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Adaptive Dimensionality-Reduction for Time-Stepping in Differential and Partial Differential Equations

Part of: Infinite-dimensional dissipative dynamical systems Numerical methods Partial differential equations, initial value and time-dependent initial-boundary value problems

Published online by Cambridge University Press:  12 September 2017

Xing Fu  and
J. Nathan Kutz
Xing Fu
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
J. Nathan Kutz*
Affiliation:
Department of Applied Mathematics, University of Washington, Seattle, WA 98195, USA
*
*Corresponding author. Email address:kutz@uw.edu (J. N. Kutz)
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Abstract

A numerical time-stepping algorithm for differential or partial differential equations is proposed that adaptively modifies the dimensionality of the underlying modal basis expansion. Specifically, the method takes advantage of any underlying low-dimensional manifolds or subspaces in the system by using dimensionality-reduction techniques, such as the proper orthogonal decomposition, in order to adaptively represent the solution in the optimal basis modes. The method can provide significant computational savings for systems where low-dimensional manifolds are present since the reduction can lower the dimensionality of the underlying high-dimensional system by orders of magnitude. A comparison of the computational efficiency and error for this method are given showing the algorithm to be potentially of great value for high-dimensional dynamical systems simulations, especially where slow-manifold dynamics are known to arise. The method is envisioned to automatically take advantage of any potential computational saving associated with dimensionality-reduction, much as adaptive time-steppers automatically take advantage of large step sizes whenever possible.

Keywords

Time-steppingdimensionality reductionproper orthogonal decomposition

MSC classification

Secondary: 74S25: Spectral and related methods 65M70: Spectral, collocation and related methods 37L65: Special approximation methods (nonlinear Galerkin, etc.)
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 10 , Issue 4 , November 2017 , pp. 872 - 894
Copyright
Copyright © Global-Science Press 2017 

