Applications of Mathematics, Vol. 67, No. 6, pp. 727-749, 2022
On surrogate learning for linear stability assessment of Navier-Stokes equations with stochastic viscosity
Bedřich Sousedík, Howard C. Elman, Kookjin Lee, Randy Price
Received February 28, 2021. Published online March 1, 2022.
Abstract: We study linear stability of solutions to the Navier-Stokes equations with stochastic viscosity. Specifically, we assume that the viscosity is given in the form of a stochastic expansion. Stability analysis requires a solution of the steady-state Navier-Stokes equation and then leads to a generalized eigenvalue problem, from which we wish to characterize the real part of the rightmost eigenvalue. While this can be achieved by Monte Carlo simulation, due to its computational cost we study three surrogates based on generalized polynomial chaos, Gaussian process regression and a shallow neural network. The results of linear stability analysis assessment obtained by the surrogates are compared to that of Monte Carlo simulation using a set of numerical experiments.
Keywords: linear stability; Navier-Stokes equations; generalized polynomial chaos; stochastic collocation; stochastic Galerkin method; Gaussian process regression; shallow neural network
References: [1] E. Åkervik, L. Brandt, D. S. Henningson, J. Hœpffner, O. Marxen, P. Schlatter: Steady solutions of the Navier-Stokes equations by selective frequency damping. Phys. Fluids 18 (2006), Article ID 068102. DOI 10.1063/1.2211705
[2] R. Andreev, C. Schwab: Sparse tensor approximation of parametric eigenvalue problems. Numerical Analysis of Multiscale Problems. Lecture Notes in Computational Science and Engineering 83. Springer, Berlin (2012), 203-241. DOI 10.1007/978-3-642-22061-6_7 | MR 3050915 | Zbl 1248.65116
[3] I. Babuška, F. Nobile, R. Tempone: A stochastic collocation method for elliptic partial differential equations with random input data. SIAM Rev. 52 (2010), 317-355. DOI 10.1137/100786356 | MR 2646806 | Zbl 1226.65004
[4] P. Benner, A. Onwunta, M. Stoll: A low-rank inexact Newton-Krylov method for stochastic eigenvalue problems. Comput. Methods Appl. Math. 19 (2019), 5-22. DOI 10.1515/cmam-2018-0030 | MR 3898202 | Zbl 1420.65043
[5] K. A. Cliffe, T. J. Garratt, A. Spence: Eigenvalues of block matrices arising from problems in fluid mechanics. SIAM J. Matrix Anal. Appl. 15 (1994), 1310-1318. DOI 10.1137/S0895479892233230 | MR 1293919 | Zbl 0807.65030
[6] K. A. Cliffe, A. Spence, S. J. Tavener: The numerical analysis of bifurcation problems with application to fluid mechanics. Acta Numerica 9 (2000), 39-131. DOI 10.1017/S0962492900000398 | MR 1883627 | Zbl 1005.65138
[7] F. Dan Foresee, M. T. Hagan: Gauss-Newton approximation to Bayesian learning. Proceedings of International Conference on Neural Networks (ICNN'97). Vol. 3. IEEE, Piscataway (1997), 1930-1935. DOI 10.1109/ICNN.1997.614194
[8] H. C. Elman, K. Meerbergen, A. Spence, M. Wu: Lyapunov inverse iteration for identifying Hopf bifurcations in models of incompressible flow. SIAM J. Sci. Comput. 34 (2012), A1584-A1606. DOI 10.1137/110827600 | MR 2970265 | Zbl 1247.65047
[9] H. C. Elman, A. Ramage, D. J. Silvester: IFISS: A computational laboratory for investigating incompressible flow problems. SIAM Rev. 56 (2014), 261-273. DOI 10.1137/120891393 | MR 3201182 | Zbl 1426.76645
