Windowed least-squares model reduction for dynamical systems. (English) Zbl 07510054
Summary: This work proposes a windowed least-squares (WLS) approach for model reduction of dynamical systems. The proposed approach sequentially minimizes the time-continuous full-order-model residual within a low-dimensional space-time trial subspace over time windows. The approach comprises a generalization of existing model reduction approaches, as particular instances of the methodology recover Galerkin, least-squares Petrov-Galerkin (LSPG), and space-time LSPG projection. In addition, the approach addresses key deficiencies in existing model reduction techniques, e.g., the dependence of LSPG and space-time LSPG projection on the time discretization and the exponential growth in time exhibited by a posteriori error bounds for both Galerkin and LSPG projection. We consider two types of space-time trial subspaces within the proposed approach: one that reduces only the spatial dimension of the full-order model, and one that reduces both the spatial and temporal dimensions of the full-order model. For each type of trial subspace, we consider two different solution techniques: direct (i.e., discretize then optimize) and indirect (i.e., optimize then discretize). Numerical experiments conducted using trial subspaces characterized by spatial dimension reduction demonstrate that the WLS approach can yield more accurate solutions with lower space-time residuals than Galerkin and LSPG projection.
MSC:
| 65-XX | Numerical analysis |
| 91-XX | Game theory, economics, finance, and other social and behavioral sciences |
Keywords:
reduced-order modeling; residual minimization; least-squares; Petrov-Galerkin; optimal control; space-timeReferences:
| [1] | Abgrall, R.; Crisovan, R., Model reduction using L1-norm minimization as an application to nonlinear hyperbolic problems, Int. J. Numer. Methods Fluids, 87, 628-651 (2018) |
| [2] | Balajewicz, M.; Tezaur, I.; Dowell, E., Minimal subspace rotation on the Stiefel manifold for stabilization and enhancement of projection-based reduced order models for the compressible Navier-Stokes equations, J. Comput. Phys., 321, 224-241 (2016) · Zbl 1349.76171 |
| [3] | Barrault, M.; Maday, Y.; Nguyen, N. C.; Patera, A. T., An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations, C. R. Acad. Sci. Paris, 339, 667-672 (2004) · Zbl 1061.65118 |
| [4] | Baumann, M.; Benner, P.; Heiland, J., Space-time Galerkin POD with application in optimal control of semilinear partial differential equations, SIAM J. Sci. Comput., 40, A1611-A1641 (2018) · Zbl 1392.35323 |
| [5] | Beattie, C.; Gugercin, S., Structure-preserving model reduction for nonlinear port-Hamiltonian systems, (2011 50th IEEE Conference on Decision and Control and European Control Conference (CDC-ECC) (2011), IEEE), 6564-6569 |
| [6] | Benner, P.; Gugercin, S.; Willcox, K., A survey of projection-based model reduction methods for parametric dynamical systems, SIAM Rev., 57, 483-531 (2015) · Zbl 1339.37089 |
| [7] | Bergmann, M.; Bruneaua, C.; Iollo, A., Enablers for robust POD models, J. Comput. Phys., 228, 516-538 (2009) · Zbl 1409.76099 |
| [8] | Berkooz, G.; Holmes, P.; Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. Fluid Mech., 25, 539-575 (1993) |
| [9] | Bochev, P.; Gunzburger, M., Finite element methods of least-squares type, SIAM Rev., 40, 789-837 (1998) · Zbl 0914.65108 |
| [10] | Broyden, C. G., A Class of Methods for Solving Nonlinear Simultaneous Equations (1965) · Zbl 0131.13905 |
| [11] | Bui-Thanh, T., Model-constrained optimization methods for reduction of parameterized large-scale systems (2007), Massachusetts Institute of Technology, PhD thesis |
