On Snapshot-Based Model Reduction Under Compatibility Conditions for a Nonlinear Flow Problem on Networks

Springer Science and Business Media LLC - 2022
Björn Liljegren-Sailer1, Nicole Marheineke1
1Universität Trier, FB IV - Mathematik, Lehrstuhl Modellierung und Numerik, Trier, Germany

Tóm tắt

AbstractThis paper is on the construction of structure-preserving, online-efficient reduced models for the barotropic Euler equations with a friction term on networks. The nonlinear flow problem finds broad application in the context of gas distribution networks. We propose a snapshot-based reduction approach that consists of a mixed variational Galerkin approximation combined with quadrature-type complexity reduction. Its main feature is that certain compatibility conditions are assured during the training phase, which make our approach structure-preserving. The resulting reduced models are locally mass conservative and inherit an energy bound and port-Hamiltonian structure. We also derive a wellposedness result for them. In the training phase, the compatibility conditions pose challenges, we face constrained data approximation problems as opposed to the unconstrained training problems in the conventional reduction methods. The training of our model order reduction consists of a principal component analysis under a compatibility constraint and, notably, yields reduced models that fulfill an optimality condition for the snapshot data. The training of our quadrature-type complexity reduction involves a semi-definite program with combinatorial aspects, which we approach by a greedy procedure. Efficient algorithmic implementations are presented. The robustness and good performance of our structure-preserving reduced models are showcased at the example of gas network simulations.

Từ khóa


Tài liệu tham khảo

Afkham, B., Hesthaven, J.: Structure preserving model reduction of parametric Hamiltonian systems. SIAM J. Sci. Comput. 39(6), A2616–A2644 (2017)

Afkham, B., Hesthaven, J.: Structure-preserving model-reduction of dissipative Hamiltonian systems. J. Sci. Comput. 81(1), 3–21 (2019)

Ali, S., Ballarin, F., Rozza, G.: Stabilized reduced basis methods for parametrized steady Stokes and Navier-Stokes equations. Comput. Math 80(11), 2399–2416 (2020)

An, S.S., Kim, T., James, D.L.: Optimizing cubature for efficient integration of subspace deformations. ACM Trans. Graph. 27(5), 1–10 (2008)

Antonelli, P., Marcati, P.: The quantum hydrodynamics system in two space dimensions. Arch. Ration. Mech. Anal. 203(2), 499–527 (2012)

Barrault, M., Maday, Y., Nguyen, N.C., Patera, A.T.: An empirical interpolation method: Application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus Mathematique 339(9), 667–672 (2004)

Ben-Israel, A., Greville, T.N.E.: Generalized Inverses: Theory and Applications, 2nd edn. Springer, New York (2001)

Benner, P., Mehrmann, V., Sorensen, D.C.: editors. Dimension Reduction of Large-Scale Systems. Lecture Notes in Computational Science and Engineering. Springer, 1 edition, (2005)

Blumensath, T., Davies, M.E.: Gradient pursuit for non-linear sparse signal modelling. In: 2008 16th European Signal Processing Conference, pages 1–5, (2008)

Boffi, D., Gastaldi, L.: editors. Mixed Finite Elements, Compatibility Conditions, and Applications. Lecture Notes in Mathematics. Springer, 1 edition, (2008)

Brouwer, J., Gasser, I., Herty, M.: Gas pipeline models revisited: Model hierarchies, nonisothermal models, and simulations of networks. Multiscale Model. Simul. 9(2), 601–623 (2011)

Buchfink, P., Bhatt, A., Haasdonk, B.: Symplectic model order reduction with non-orthonormal bases. Math. Comp. Appl 24(2), 43 (2019)

Carlberg, K., Bou-Mosleh, C., Farhat, C.: Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations. Int. J. Numer. Methods Eng. 86(2), 155–181 (2011)

Carlberg, K., Tuminaro, R., Boggs, P.: Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics. SIAM J. Sci. Comput. 37(2), B153–B184 (2015)

