# PSD-Greedy Basis Generation for Structure-Preserving Model Order Reduction of Hamiltonian Systems

## Main Article Content

## Abstract

Hamiltonian systems are central in the formulation of non-dissipative physical systems. They are characterized by a phase-space, a symplectic form and a Hamiltonian function. In numerical simulations of Hamiltonian systems, algorithms show improved accuracy when the symplectic structure is preserved [10]. For structure-preserving model order reduction (MOR) of Hamiltonian systems, symplectic MOR [17, 14, 11, 3, 16] can be used. It is based on a reduced-order basis that is symplectic, which requires symplectic basis generation techniques. In our work, we discuss greedy algorithms for symplectic basis generation. We complement the procedure presented in [14] with ideas of the POD-greedy [9], which results in a new greedy symplectic basis generation technique, the PSD-greedy. Inspired by POD-greedy, we use compression techniques in the greedy iterations to enrich the basis iteratively. We prove that this algorithm computes a symplectic basis when symplectic techniques are used for compression. In the numerical experiments, we compare the discussed methods for a linear elasticity problem. The results show that improvements of up to one order of magnitude in the relative reduction error are achievable with the new basis generation technique compared to the existing greedy approach from [14].

## Article Details

How to Cite

Buchfink, P., Haasdonk, B., & Rave, S.
(2020).
PSD-Greedy Basis Generation for Structure-Preserving Model Order Reduction of Hamiltonian Systems.

*Proceedings Of The Conference Algoritmy,*, 151 - 160. Retrieved from http://www.iam.fmph.uniba.sk/amuc/ojs/index.php/algoritmy/article/view/1577/829
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Articles

## References

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[13] H. P. Langtangen and A. Logg. Solving PDEs in Python. Springer, 2017.

[14] B. Maboudi Afkham and J. Hesthaven. Structure Preserving Model Reduction of Parametric Hamiltonian Systems. SIAM Journal on Scientific Computing, 39(6):A2616–A2644, 2017.

[15] R. Milk, S. Rave, and F. Schindler. pyMOR – Generic Algorithms and Interfaces for Model Order Reduction. SIAM Journal on Scientific Computing, 38(5):S194–S216, 2016.

[16] C. Pagliantini. Dynamical Reduced Basis Methods for Hamiltonian Systems. EPFL Infoscience, 2019. Preprint.

[17] L. Peng and K. Mohseni. Symplectic Model Reduction of Hamiltonian Systems. SIAM Journal on Scientific Computing, 38(1):A1–A27, 2016.

[18] A. Salam. On theoretical and numerical aspects of symplectic Gram–Schmidt-like algorithms. Numerical Algorithms, 39:437–462, 2005.

[19] L. Sirovich. Turbulence the dynamics of coherent structures. Part I: coherent structures. Quarterly of Applied Mathematics, 45(3):561–571, 1987.

[20] K. Veroy, C. Prud’Homme, D. V. Rovas, and A. T. Patera. A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003.

[21] H. Xu. An SVD-like matrix decomposition and its applications. Linear Algebra and its Applications, 368:1 – 24, 2003.

[22] H. Xu. A Numerical Method for Computing an SVD-like Decomposition. SIAM Journal on Matrix Analysis and Applications, 26(4):1058–1082, 2005.

[2] P. Buchfink. Structure-Preserving Model Order Reduction of Hamiltonian Systems for Linear Elasticity. ARGESIM Report, 55:35–36, 2018. Mathmod 2018 Extended Abstracts.

[3] P. Buchfink, A. Bhatt, and B. Haasdonk. Symplectic Model Order Reduction with Non- Orthonormal Bases. Mathematical and Computational Applications, 24(2), 2019.

[4] K. Carlberg, R. Tuminaro, and P. Boggs. Preserving lagrangian structure in nonlinear model reduction with application to structural dynamics. SIAM Journal on Scientific Computing, 37(2):B153–B184, 2015.

[5] A. C. da Silva. Introduction to Symplectic and Hamiltonian Geometry. Notes for a Short Course at IMPA, 2007.

[6] A. Ern and J.-L. Guermond. Theory and practice of finite elements, volume 159 of Applied Mathematical Sciences. Springer-Verlag, New York, 2004.

[7] M. Grepl. Reduced-basis Approximations and a Posteriori Error Estimation for Parabolic Partial Differential Equations. PhD thesis, MIT, May 2005.

[8] B. Haasdonk. Convergence Rates of the POD-Greedy Method. ESAIM: Mathematical Modelling and Numerical Analysis, 47(3):859–873, 2013.

[9] B. Haasdonk and M. Ohlberger. Reduced basis method for finite volume approximations of parametrized linear evolution equations. ESAIM: Mathematical Modelling and Numerical Analysis, 42(2):277–302, 2008.

[10] E. Hairer, G. Wanner, and C. Lubich. Geometric Numerical Integration: Structure-Preserving Algorithms for Ordinary Differential Equations. Springer, Berlin, Heidelberg, 2006.

[11] J. S. Hesthaven and C. Pagliantini. Structure-Preserving Reduced Basis Methods for Hamiltonian Systems with a Nonlinear Poisson Structure. EPFL Infoscience, 2018. Preprint.

[12] Y. Kajee, J.-P. V. Pelteret, and B. D. Reddy. The biomechanics of the human tongue. International Journal for Numerical Methods in Biomedical Engineering, 29:492–514, 2013.

[13] H. P. Langtangen and A. Logg. Solving PDEs in Python. Springer, 2017.

[14] B. Maboudi Afkham and J. Hesthaven. Structure Preserving Model Reduction of Parametric Hamiltonian Systems. SIAM Journal on Scientific Computing, 39(6):A2616–A2644, 2017.

[15] R. Milk, S. Rave, and F. Schindler. pyMOR – Generic Algorithms and Interfaces for Model Order Reduction. SIAM Journal on Scientific Computing, 38(5):S194–S216, 2016.

[16] C. Pagliantini. Dynamical Reduced Basis Methods for Hamiltonian Systems. EPFL Infoscience, 2019. Preprint.

[17] L. Peng and K. Mohseni. Symplectic Model Reduction of Hamiltonian Systems. SIAM Journal on Scientific Computing, 38(1):A1–A27, 2016.

[18] A. Salam. On theoretical and numerical aspects of symplectic Gram–Schmidt-like algorithms. Numerical Algorithms, 39:437–462, 2005.

[19] L. Sirovich. Turbulence the dynamics of coherent structures. Part I: coherent structures. Quarterly of Applied Mathematics, 45(3):561–571, 1987.

[20] K. Veroy, C. Prud’Homme, D. V. Rovas, and A. T. Patera. A posteriori error bounds for reduced-basis approximation of parametrized noncoercive and nonlinear elliptic partial differential equations. In Proceedings of the 16th AIAA Computational Fluid Dynamics Conference, 2003.

[21] H. Xu. An SVD-like matrix decomposition and its applications. Linear Algebra and its Applications, 368:1 – 24, 2003.

[22] H. Xu. A Numerical Method for Computing an SVD-like Decomposition. SIAM Journal on Matrix Analysis and Applications, 26(4):1058–1082, 2005.