| [12] | Bui-Thanh, T.; Willcox, K.; Ghattas, O., Model reduction for large-scale systems with high-dimensional parametric input space, SIAM J. Sci. Comput., 30, 3270-3288 (2008) · Zbl 1196.37127 |
| [13] | Bui-Thanh, T.; Willcox, K.; Ghattas, O., Parametric reduced-order models for probabilistic analysis of unsteady aerodynamic applications, AIAA J., 46, 2520-2529 (2008) |
| [14] | Caiazzo, A.; Iliescu, T.; John, V.; Schyschlowa, S., A numerical investigation of velocity-pressure reduced order models for incompressible flows, J. Comput. Phys., 259, 598-616 (2014) · Zbl 1349.76050 |
| [15] | Carlberg, K., Model reduction of nonlinear mechanical systems via optimal projection and tensor approximation (2011), Stanford University, PhD thesis |
| [16] | Carlberg, K., Adaptive h-refinement for reduced-order models, Int. J. Numer. Methods Eng., 102, 1192-1210 (2015) · Zbl 1352.65136 |
| [17] | Carlberg, K.; Barone, M.; Antil, H., Galerkin v. least-squares Petrov-Galerkin projection in nonlinear model reduction, J. Comput. Phys., 330, 693-734 (2017) · Zbl 1378.65145 |
| [18] | 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, 155-181 (2011) · Zbl 1235.74351 |
| [19] | Carlberg, K.; Choi, Y.; Sargsyan, S., Conservative model reduction for finite-volume models, J. Comput. Phys., 371, 280-314 (2018) · Zbl 1415.65208 |
| [20] | Carlberg, K.; Farhat, C.; Cortial, J.; Amsallem, D., The GNAT method for nonlinear model reduction: effective implementation and application to computational fluid dynamics and turbulent flows, J. Comput. Phys., 242, 623-647 (2013) · Zbl 1299.76180 |
| [21] | Carlberg, K.; Tuminaro, R.; Boggs, P., Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics, SIAM J. Sci. Comput., 37, B153-B184 (2015) · Zbl 1320.65193 |
| [22] | Ceze, M.; Diosady, L.; Murman, S. M., Development of a High-Order Space-Time Matrix-Free Adjoint Solver (2016) |
| [23] | J. Chan, Entropy stable reduced order modeling of nonlinear conservation laws, arXiv e-print, 2019. |
| [24] | Chaturantabut, S.; Beattie, C.; Gugercin, S., Structure-preserving model reduction for nonlinear port-Hamiltonian systems, SIAM J. Sci. Comput., 38, B837-B865 (2016) · Zbl 1376.37120 |
| [25] | Choi, Y.; Carlberg, K., Space-time least-squares Petrov-Galerkin projection for nonlinear model reduction, SIAM J. Sci. Comput., 41, A26-A58 (2019) · Zbl 1405.65140 |
| [26] | Constantine, P. G.; Wang, Q., Residual minimizing model interpolation for parameterized nonlinear dynamical systems, SIAM J. Sci. Comput. (2012) · Zbl 1259.65110 |
| [27] | Conway, B. A., A survey of methods available for the numerical optimization of continuous dynamic systems, J. Optim. Theory Appl., 152, 271-306 (2012) · Zbl 1237.90247 |
| [28] | Drmac, Z.; Gugercin, S., A new selection operator for the discrete empirical interpolation method—improved a priori error bound and extensions, J. Sci. Comput., 38, A631-A648 (2016) · Zbl 1382.65193 |
| [29] | P. Etter, K. Carlberg, Online adaptive basis refinement and compression for reduced-order models via vector-space sieving, arXiv e-print, 2019. |
| [30] | Everson, R.; Sirovich, L., Karhunen-Loève procedure for gappy data, J. Opt. Soc. Am. A, 12, 1657-1664 (1995) |
| [31] | 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. Methods Eng., 98, 625-662 (2014) · Zbl 1352.74348 |
| [32] | Gugercin, S.; Antoulas, A.; Beattie, C., \( \mathcal{H}_2\) model reduction for large-scale linear dynamical systems, SIAM J. Matrix Anal. Appl., 30, 609-638 (2008) · Zbl 1159.93318 |
| [33] | Gunzburger, M. D.; Bochev, P. B., Least-Squares Finite Element Methods (2009), Springer: Springer New York, NY · Zbl 1168.65067 |
