. SIAM J. Sci. Comput., 38(5):A3318–A3356, 2016.
[25] C. Weber and P. Werner. A local compactness theorem for Maxwell’s equations. Mathematical Methods in the Applied Sciences, 2(1):12–25, 1980.
[26] N. Weck. Maxwell’s boundary value problem on Riemannian manifolds with nonsmooth boundaries. Journal of Mathematical Analysis and Applications, 46(2):410–437, 1974.