Balicki, L.

Mlinarić, P.

Rave, S.

Saak, J.

English

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MPG.PuRe

Received: 31 May 2019 Accepted: 28 June 2019DOI: 10.1002/pamm.201900459System-theoretic model order reduction with pyMORLinus Balicki1,∗, Petar Mlinaric´2,∗∗, Stephan Rave3,∗∗∗, and Jens Saak2,†1 Otto von Guericke University Magdeburg, Universitätsplatz 2, 39106 Magdeburg, Germany2 Max Planck Institute for Dynamics of Complex Technical Systems, Sandtorstraße 1, 39106 Magdeburg, Germany3 University of Münster, Orleans-Ring 10, 48149 Münster, GermanyThis paper shows recent developments in pyMOR, in particular the addition of system-theoretic methods. All methods areimplemented using pyMOR’s abstract interfaces, which allows the application to partial differential equation (PDE) modelsimplemented with third-party libraries. We demonstrate this by applying balanced truncation to a PDE model discretized inFEniCS.c© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim1 IntroductionpyMOR is a free software library for building model order reduction (MOR) applications in the Python programming lan-guage. All of its algorithms are based on abstract interfaces which allow seamless integration with external partial differentialequation (PDE) solver packages (see [1]). Its initial focus was on reduced basis methods for linear and nonlinear parameterizedPDEs. We present here recent developments in system-theoretic MOR methods and related algorithms.In Section 2, we give a brief overview of linear systems and balanced truncation (BT). Next, we describe pyMOR’s abstractinterfaces and the implementation of the methods in Section 3. We demonstrate BT applied to a PDE model discretized usingFEniCS in Section 4.2 System-theoretic model order reductionWe consider continuous-time, linear time-invariant systemsEx˙(t) = Ax(t) +Bu(t), y(t) = Cx(t), x(0) = 0, (1)with E,A ∈ Rn×n (E invertible), B ∈ Rn×m, C ∈ Rp×n, input u(t) ∈ Rm, state x(t) ∈ Rn, and output y(t) ∈ Rp. Thegoal of MOR is to find a reduced-order model (ROM)Eˆ ˙ˆx(t) = Aˆxˆ(t) + Bˆu(t), yˆ(t) = Cˆxˆ(t), xˆ(0) = 0, (2)with Eˆ, Aˆ ∈ Rr×r, Bˆ ∈ Rr×m, Cˆ ∈ Rp×r, for some r  n such that ‖y − yˆ‖ is small for every input u in some functionspace. BT is a projection-based method where Eˆ, Aˆ, Bˆ, Cˆ are obtained by Petrov-Galerkin projection of the original systemmatrices on appropriate ansatz and test spaces. The main computational cost in finding these spaces lies in solving two large-scale Lyapunov equations: APET + EPAT +BBT = 0 and ATQE + ETQA+ CTC = 0. Low-rank approximations forP and Q are computed using the low-rank alternating directions implicit (LR-ADI) method. The method consists of solvinga sequence of linear systems (piE +A)xi = bi with shifts pi.3 Generic interfaces for model order reductionpyMOR’s PDE solver integration is based on three abstract interfaces: VectorArrays, Operators, and Models. EachVectorArray is an element of a VectorSpace, Operator is a mapping (possibly nonlinear or parametric) betweentwo VectorSpaces, and a Model is a structured collection of Operators. The interfaces are completely agnostic of thesolver’s internal data representation. All reduction algorithms in pyMOR are implemented in terms of these interfaces.In particular, in pyMOR 0.5, LTISystem is a model representing (1), where E,A,B,C are Operators. In case of theexample in Section 4, these operators come from a discretization in FEniCS. Via linear combination and the Operator’sapply_inverse method, the LR-ADI algorithm instructs FEniCS to solve (piE +A)xi = bi.∗ linus.balicki@ovgu.de∗∗ Corresponding author: mlinaric@mpi-magdeburg.mpg.de∗∗∗ stephan.rave@uni-muenster.de† saak@mpi-magdeburg.mpg.deThis is an open access article under the terms of the Creative Commons Attribution License 4.0, which permits use,distribution and reproduction in any medium, provided the original work is properly cited.PAMM · Proc. Appl. Math. Mech. 2019;19:e201900459. www.gamm-proceedings.com 1 of 2https://doi.org/10.1002/pamm.201900459 c© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim2 of 2 Section 22: Scientific computingFig. 1: Left column: Upper and lower bounds for the relative H∞ reduction error when using balanced truncation. Middle column: Full-order model snapshot at time t = 10 and its reconstruction error using a reduced-order model of order 10. Right column: The full-orderand reduced-order models’ outputs and the error.4 Numerical exampleAs an example, we consider the heat equation over a cross section of a heat sink (see [2] for the problem description). Finiteelement discretization, using FEniCS, leads to a system of the form (1) with n = 12 296, m = 1, and p = 1. The top-middleplot in Figure 1 shows the state of the full-order model at t = 10 when the input is u(t) = sin(pi3 t)2. The average run timeis about 8 seconds on a laptop (model: Vaio VJS132C11L; processor: Intel c© CoreTM i7-8550U CPU @ 1.80 GHz, 4 cores,8 logical processors, hyper-threading activated; RAM: 8 GB; cache: 8 MB).We applied BT (using pyMOR 0.5) to get a ROM of order 10, with a run time of about 7 seconds. Based on Hankel singularvalues, we find the relative H∞-error lies in the interval [7.27 × 10−5, 1.23 × 10−4], shown also in the left plot of Figure 1.The relativeH2-error can be computed and its value is 7.37×10−3. In the bottom row of Figure 1, we see the state and outputerrors of the ROM. As expected, we see that the output is approximated better than the state. The average run time of thesimulation is about 15 milliseconds, which is roughly 500 times faster then for the full-order model.The entire script to generate these results can be found in [2].5 Conclusion and outlookWe demonstrated BT applied to PDE models discretized in the PDE solver package FEniCS. This is possible due to theLR-ADI method’s implementation using pyMOR’s abstract interfaces.The example used FEniCS, but other solver packages are also possible. Currently also deal.II, DUNE, and NGSolve aresupported. Custom (domain specific) solvers can be easily integrated with pyMOR.In pyMOR 0.5, further variants of BT for first-order systems are available, together with a few methods for second-ordersystems. Furthermore, interpolatory methods for first-order and second-order systems are also included.Some PDE solver packages do not support complex numbers (e.g., FEniCS). In the example, we considered a system withsymmetric E and A, where real shifts pi were sufficient. For general systems, we plan to add support to pyMOR for solvinglinear systems with complex shifts based on real linear algebra to further enable applicability of system-theoretic methods tomodels defined with such PDE solver packages.Acknowledgements This work was funded by the Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Ger-many’s Excellence Strategy - EXC 2044 - 390685587, Mathematics Münster:Dynamics - Geometry - Structure, the German Federal Ministryof Education and Research (BMBF) under contract 05M18PMA, and DFG “pyMOR - Sustainable Software for Model Order Reduction”project (RA 3055/1-1, SA 3477/1-1).References[1] R. Milk, S. Rave, and F. Schindler, SIAM J. Sci. Comput. 38(5), pp. S194–S216 (2016).[2] P. Mlinaric´ and S. Rave, pyMOR demo for balanced truncation applied to a heat sink model in FEniCS, Zenodo (2019), http://doi.org/10.5281/zenodo.3232900.c© 2019 The Authors Proceedings in Applied Mathematics & Mechanics published by Wiley-VCH Verlag GmbH & Co. KGaA Weinheim www.gamm-proceedings.com

System-Theoretic Model Order Reduction with pyMOR

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