| [34] | Hughes, T. J.; Franca, L. P.; Hulbert, G. M., A new finite element formulation for computational fluid dynamics: Viii. The Galerkin/least-squares method for advective-diffusive equations, Comput. Methods Appl. Mech. Eng., 73, 173-189 (1989) · Zbl 0697.76100 |
| [35] | Hughes, T. J.; Hulbert, G. M., Space-time finite element methods for elastodynamics: formulations and error estimates, Comput. Methods Appl. Mech. Eng., 66, 339-363 (1988) · Zbl 0616.73063 |
| [36] | Hughes, T. J.; Stewart, J. R., A space-time formulation for multiscale phenomena, J. Comput. Appl. Math., 74, 217-229 (1996) · Zbl 0869.65061 |
| [37] | Iliescu, T.; Wang, Z., Variational multiscale proper orthogonal decomposition: Navier-Stokes equations, Numer. Methods Partial Differ. Equ., 30, 641-663 (2014) · Zbl 1452.76048 |
| [38] | Kalashnikova, I.; Arunajatesan, S.; Barone, M. F.; van Bloemen Waanders, B. G.; Fike, J. A., Reduced order modeling for prediction and control of large-scale systems (May 2014), Report SAND2014-4693, Sandia |
| [39] | Kirk, D. E., Optimal Control Theory: An Introduction (2007), Prentice Hall |
| [40] | Knoll, D.; Keyes, D., Jacobian-free Newton—Krylov methods: a survey of approaches and applications, J. Comput. Phys., 193, 357-397 (2004) · Zbl 1036.65045 |
| [41] | Lall, S.; Krysl, P.; Marsden, J. E., Structure-preserving model reduction for mechanical systems, Complexity and Nonlinearity in Physical Systems – a Special Issue to Honor Alan Newell. Complexity and Nonlinearity in Physical Systems – a Special Issue to Honor Alan Newell, Phys. D, Nonlinear Phenom., 184, 304-318 (2003) · Zbl 1041.70011 |
| [42] | K. Lee, K. Carlberg, Model reduction of dynamical systems on nonlinear manifolds using deep convolutional autoencoders, arXiv e-print, 2018. |
| [43] | LeGresley, P.; Alonso, J., Airfoil Design Optimization Using Reduced Order Models Based on Proper Orthogonal Decomposition (2000) |
| [44] | LeGresley, P.; Alonso, J., Dynamic Domain Decomposition and Error Correction for Reduced Order Models (2003) |
| [45] | LeGresley, P.; Alonso, J., Investigation of Non-linear Projection for POD Based Reduced Order Models for Aerodynamics (2001) |
| [46] | Lenhart, S.; Workman, J., Optimal Control Applied to Biological Models (2007), Chapman & Hall/CRC Press · Zbl 1291.92010 |
| [47] | McAsey, M.; Mou, L.; Han, W., Convergence of the forward-backward sweep method in optimal control, Comput. Optim. Appl., 53, 207-226 (2012) · Zbl 1258.49040 |
| [48] | Moore, B., Principal component analysis in linear systems: controllability, observability, and model reduction, IEEE Trans. Autom. Control, 26, 17-32 (1981) · Zbl 0464.93022 |
| [49] | Morrison, D.; Riley, J.; Zancanaro, J., Multiple shooting method for two-point boundary value problems, Commun. ACM, 5, 613-614 (1962) · Zbl 0106.31903 |
| [50] | Mullis, C. T.; Roberts, R. A., Synthesis of minimum roundoff noise fixed point digital filters, IEEE Trans. Circuits Syst., 23, 551-562 (1976) · Zbl 0342.93066 |
| [51] | Ngoc Cuong, N.; Veroy, K.; Patera, A. T., Certified Real-Time Solution of Parametrized Partial Differential Equations, 1529-1564 (2005), Springer Netherlands: Springer Netherlands Dordrecht |
| [52] | Parish, E.; Wentland, C.; Duraisamy, K., The Adjoint Petrov-Galerkin method for non-linear model reduction, Comput. Methods Appl. Mech. Eng., 365, Article 112991 pp. (2020) · Zbl 1442.76069 |
| [53] | Pedregosa, F.; Varoquaux, G.; Gramfort, A.; Michel, V.; Thirion, B.; Grisel, O.; Blondel, M.; Prettenhofer, P.; Weiss, R.; Dubourg, V.; Vanderplas, J.; Passos, A.; Cournapeau, D.; Brucher, M.; Perrot, M.; Duchesnay, E., Scikit-learn: machine learning in Python, J. Mach. Learn. Res., 12, 2825-2830 (2011) · Zbl 1280.68189 |
| [54] | Peherstorfer, B.; Willcox, K., Online adaptive model reduction for nonlinear systems via low-rank updates, J. Sci. Comput., 37, A2123-A2150 (2015) · Zbl 1323.65102 |