Celledoni, E., Grimm, V., McLachlan, R.I., McLaren, D.I., O’Neale, D., Owren, B., Quispel, G.R.W.: Preserving energy resp. dissipation in numerical PDEs using the Average Vector Field method. J. Comput. Phys. 231(20), 6770–6789 (2012)

Chaturantabut, S., Beattie, C., Gugercin, S.: Structure-preserving model reduction for nonlinear port-Hamiltonian systems. SIAM J. Sci. Comput. 38(5), B837–B865 (2016)

Chaturantabut, S., Sorensen, D.C.: Nonlinear model reduction via discrete empirical interpolation. SIAM J. Sci. Comput. 32(5), 2737–2764 (2010)

Chaturantabut, S., Sorensen, D.C.: A state space error estimate for POD-DEIM nonlinear model reduction. SIAM J. Numer. Anal. 50(1), 46–63 (2012)

Christiansen, S.H., Munthe-Kaas, H.Z., Owren, B.: Topics in structure-preserving discretization. Acta Numerica 20, 1–119 (2011)

Domschke, P., Dua, A., Stolwijk, J.J., Lang, J., Mehrmann, V.: Adaptive refinement strategies for the simulation of gas flow in networks using a model hierarchy. Electron. Trans. Numer. Anal. 48, 97–113 (2018)

Domschke, P., Kolb, O., Lang, J.: Adjoint-based error control for the simulation and optimization of gas and water supply networks. Appl. Math. Comput. 259, 1003–1018 (2015)

Egger, H.: A robust conservative mixed finite element method for compressible flow on pipe networks. SIAM J. Sci. Comput 40(1), A108–A129 (2018)

Egger, H., Kugler, T.: Damped wave systems on networks: Exponential stability and uniform approximations. Numer. Math. 138(4), 839–867 (2018)

Egger, H., Kugler, T., Liljegren-Sailer, B.: Stability preserving approximations of a semilinear hyperbolic gas transport model. In: Hyperbolic Problems: Theory, Numerics, Applications, volume 10, pages 427–433. AIMS Series on Appl. Math, (2020)

Egger, H., Kugler, T., Liljegren-Sailer, B., Marheineke, N., Mehrmann, V.: On structure-preserving model reduction for damped wave propagation in transport networks. SIAM J. Sci. Comput. 40(1), A331–A365 (2018)

Fareed, H., Singler, J.R., Zhang, Y., Shen, J.: Incremental proper orthogonal decomposition for PDE simulation data. Comput. Math. Appl. 75(6), 1942–1960 (2018)

Farhat, C., Avery, P., Chapman, T., Cortial, J.: Dimensional reduction of nonlinear finite element dynamic models with finite rotations and energy-based mesh sampling and weighting for computational efficiency. Int. J. Numer. Meth. Eng. 98(9), 625–662 (2014)

Fisher, T.C., Carpenter, M.H.: High-order entropy stable finite difference schemes for nonlinear conservation laws: Finite domains. J. Comput. Phys. 252, 518–557 (2013)

Giesselmann, J., Lattanzio, C., Tzavaras, A.E.: Relative energy for the Korteweg theory and related Hamiltonian flows in gas dynamics. Arch. Ration. Mech. Anal. 223(3), 1427–1484 (2017)

Gotsman, C., Toledo, S.: On the computation of null spaces of sparse rectangular matrices. SIAM J. Matrix Anal. Appl. 30(2), 445–463 (2008)

Grundel, S., Jansen, L., Hornung, N., Clees, T., Tischendorf, C., Benner, P.: Model order reduction of differential algebraic equations arising from the simulation of gas transport networks. In: Progress in Differential-Algebraic Equations: Deskriptor 2013, pages 183–205. Springer, (2014)

Gugercin, S., Polyuga, R.V., Beattie, C.A., van der Schaft, A.: Interpolation-based $$H_2$$ model reduction for port-Hamiltonian systems. In: Proceedings of the 48th IEEE Conference on Decision and Control, and the 28th Chinese Control Conference, Shanghai, pages 5362–5369, (2009)

Hairer, E., Lubich, C., Wanner, G.: Geometric Numerical Integration. Springer, 2nd edn. Springer, Heidelberg, Berlin (2006)