[10] H. C. Elman, D. J. Silvester: Collocation methods for exploring perturbations in linear stability analysis. SIAM J. Sci. Comput. 40 (2018), A2667-A2693. DOI 10.1137/17M1117689 | MR 3846294 | Zbl 1398.65176
[11] H. C. Elman, D. J. Silvester, A. J. Wathen: Finite Elements and Fast Iterative Solvers: With Applications in Incompressible Fluid Dynamics. Numerical Mathematics and Scientific Computation. Oxford University Press, Oxford (2014). DOI 10.1093/acprof:oso/9780199678792.001.0001 | MR 3235759 | Zbl 1304.76002
[12] C. B. Erickson, B. E. Ankenman, S. M. Sanchez: Comparison of Gaussian process modeling software. Eur. J. Oper. Res. 266 (2018), 179-192. DOI 10.1016/j.ejor.2017.10.002 | MR 3736988 | Zbl 1403.62006
[13] T. Gerstner, M. Griebel: Numerical integration using sparse grids. Numer. Algorithms 18 (1998), 209-232. DOI 10.1023/A:1019129717644 | MR 1669959 | Zbl 0921.65022
[14] R. G. Ghanem: The nonlinear Gaussian spectrum of log-normal stochastic processes and variables. J. Appl. Mech. 66 (1999), 964-973. DOI 10.1115/1.2791806
[15] R. G. Ghanem, P. D. Spanos: Stochastic Finite Elements: A Spectral Approach. Springer, New York (1991). DOI 10.1007/978-1-4612-3094-6 | MR 1083354 | Zbl 0722.73080
[16] V. Girault, P.-A. Raviart: Finite Element Methods for Navier-Stokes Equations: Theory and Algorithms. Springer Series in Computational Mathematics 5. Springer, Berlin (1986). DOI 10.1007/978-3-642-61623-5 | MR 0851383 | Zbl 0585.65077
[17] W. J. F. Govaerts: Numerical Methods for Bifurcations of Dynamical Equilibria. SIAM, Philadelphia (2000). DOI 10.1137/1.9780898719543 | MR 1736704 | Zbl 0935.37054
[18] K. Lee, H. C. Elman, B. Sousedík: A low-rank solver for the Navier-Stokes equations with uncertain viscosity. SIAM/ASA J. Uncertain. Quantif. 7 (2019), 1275-1300. DOI 10.1137/17M1151912 | MR 4025775 | Zbl 1442.35568
[19] K. Lee, B. Sousedík: Inexact methods for symmetric stochastic eigenvalue problems. SIAM/ASA J. Uncertain. Quantif. 6 (2018), 1744-1776. DOI 10.1137/18M1176026 | MR 3892437 | Zbl 1405.65052
[20] O. P. Le Maître, O. M. Knio: Spectral Methods for Uncertainty Quantification: With Applications to Computational Fluid Dynamics. Scientific Computation. Springer, Dordrecht (2010). DOI 10.1007/978-90-481-3520-2 | MR 2605529 | Zbl 1193.76003
[21] J. C. Loiseau, M. A. Bucci, S. Cherubini, J.-C. Robinet: Time-stepping and Krylov methods for large-scale instability problems. Computational Modelling of Bifurcations and Instabilities in Fluid Dynamics. Computational Methods in Applied Sciences 50. Springer, Cham (2019), 33-73. DOI 10.1007/978-3-319-91494-7_2 | MR 3822594
[22] D. J. C. MacKay: Bayesian interpolation. Neural Comput. 4 (1992), 415-447. DOI 10.1162/neco.1992.4.3.415
[23] H. G. Matthies, A. Keese: Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations. Comput. Methods Appl. Mech. Eng. 194 (2005), 1295-1331. DOI 10.1016/j.cma.2004.05.027 | MR 2121216 | Zbl 1088.65002
[24] E. Novak, K. Ritter: High dimensional integration of smooth functions over cubes. Numer. Math. 75 (1996), 79-97. DOI 10.1007/s002110050231 | MR 1417864 | Zbl 0883.65016
[25] A. O'Hagan: Polynomial chaos: A tutorial and critique from a statistician's perspective. Available at http://tonyohagan.co.uk/academic/pdf/Polynomial-chaos.pdf (2013).