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References

[1] Quarteroni, A. and Rozza, G., Reduced Order Methods for Modeling and Computational Reduction, Springer, 2014.CrossRefGoogle Scholar
[2] Benner, P., Gugercin, S. and Willcox, K., A Survey of Projection-Based Model Reduction Methods for Parametric Dynamical Systems, SIAM Rev., 57(4) (2015), pp. 483–531.Google Scholar
[3] Pearson, K., On lines and planes of closest fit to systems of points in space, Phil. Mag., 2 (1901), pp. 559–572.CrossRefGoogle Scholar
[4] Hotelling, H., Analysis of a complex of statistical variables into principal components, J. Educational Psychol., 23 (1933), pp. 498–520.CrossRefGoogle Scholar
[5] Karhunen, K., Über lineare Methoden in der Wahrscheinlichkeitsrechnung, Ann. Acad. Sci. Fennicae. Ser. A. I. Math.-Phys., 3(37) (1947), pp. 1–79.Google Scholar
[6] Loéve, M., Probability Theory, 4th ed., Graduate Texts in Mathematics 46, Springer-Verlag, 1978.Google Scholar
[7] Sirovich, L., Turbulence and the dynamics of coherent structures. I - Coherent structures, II - Symmetries and transformations, III - Dynamics and scaling, Quart. Applied Math., 45 (1987), pp. 561–590.CrossRefGoogle Scholar
[8] Holmes, P., Lumley, J. L. and Berkooz, G., Turbulence, Coherent Structures, Dynamical Systems and Symmetry, Cambridge, 1998.Google Scholar
[9] Kutz, J. N., Data-Driven Modeling and Scientific Computing, Oxford, 2013.Google Scholar
[10] McCormick, S., Multigrid Methods(Frontiers in Applied Mathematics), SIAM, 1987.Google Scholar
[11] Engquist, B., Runborg, O. and Tsai, Y. H., Numerical Analysis of Multiscale Computations: Proceedings of a Winter Workshop at the Banff International Research Station, Lect. Notes Comp. Sci., Springer, New York, 2009.Google Scholar
[12] Martinsson, P. G., Rokhlin, V. and Tygert, M., A randomized algorithm for the approximation of matrices, App. Comp. Harmonic Ana., 30(1) (2011), pp. 47–68.CrossRefGoogle Scholar
[13] Halko, N., Martinsson, P. and Tropp, J., Finding Structure with Randomness: Probabilistic Algorithms for Constructing Approximate Matrix Decompositions, SIAM Rev., 53(2) (2011), pp. 217–288.Google Scholar
[14] Halko, N., Martinsson, P. G., Shkolnisky, Y. and Tygert, M., An algorithm for the principal component analysis of large data sets, SIAM J. Sci. Comput., 33(5) (2011), pp. 2580–2594.Google Scholar
[15] Xia, J., Chandrasekaran, S., Gu, M. and Li, X. S., Fast algorithms for hierarchically semiseparable matrices, Numer. Linear Algebra Appl., 17(6) (2010), pp. 953–976.CrossRefGoogle Scholar
[16] Cross, M. and Hohenberg, P., Pattern formation outside of equilibrium, Rev. Modern Phys., 65 (1993), pp. 851–1112.CrossRefGoogle Scholar
[17] Schaeffer, H., Osher, S., Caflisch, R. and Hauck, C., Sparse dynamics for partial differential equations, Proceedings of the National Academy of Sciences of the United States of America, 110(17) (2013), pp. 6634–6639.CrossRefGoogle ScholarPubMed
[18] Jokar, S., Mehrmann, V., Pfetsch, M., Yserentant, H., Sparse approximate solution of partial differential equations, J. Applied Numerical Mathematics archive, 60(4) (2010), pp. 452–472.Google Scholar
[19] Williams, M., Schmid, P. and Kutz, J. N., Hybrid reduced-order integration with proper orthogonal decomposition and dynamic mode decomposition, SIAM Multiscale Model. Simul., 11(2) (2013), pp. 522–544.Google Scholar
[20] Barrault, M., Maday, Y., Nguyen, N. C. and Patera, A. T., An empirical interpolation method: application to efficient reduced-basis discretization of partial differential equations, Comptes Rendus Mathematique, 339(9) (2004), pp. 667–672.Google Scholar
[21] Chaturantabut, S. and Sorensen, D. C., Nonlinear model reduction via discrete empirical interpolation, SIAM J. Sci. Comput., 32(5), (2010), pp. 2737–2764.Google Scholar
[22] Amsallem, D., Zahr, M., Farhat, C., Nonlinear Model Order Reduction Based on Local Reduced-Order Bases, Int. J. Numer. Meth. Eng., 92(10) (2012), pp. 891–916.Google Scholar
[23] Everson, R. and Sirovich, L., The Karhunen-Loéve procedure for gappy data, J. Opt. Soc. Am. A, 12(8) (1995), pp. 1657–1664.Google Scholar
[24] Wilcox, K., Unsteady flow sensing and estimation via the happy proper orthogonal decomposition, Comput. Fluids, 35(2) (2006), pp. 208–226.CrossRefGoogle Scholar
[25] Carlberg, K., Adaptive h-refinement for reduced-order models, Int. J. Numer. Meth. Eng., 102(5) (2015), pp. 1192–1210.Google Scholar
[26] Dychelynck, D., Vincent, F. and Cantournet, S., Multidimensional a priori hyper-reduction of mechanical models involving internal variables, Comput. Methods Appl. Mech. Engrg, 225-228(17-20) (2012), pp. 28–43.Google Scholar
[27] Kim, T. and James, D. L., Skipping steps in deformable simulation with online model reduction, Acm Transactions on Graphics, 28(5) (2009), pp. 1–9.Google Scholar
[28] Oxberry, G., Kostova-Vassilevska, T., Arrighi, W. and Chand, K., Limited-memory adaptive snapshot selection for proper orthogonal decomposition, Int. J. Numer. Meth. Eng., 109 (2017), pp. 198–217.Google Scholar
[29] Varshney, A., Pitchaiah, S. and Armaou, A., Feedback Control of Dissipative PDE Systems using Adaptive Model Reduction, AIChE Journal, 55(4) (2010), pp. 906-918.Google Scholar
[30] Trefethen, N. L. and Bau, D., Numerical Linear Algebra, SIAM, 1997.Google Scholar
[31] Carlberg, K., Barone, M. and Harbir, A., Galerkin v. least-squares Petrov-Galerkin projection in nonlinear model reduction, J. Comput. Phys., 26(2) (2016), pp. 24–31.Google Scholar
[32] Homescu, C., Petzold, L., Serban, R., Error Estimation for Reduced-Order Models of Dynamical Systems, SIAM J. Numer. Anal., 43(4) (2006), pp. 1693–1714.Google Scholar
[33] Peherstorfer, B. and Willcox, K., Online adaptive model reduction for nonlinear systems via low-rank updates, SIAM J. Sci. Comput., 37(4) (2015), pp. 2123–2150.Google Scholar
[34] Baker, C., Gallivan, K. and Van Dooren, P., Low-rank incremental methods for computing dominant singular subspaces, Linear Algebra Appl., 436(8) (2012), pp. 2866–2888.Google Scholar
[35] Brand, M., Fast low-rank modifications of the thin singular value decomposition, Linear Algebra Appl., 415(1) (2006), pp. 20–30.Google Scholar
[36] Halko, N., Martinsson, P. G. and Tropp, J., Finding structure with randomness: Probabilistic algorithms for constructing approximate matrix decompositions, SIAM Rev., 53(2) (2011), pp. 217–288.Google Scholar
[37] Akhtar, I., Wang, Z., Borggaard, J. and Iliescu, T., A new closure strategy for proper orthogonal decomposition reduced-order models, J. Comput. Nonlin. Dyn., 7(3) (2012), 034503.Google Scholar
[38] Östh, J., Noack, B., Krajnović, S., Barros, D. and Borée, J., On the need for a nonlinear subscale turbulence term in POD models as exemplified for a high Reynolds number flow over an Ahmed body, J. Fluid Mech., 747(3) (2014), pp. 518–544.CrossRefGoogle Scholar
[39] Kutz, J. N. and Sandstede, B., Theory of passive harmonic mode-locking using wave-guide arrays, Optics Express, 16(2) (2008), pp. 636–650.CrossRefGoogle Scholar
[40] Williams, M., Shlizerman, E. and Kutz, J. N., The multi-pulsing transition in mode-locked lasers: a low-dimensional approach using waveguide arrays, J. Opt. Soc. Am. B, 27(12) (2010), pp. 2471–2481.CrossRefGoogle Scholar
[41] Rowley, C. W. and Marsden, J. E., Reconstruction Equations and the Karhunen-Loeve Expansion, Physica D, 2000.CrossRefGoogle Scholar
[42] Kutz, J. N., Mode-Locked Soliton Lasers, SIAM Review, 2006.Google Scholar
[43] Ding, E., Shlizerman, E. and Kutz, J. N., Generalized master equation for high-energy passive mode-locking: The sinusoidal Ginzburg-Landau equation, IEEE J. Quantum Electron., 47(5) (2011), pp. 705–714.Google Scholar
[44] Ding, E., Lu, K. and Kutz, J. N., Stability analysis of cavity solitons governed by the cubic-quintic Ginzburg-Landau equation, J. Physics B, 44(6) (2011), pp. 157–160.Google Scholar