zbMATH — the first resource for mathematics

Decreasing the temporal complexity for nonlinear, implicit reduced-order models by forecasting. (English) Zbl 1425.65073
Summary: Implicit numerical integration of nonlinear ODEs requires solving a system of nonlinear algebraic equations at each time step. Each of these systems is often solved by a Newton-like method, which incurs a sequence of linear-system solves. Most model-reduction techniques for nonlinear ODEs exploit knowledge of a system’s spatial behavior to reduce the computational complexity of each linear-system solve. However, the number of linear-system solves for the reduced-order simulation often remains roughly the same as that for the full-order simulation.
We propose exploiting knowledge of the model’s temporal behavior to (1) forecast the unknown variable of the reduced-order system of nonlinear equations at future time steps, and (2) use this forecast as an initial guess for the Newton-like solver during the reduced-order-model simulation. To compute the forecast, we propose using the Gappy POD technique. The goal is to generate an accurate initial guess so that the Newton solver requires many fewer iterations to converge, thereby decreasing the number of linear-system solves in the reduced-order-model simulation.

65L05 Numerical methods for initial value problems
Full Text: DOI arXiv
[1] 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. Math. Acad. Sci. Paris, 339, 667-672 (2004) · Zbl 1061.65118
[2] Chaturantabut, S.; Sorensen, D. C., Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32, 2737-2764 (2010) · Zbl 1217.65169
[3] Everson, R.; Sirovich, L., Karhunen-Loève procedure for gappy data, J. Opt. Soc. Am. A, 12, 1657-1664 (1995)
[4] 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
[5] Diggle, P., Time Series: A Biostatistical Introduction (1990), Clarendon Press · Zbl 0727.62083
[6] Gourieroux, C., ARCH Models and Financial Applications (1997), Springer-Verlag · Zbl 0880.62107
[7] Engle, R. F., Autoregressive conditional heteroscedasticity with estimates of the variance of U.K. inflation, Econometrica, 50, 987-1008 (1982) · Zbl 0491.62099
[8] Graybill, F. A., An Introduction to Linear Statistical Models (1961), McGraw-Hill: McGraw-Hill New York · Zbl 0121.35605
[9] Hollander, M.; Wolfe, D. A., Nonparametric Statistical Methods (1973), Wiley: Wiley New York · Zbl 0277.62030
[10] Percival, D. B.; Walden, A. T., Spectral Analysis for Physical Applications (1993), Cambridge University Press · Zbl 0796.62077
[11] Holt, C. C., Forecasting seasonals and trends by exponentially weighted moving averages, Int. J. Forecast., 20, 5-10 (2004)
[12] Winters, P. R., Forecasting sales by exponentially weighted moving averages, Manage. Sci., 6, 324-342 (1960) · Zbl 0995.90562
[13] Brown, P.; Byrne, G.; Hindmarsh, A., VODE: A variable-coefficient ODE solver, SIAM J. Scient. Stat. Comput., 10, 1038-1051 (1989) · Zbl 0677.65075
[14] Nievergelt, J., Parallel methods for integrating ordinary differential equations, Comm. ACM, 7, 731-733 (1964) · Zbl 0134.32804
[15] Gander, M. J., A waveform relaxation algorithm with overlapping splitting for reaction diffusion equations, Numer. Linear Algebra Appl., 6, 125-145 (1999) · Zbl 0983.65107
[16] Horton, G.; Vandewalle, S., A space-time multigrid methods for parabolic partial differential equaions, SIAM J. Sci. Comput., 16, 848-864 (1995) · Zbl 0828.65105
[17] Lions, J.; Maday, Y.; Turinici, G., A “parareal” in time discretization of PDEs, C. R. Acad. Sci. Ser. I Math., 332, 661-668 (2001) · Zbl 0984.65085
[18] Cortial, J., Time-parallel methods for accelerating the solution of structural dynamics problems (2011), Stanford University, (Ph.D. thesis)