[26] N. E. Owen, P. Challenor, P. P. Menon, S. Bennani: Comparison of surrogate-based uncertainty quantification methods for computationally expensive simulators. SIAM/ASA J. Uncertain. Quantif. 5 (2017), 403-435. DOI 10.1137/15M1046812 | MR 3639591 | Zbl 06736509
[27] G. C. Y. Peng, M. Alber, A. Buganza Tepole et al.: Multiscale modeling meets machine learning: What can we learn? Arch. Comput. Methods Eng. 28 (2021), 1017-1037. DOI 10.1007/s11831-020-09405-5 | MR 4246233
[28] C. E. Powell, D. J. Silvester: Preconditioning steady-state Navier-Stokes equations with random data. SIAM J. Sci. Comput. 34 (2012), A2482-A2506. DOI 10.1137/120870578 | MR 3023713 | Zbl 1256.35216
[29] C. E. Rasmussen, C. K. I. Williams: Gaussian Processes for Machine Learning. Adaptive Computation and Machine Learning. MIT Press, Cambridge (2006). DOI 10.7551/mitpress/3206.001.0001 | MR 2514435 | Zbl 1177.68165
[30] P. J. Schmid, D. S. Henningson: Stability and Transition in Shear Flows. Applied Mathematical Sciences 142. Springer, New York (2001). DOI 10.1007/978-1-4613-0185-1 | MR 1801992 | Zbl 0966.76003
[31] J. Sirignano, K. Spiliopoulos: DGM: A deep learning algorithm for solving partial differential equations. J. Comput. Phys. 375 (2018), 1339-1364. DOI 10.1016/j.jcp.2018.08.029 | MR 3874585 | Zbl 1416.65394
[32] B. Sousedík, H. C. Elman: Stochastic Galerkin methods for the steady-state Navier-Stokes equations. J. Comput. Phys. 316 (2016), 435-452. DOI 10.1016/j.jcp.2016.04.013 | MR 3494362 | Zbl 1349.76268
[33] B. Sousedík, K. Lee: Stochastic Galerkin methods for linear stability analysis of systems with parametric uncertainty. Avaliable at https://arxiv.org/abs/2202.12485 (2022), 27 pages.
[34] M. L. Stein: Interpolation of Spatial Data: Some Theory for Kriging. Springer Series in Statistics. Springer, New York (1999). DOI 10.1007/978-1-4612-1494-6 | MR 1697409 | Zbl 0924.62100
[35] T. P. Vogl, J. K. Mangis, A. K. Rigler, W. T. Zink, D. L. Alkon: Accelerating the convergence of the back-propagation method. Biol. Cybern. 59 (1988), 257-263. DOI 10.1007/BF00332914
[36] D. Xiu: Numerical Methods for Stochastic Computations: A Spectral Method Approach. Princeton University Press, Princeton (2010). DOI 10.1515/9781400835348 | MR 2723020 | Zbl 1210.65002
[37] D. Xiu, G. E. Karniadakis: The Wiener-Askey polynomial chaos for stochastic differential equations. SIAM J. Sci. Comput. 24 (2002), 619-644. DOI 10.1137/S1064827501387826 | MR 1951058 | Zbl 1014.65004
[38] L. Yan, X. Duan, B. Liu, J. Xu: Gaussian processes and polynomial chaos expansion for regression problem: Linkage via the RKHS and comparison via the KL divergence. Entropy 20 (2018), Article ID 191, 22 pages. DOI 10.3390/e20030191
Affiliations: Bedřich Sousedík, Department of Mathematics and Statistics, University of Maryland, Baltimore County, 1000 Hilltop Circle, Baltimore, MD 21250, USA, e-mail: sousedik@umbc.edu; Howard C. Elman, Department of Computer Science and Institute for Advanced Computer Studies, Iribe Center, 8125 Paint Branch Dr, University of Maryland, College Park, MD 20742, USA, e-mail: helman@umd.edu; Kookjin Lee, School of Computing and Augmented Intelligence, Arizona State University, 699 S Mill Ave, Tempe, AZ 85281, USA, e-mail: kookjin.lee@asu.edu; Randy Price, Center for Mathematics and Artificial Intelligence and Center for Computational Fluid Dynamics, George Mason University, 4400 University Drive, MS: 3F2, Exploratory Hall, room 4102, Fairfax, VA 22030, USA, e-mail: rprice25@gmu.edu