Hartman, P.: Ordinary Differential Equations, 2nd edn. Society for Industrial and Applied Mathematics, Philadelphia (2002)

Hernandez, J.A., Caicedo, M.A., Ferrer, A.: Dimensional hyper-reduction of nonlinear finite element models via empirical cubature. Comput. Methods Appl. Mech. Engrg. 313, 687–722 (2017)

Herran-Gonzalez, A., De La Cruz, J.M., Andres-Toro, B.D., Risco-Martin, J.L.: Modeling and simulation of a gas distribution pipeline network. Appl. Math. Model. 33(3), 1584–1600 (2009)

Hesthaven, J.S., Rozza, G., Stamm, B.: Certified Reduced Basis Methods for Parametrized Partial Differential Equations. SpringerBriefs in Mathematics. Springer, (2016)

Himpe, C., Grundel, S., Benner, P.: Model order reduction for gas and energy networks. J. Ind. Math 11(13), 2190–5983 (2021)

Karper, T.: Convergent finite differences for 1D viscous isentropic flow in Eulerian coordinates. Discrete Contin. Dyn. Syst. 7, 993–1023 (2014)

Koch, T., Hiller, B., Pfetsch, M., Schewe, L.: Evaluating Gas Network Capacities. MOS-SIAM, (2015)

Kotyczka, P., Lefevre, L.: Discrete-time port-Hamiltonian systems: A definition based on symplectic integration. Systems & Control Letters 133, 104530 (2019)

Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for parabolic systems. Numer. Math. 90, 117–148 (2001)

Kunisch, K., Volkwein, S.: Galerkin proper orthogonal decomposition methods for a general equation in fluid dynamics. SIAM J. Numer. Anal. 40(2), 492–515 (2002)

LeVeque, R.J.: Finite Volume Methods for Hyperbolic Problems. Cambridge University Press, Cambridge (2002)

Liljegren-Sailer, B.: On Port-Hamiltonian Modeling and Structure-Preserving Model Reduction. PhD thesis, Universität Trier, (2020)

Liljegren-Sailer, B.: Code for the paper “On port-Hamiltonian approximation of a nonlinear flow problem on networks”. (2022). https://doi.org/10.5281/zenodo.6372667

Liljegren-Sailer, B., Marheineke, N.: On port-Hamiltonian approximation of a nonlinear flow problem on networks. SIAM J. Sci. Comput., in press, (2022)

Nguyen, T.T., Idier, J., Soussen, C., Djermoune, E.-H.: Non-negative orthogonal greedy algorithms. IEEE Transactions on Signal Processing 67(21), 5643–5658 (2019)

Peng, L., Mohseni, K.: Symplectic model reduction of Hamiltonian systems. SIAM J. Sci. Comput. 38(1), A1–A27 (2016)

Qiu, Y., Grundel, S., Stoll, M., Benner, P.: Efficient numerical methods for gas network modeling and simulation. Netw. Heterog. Media 15(4), 653–679 (2020)

Rockafellar, R., Wets, R.: Variational Analysis. Berlin, New York, Heidelberg (1998)

Rozza, G., Huynh, D.B.P., Manzoni, A.: Reduced basis approximation and a posteriori error estimation for Stokes flows in parametrized geometries: roles of the inf-sup stability constants. Numer. Mat. 125(1), 115–152 (2013)

Schmidt, M., Aßmann, D., Burlacu, R., Humpola, J., Joormann, I., Kanelakis, N., Koch, T., Oucherif, D., Pfetsch, M., Schewe, L., Schwarz, R., Sirvent, M.: GasLib - A Library of Gas Network Instances. Data 2(4), 40 (2017)

Wolf, T., Lohmann, B., Eid, R., Kotyczka, P.: Passivity and structure preserving order reduction of linear port-Hamiltonian systems using Krylov subspaces. Eur. J. Control 16(4), 401–406 (2010)

Wolkowicz, H., Saigal, R., Vandenberghe, L.: editors. Convex Analysis on Symmetric Matrices. Springer, 1 edition, (2000)

Thông tin
Thông tin xuất bản

Springer Science and Business Media LLC

Thông tin tác giả