Reduced-Order Modeling with Time-Dependent Bases for PDEs with Stochastic Boundary Conditions
Abstract.
Low-rank approximation using time-dependent bases (TDBs) has proven effective for reduced-order modeling of stochastic partial differential equations (SPDEs). In these techniques, the random field is decomposed to a set of deterministic TDBs and time-dependent stochastic coefficients. When applied to SPDEs with nonhomogeneous stochastic boundary conditions (BCs), appropriate BC must be specified for each of the TDBs. However, determining BCs for TDB is not trivial because (i) the dimension of the random BCs is different than the rank of the TDB subspace and (ii) TDB in most formulations must preserve orthonormality or orthogonality constraints, and specifying BCs for TDB should not violate these constraints in the space-discretized form. In this work, we present a methodology for determining the boundary conditions for TDBs at no additional computational cost beyond that of solving the same SPDE with homogeneous BCs. Our methodology is informed by the fact the TDB evolution equations are the optimality conditions of a variational principle. We leverage the same variational principle to derive an evolution equation for the value of TDB at the boundaries. The presented methodology preserves the orthonormality or orthogonality constraints of TDBs. We present the formulation for the dynamically biorthonormal decomposition [P. Patil and H. Babaee, J. Comput. Phys., (2020), 109511] as well as the dynamically orthogonal decomposition [T. P. Sapsis and P. F. Lermusiaux, Phys. D, 238 (2009), pp. 2347–2360]. We show that the presented methodology can be applied to stochastic Dirichlet, Neumann, and Robin boundary conditions. We assess the performance of the presented method for linear advection-diffusion equation, Burgers’ equation, and 2D advection-diffusion equation with constant and temperature-dependent conduction coefficient.
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References
1.
P. Patil and H. Babaee, Real-time reduced-order modeling of stochastic partial differential equations via time-dependent subspaces, J. Comput. Phys., 415 (2020), 109511.
2.
T. P. Sapsis and P. F. Lermusiaux, Dynamically orthogonal field equations for continuous stochastic dynamical systems, Phys. D, 238 (2009), pp. 2347–2360.
3.
M. B. Giles, Multilevel Monte Carlo path simulation, Oper. Res., 56 (2008), pp. 607–617.
4.
A. Barth, C. Schwab, and N. Zollinger, Multi-level Monte Carlo finite element method for elliptic PDEs with stochastic coefficients, Numer. Math., 119 (2011), pp. 123–161.
5.
F. Y. Kuo, C. Schwab, and I. H. Sloan, Quasi-Monte Carlo finite element methods for a class of elliptic partial differential equations with random coefficients, SIAM J. Numer. Anal., 50 (2012), pp. 3351–3374.
6.
R. Chen and J. S. Liu, Mixture Kalman filters, J. R. Stat. Soc. Ser. B Stat. Methodol., 62 (2000), pp. 493–508.
7.
R. Van Der Merwe and E. Wan, Gaussian mixture sigma-point particle filters for sequential probabilistic inference in dynamic state-space models, in Proceedings of the 2003 IEEE International Conference on Acoustics, Speech, and Signal Processing, IEEE, 2003, pp. VI–701.
8.
R. G. Ghanem and P. D. Spanos, Stochastic Finite Elements: A Spectral Approach, Courier Corporation, North Chelmsford, MA, 2003.
9.
X. Wan and G. E. Karniadakis, Multi-element generalized polynomial chaos for arbitrary probability measures, SIAM J. Sci. Comput., 28 (2006), pp. 901–928.
10.
D. Xiu and J. S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), pp. 1118–1139.
11.
D. Xiu and G. E. Karniadakis, The Wiener–Askey polynomial chaos for stochastic differential equations, SIAM J. Sci. Comput., 24 (2002), pp. 619–644.
12.
J. Foo and G. E. Karniadakis, Multi-element probabilistic collocation method in high dimensions, J. Comput. Phys., 229 (2010), pp. 1536–1557.
13.
J. Foo, X. Wan, and G. E. Karniadakis, The multi-element probabilistic collocation method (ME-pCM): Error analysis and applications, J. Comput. Phys., 227 (2008), pp. 9572–9595.
14.
I. Babuška, F. Nobile, and R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J. Numer. Anal., 45 (2007), pp. 1005–1034.