[19] Farhat, C.; Chandesris, M., Time-decomposed parallel time-integrators: theory and feasibility studies for fluid, structure, and fluid-structure applications, Int. J. Numer. Methods Engrg., 58, 1397-1434 (2003) · Zbl 1032.74701
[20] Harden, C., Real time computing with the parareal algorithm (2008), Florida State University, (Ph.D. thesis)
[21] Bos, R.; Bombois, X.; Van den Hof, P., Accelerating large-scale non-linear models for monitoring and control using spatial and temporal correlations, Proc. Amer. Control Conf., 4, 3705-3710 (2004)
[22] Ryckelynck, D., A priori hyperreduction method: an adaptive approach, J. Comput. Phys., 202, 346-366 (2005) · Zbl 1288.65178
[23] Kim, T.; James, D., Skipping steps in deformable simulation with online model reduction, ACM Trans. Graphics, 28, 1-9 (2009)
[24] Hairer, E.; Wanner, G., Solving Ordinary Differential Equations II: Stiff and Differential-algebraic Problems (2002), Springer Verlag
[25] Golub, G. H.; Van Loan, C. F., Matrix Computations (1996), Johns Hopkins University Press · Zbl 0865.65009
[26] Krysl, P.; Lall, S.; Marsden, J. E., Dimensional model reduction in non-linear finite elements dynamics of solids and structures, Int. J. Numer. Meth. Engng, 51, 479-504 (2001) · Zbl 1013.74071
[27] Carlberg, K.; Bou-Mosleh, C.; Farhat, C., Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations, Internat. J. Numer. Methods Engrg., 86, 155-181 (2011) · Zbl 1235.74351
[28] 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
[29] Bui-Thanh, T.; Willcox, K.; Ghattas, O., Parametric reduced-order models for probabilistic analysis of unsteady aerodynamic applications, AIAA J., 46, 2520-2529 (2008)
[30] LeGresley, P. A., Application of Proper Orthogonal Decomposition (POD) to design decomposition methods (2006), Stanford University, (Ph.D. thesis)
[31] Astrid, P.; Weiland, S.; Willcox, K.; Backx, T., Missing point estimation in models described by proper orthogonal decomposition, IEEE Trans. Automat. Control, 53, 2237-2251 (2008) · Zbl 1367.93110
[32] Galbally, D.; Fidkowski, K.; Willcox, K.; Ghattas, O., Non-linear model reduction for uncertainty quantification in large-scale inverse problems, International Journal for Numerical Methods in Engineering, 81, 1581-1608 (2009) · Zbl 1183.76837
[33] Drohmann, M.; Haasdonk, B.; Ohlberger, M., Reduced basis approximation for nonlinear parametrized evolution equations based on empirical operator interpolation, SIAM Journal on Scientific Computing, 34, A937-A969 (2012) · Zbl 1259.65133
[35] Bui-Thanh, T.; Damodaran, M.; Willcox, K., Aerodynamic data reconstruction and inverse design using proper orthogonal decomposition, AIAA J., 42, 1505-1516 (2004)
[36] Venturi, D.; Karniadakis, G. E., Gappy data and reconstruction procedures for flow past a cylinder, J. Fluid Mech., 519, 315-336 (2004) · Zbl 1065.76159
[37] Willcox, K., Unsteady flow sensing and estimation via the gappy proper orthogonal decomposition, Comput. & Fluids, 35, 208-226 (2006) · Zbl 1160.76394
[39] Carlberg, K.; Tuminaro, R.; Boggs, P., Preserving Lagrangian structure in nonlinear model reduction with application to structural dynamics, SIAM J. Sci. Comput., in press (2015) · Zbl 1320.65193
[40] Chowdhury, I.; Dasgupta, S., Computation of Rayleigh damping coefficients for large systems, Electron. J. Geotech. Eng., 8 (2003)
[42] McKay, M. D.; Beckman, R. J.; Conover, W. J., Comparison of three methods for selecting values of input variables in the analysis of output from a computer code, Technometrics, 21, 239-245 (1979) · Zbl 0415.62011
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.