| [55] | Pironneau, O.; Liou, J.; Tezduyar, T., Characteristic-Galerkin and Galerkin/least-squares space-time formulations for the advection-diffusion equation with time-dependent domains, Comput. Methods Appl. Mech. Eng., 100, 117-141 (1992) · Zbl 0761.76073 |
| [56] | Prud’homme, C.; Rovas, D. V.; Veroy, K.; Machiels, L.; Maday, Y.; Patera, A. T.; Turinici, G., Reliable real-time solution of parametrized partial differential equations: reduced-basis output bound methods, J. Fluids Eng., 124, 70-80 (2001) |
| [57] | Rac, A., A survey of numerical methods for optimal control, Adv. Astronaut. Sci., 135 (2010) |
| [58] | Rovas, D. V., Reduced-basis output bound methods for parametrized partial differential equations (2003), Massachusetts Institute of Technology, PhD thesis |
| [59] | Rowley, C. W.; Colonius, T.; Murray, R. M., Model reduction for compressible flows using POD and Galerkin projection, Phys. D, Nonlinear Phenom., 189, 115-129 (2004) · Zbl 1098.76602 |
| [60] | Rozza, G.; Huynh, D. B.P.; Patera, A. T., Reduced basis approximation and a posteriori error estimation for affinely parametrized elliptic coercive partial differential equations, Arch. Comput. Methods Eng., 15, 229 (2008) · Zbl 1304.65251 |
| [61] | Rusanov, V., On difference schemes of third order accuracy for nonlinear hyperbolic systems, J. Comput. Phys., 5, 507-516 (1970) · Zbl 0217.21703 |
| [62] | Saad, Y.; Schultz, M. H., Gmres: a generalized minimal residual algorithm for solving nonsymmetric linear systems, SIAM J. Sci. Stat. Comput., 7, 856-869 (1986) · Zbl 0599.65018 |
| [63] | San, O.; Iliescu, T., Proper orthogonal decomposition closure models for fluid flows: Burgers equation, Comput. Methods Appl. Mech. Eng., 5, 217-237 (2014) · Zbl 1463.65245 |
| [64] | San, O.; Iliescu, T., A stabilized proper orthogonal decomposition reduced-order model for large scale quasigeostrophic ocean circulation, Adv. Comput. Math., 41, 1289-1319 (2015) · Zbl 1327.37028 |
| [65] | San, O.; Maulik, R., Neural network closures for nonlinear model order reduction, Adv. Comput. Math., 44, 1717-1750 (2018) · Zbl 1404.37101 |
| [66] | Walther, A.; Griewank, A., Getting started with ADOL-C, (Combinatorial Scientific Computing (2012), Chapman-Hall CRC Computational Science) |
| [67] | Urban, K.; Patera, A. T., A new error bound for reduced basis approximation of parabolic partial differential equations, C. R. Math., 350, 203-207 (2012) · Zbl 1242.35157 |
| [68] | Veroy, K.; Patera, A. T., Certified real-time solution of the parametrized steady incompressible Navier-Stokes equations: rigorous reduced-basis a posteriori error bounds, Int. J. Numer. Methods Fluids, 47, 773-788 (2005) · Zbl 1134.76326 |
| [69] | Veroy, K.; Prud’homme, C.; Rovas, D.; Patera, A., A Posteriori Error Bounds for Reduced-Basis Approximation of Parametrized Noncoercive and Nonlinear Elliptic Partial Differential Equations (2003) |
| [70] | Wang, Q.; Ripamonti, N.; Hesthaven, J. S., Recurrent neural network closure of parametric POD-Galerkin reduced-order models based on the Mori-Zwanzig formalism, J. Comput. Phys. (2019) |
| [71] | Wang, Z., Reduced-Order Modeling of Complex Engineering and Geophysical Flows: Analysis and Computations (2012), Virginia Polytechnic Institute and State University, PhD thesis |
| [72] | Wentland, C. R.; Huang, C.; Duraisamy, K., Closure of Reacting Flow Reduced-Order Models via the Adjoint Petrov-Galerkin Method (2019) |
| [73] | Yano, M.; Patera, A. T.; Urban, K., A space-time hp-interpolation-based certified reduced basis method for Burgers equation, Math. Models Methods Appl. Sci., 24, 1903-1935 (2014) · Zbl 1295.65098 |
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