15.
B. Ganapathysubramanian and N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, J. Comput. Phys., 225 (2007), pp. 652–685.
16.
X. Yang, M. Choi, G. Lin, and G. E. Karniadakis, Adaptive ANOVA decomposition of stochastic incompressible and compressible flows, J. Comput. Phys., 231 (2012), pp. 1587–1614.
17.
H. Babaee, X. Wan, and S. Acharya, Effect of uncertainty in blowing ratio on film cooling effectiveness, J. Heat Transf., 136 (2014).
18.
D. Zhang, H. Babaee, and G. E. Karniadakis, Stochastic domain decomposition via moment minimization, SIAM J. Sci. Comput., 40 (2018), pp. A2152–A2173.
19.
P. Frauenfelder, C. Schwab, and R. A. Todor, Finite elements for elliptic problems with stochastic coefficients, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 205–228.
20.
H. G. Matthies and A. Keese, Galerkin methods for linear and nonlinear elliptic stochastic partial differential equations, Comput. Methods Appl. Mech. Engrg., 194 (2005), pp. 1295–1331.
21.
C. Schwab and R.-A. Todor, Sparse finite elements for elliptic problems with stochastic loading, Numer. Math., 95 (2003), pp. 707–734.
22.
O. P. L. Maître, O. M. Knio, H. N. Najm, and R. G. Ghanem, A stochastic projection method for fluid flow. I: Basic formulation, J. Comput. Phys., 173 (2001), pp. 481–511.
23.
M. Jardak, C.-H. Su, and G. E. Karniadakis, Spectral polynomial chaos solutions of the stochastic advection equation, J. Sci. Comput., 17 (2002), pp. 319–338.
24.
D. Xiu and G. E. Karniadakis, Modeling uncertainty in flow simulations via generalized polynomial chaos, J. Comput. Phys., 187 (2003), pp. 137–167.
25.
A. J. Chorin, Gaussian fields and random flow, J. Fluid Mech., 63 (1974), pp. 21–32.
26.
O. M. Knio and O. Le Maître, Uncertainty propagation in CFD using polynomial chaos decomposition, Fluid Dyn. Res., 38 (2006), 616.
27.
S. Das, R. Ghanem, and S. Finette, Polynomial chaos representation of spatio-temporal random fields from experimental measurements, J. Comput. Phys., 228 (2009), pp. 8726–8751.
28.
X. Wan and G. E. Karniadakis, Long-term behavior of polynomial chaos in stochastic flow simulations, Comput. Methods Appl. Mech. Engrg., 195 (2006), pp. 5582–5596.
29.
M. Branicki and A. J. Majda, Fundamental limitations of polynomial chaos for uncertainty quantification in systems with intermittent instabilities, Commun. Math. Sci., 11 (2013), pp. 55–103.
30.
C. Pettit and P. Beran, Spectral and multiresolution Wiener expansions of oscillatory stochastic processes, J. Sound Vib., 294 (2006), pp. 752–779.
31.
T. P. Sapsis, Dynamically Orthogonal Field Equations for Stochastic Fluid Flows and Particle Dynamics, Ph.D. thesis, Massachusetts Institute of Technology, 2011.
32.
H. Babaee, M. Choi, T. P. Sapsis, and G. E. Karniadakis, A robust bi-orthogonal/dynamically-orthogonal method using the covariance pseudo-inverse with application to stochastic flow problems, J. Comput. Phys., 344 (2017), pp. 303–319.
33.
M. Cheng, T. Y. Hou, and Z. Zhang, A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations I: Derivation and algorithms, J. Comput. Phys., 242 (2013), pp. 843–868.
34.
M. Cheng, T. Y. Hou, and Z. Zhang, A dynamically bi-orthogonal method for time-dependent stochastic partial differential equations II: Adaptivity and generalizations, J. Comput. Phys., 242 (2013), pp. 753–776.
35.
M. Choi, T. P. Sapsis, and G. E. Karniadakis, On the equivalence of dynamically orthogonal and bi-orthogonal methods: Theory and numerical simulations, J. Comput. Phys., 270 (2014), pp. 1–20.
36.
H. Babaee, An observation-driven time-dependent basis for a reduced description of transient stochastic systems, Proc. A, 475 (2019), 20190506.
37.
H. Babaee and T. Sapsis, A minimization principle for the description of modes associated with finite-time instabilities, Proc. A, 472 (2016), 20150779.
38.
H. Babaee, M. Farazmand, T. Sapsis, and G. Haller, Computing finite-time Lyapunov exponents with optimally time dependent reduction, in APS Division of Fluid Dynamics Meeting Abstracts, American Physical Society, College Park, MD, 2016, pp. L8–003.
39.
H. Babaee, M. Farazmand, G. Haller, and T. P. Sapsis, Reduced-order description of transient instabilities and computation of finite-time Lyapunov exponents, Chaos, 27 (2017), 063103.
40.
A. Blanchard, S. Mowlavi, and T. P. Sapsis, Control of linear instabilities by dynamically consistent order reduction on optimally time-dependent modes, Nonlinear Dyn., 95 (2019), pp. 2745–2764.
41.
T. P. Sapsis, New perspectives for the prediction and statistical quantification of extreme events in high-dimensional dynamical systems, Philos. Trans. Roy. Soc. A, 376 (2018), 20170133.
42.
M. Donello, M. Carpenter, and H. Babaee, Computing sensitivities in evolutionary systems: A real-time reduced order modeling strategy, SIAM J. Sci. Comput., 44 (2022), pp. A128–A149, https://doi.org/10.1137/20M1388565.
43.
A. Nouri, H. Babaee, P. Givi, H. Chelliah, and D. Livescu, Skeletal model reduction with forced optimally time dependent modes, Combust. Flame, 235 (2022), 111684.
44.
D. Ramezanian, A. G. Nouri, and H. Babaee, On-the-fly reduced order modeling of passive and reactive species via time-dependent manifolds, Comput. Methods Appl. Mech. Engrg., 382 (2021), 113882.
45.
M. H. Naderi and H. Babaee, Adaptive sparse interpolation for accelerating nonlinear stochastic reduced-order modeling with time-dependent bases, Comput. Methods Appl. Mech. Engrg., 405 (2023), 115813, https://doi.org/10.1016/j.cma.2022.115813.
46.
M. H. Beck, A. Jäckle, G. A. Worth, and H.-D. Meyer, The multiconfiguration time-dependent Hartree (MCTDH) method: A highly efficient algorithm for propagating wavepackets, Phys. Rep., 324 (2000), pp. 1–105.
47.
C. Bardos, F. Golse, A. D. Gottlieb, and N. J. Mauser, Mean field dynamics of fermions and the time-dependent Hartree-Fock equation, J. Math. Pures Appl., 82 (2003), pp. 665–683.
48.
O. Koch and C. Lubich, Dynamical low-rank approximation, SIAM J. Matrix Anal. Appl., 29 (2007), pp. 434–454.
49.
O. Koch and C. Lubich, Dynamical tensor approximation, SIAM J. Matrix Anal. Appl., 31 (2010), pp. 2360–2375.
50.
E. Musharbash and F. Nobile, Dual dynamically orthogonal approximation of incompressible Navier Stokes equations with random boundary conditions, J. Comput. Phys., 354 (2018), pp. 135–162.
51.
X. Wan and G. E. Karniadakis, An adaptive multi-element generalized polynomial chaos method for stochastic differential equations, J. Comput. Phys., 209 (2005), pp. 617–642.
52.
G. E. Karniadakis and S. J. Sherwin, Spectral/hp Element Methods for Computational Fluid Dynamics, Oxford University Press, Oxford, 2005.
Information & Authors
Information
Published In
SIAM/ASA Journal on Uncertainty Quantification
Pages: 727 - 756
DOI: 10.1137/21M1468097
ISSN (online): 2166-2525
Copyright
© 2023 Society for Industrial and Applied Mathematics and American Statistical Association.
History
Submitted: 29 December 2021
Accepted: 14 March 2023
Published online: 10 July 2023
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Funding Information
Air Force Office of Scientific Research (AFOSR): FA9550-21-1-0247
National Science Foundation (NSF): 2042918
Funding: This work has been funded by Air Force Office of Scientific Research award (Program Manager: Fariba Faharoo), FA9550-21-1-0247, and by the U.S. National Science Foundation under grant 2042918. This research was supported in part by the University of Pittsburgh Center for Research Computing through the resources provided.
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