A fully discrete HDG ensemble Monte Carlo algorithm for a heat equation under uncertainty

JinJun Yong; Changlun Ye; Xianbing Luo; JinJun Yong; Changlun Ye; Xianbing Luo

doi:10.3934/nhm.2025005

Networks and Heterogeneous Media

2025, Volume 20, Issue 1: 65-88. doi: 10.3934/nhm.2025005

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A fully discrete HDG ensemble Monte Carlo algorithm for a heat equation under uncertainty

1.
School of Mathematics and Statistics, Guizhou University, Guiyang, 550025, China
2.
School of Mathematics and Big Data, Guizhou Key Laboratory of Artificial Intelligence and Brain-inspired Computing, Guizhou Education University, Guiyang, 550018, China
3.
School of Mathematical Sciences, Guizhou Normal University, Guiyang, 550001, China

Received: 09 October 2024 Revised: 02 January 2025 Accepted: 13 January 2025 Published: 20 January 2025
65C05, 65C20, 65M60

This paper has introduced a novel fully discrete hybridizable discontinuous Galerkin (HDG) ensemble Monte Carlo method (FEMC-HDG) tailored for solving the heat equation with random diffusion and Robin coefficients. The FEMC-HDG method solves a single linear system with multiple right-hand side vectors per time step. We established stability analysis and error estimates that are optimal in the spatial and first-order accuracy in time for the $ L^{\infty}(0, T, L^2(D)) $-norm error estimate. Numerical experiments were included to confirm the theoretical convergence and showcase the method's efficiency.
- hybridizable discontinuous Galerkin,
- ensemble approach,
- Monte Carlo method,
- diffusion coefficient,
- Robin boundary coefficient,
- optimal
Citation: JinJun Yong, Changlun Ye, Xianbing Luo. A fully discrete HDG ensemble Monte Carlo algorithm for a heat equation under uncertainty[J]. Networks and Heterogeneous Media, 2025, 20(1): 65-88. doi: 10.3934/nhm.2025005

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Abstract

This paper has introduced a novel fully discrete hybridizable discontinuous Galerkin (HDG) ensemble Monte Carlo method (FEMC-HDG) tailored for solving the heat equation with random diffusion and Robin coefficients. The FEMC-HDG method solves a single linear system with multiple right-hand side vectors per time step. We established stability analysis and error estimates that are optimal in the spatial and first-order accuracy in time for the $ L^{\infty}(0, T, L^2(D)) $-norm error estimate. Numerical experiments were included to confirm the theoretical convergence and showcase the method's efficiency.

References

[1]	K. Liu, B. M. Riviere, Discontinuous Galerkin methods for elliptic partial differential equations with random coefficients, Int. J. Comput. Math., 90 (2013), 2477–2490. https://doi.org/10.1080/00207160.2013.784280 doi: 10.1080/00207160.2013.784280
[2]	G. J. Lord, C. E. Powell, T. Shardlow, An Introduction to Computational Stochastic PDEs, New York: Cambridge University Press, 2014. https://doi.org/10.1017/CBO9781139017329
[3]	M. Gunzburger, C. G. Webster, G. Zhang, Stochastic finite element methods for partial differential equations with random input data, Acta Numer., 23 (2014), 521–650. https://doi.org/10.1017/S0962492914000075 doi: 10.1017/S0962492914000075
[4]	I. Babuska, F. Nobile, R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM Rev., 52 (2010), 317–355. https://doi.org/10.1137/100786356 doi: 10.1137/100786356
[5]	B. Ganapathysubramanian, N. Zabaras, Sparse grid collocation schemes for stochastic natural convection problems, J. Comput. Phys., 225 (2007), 652–685. https://doi.org/10.1016/j.jcp.2006.12.014 doi: 10.1016/j.jcp.2006.12.014
[6]	D. Xiu, J. S. Hesthaven, High-order collocation methods for differential equations with random inputs, SIAM J. Sci. Comput., 27 (2005), 1118–1139. https://doi.org/10.1137/040615201 doi: 10.1137/040615201
[7]	X. Zhu, E. M. Linebarger, D. Xiu, Multi-fidelity stochastic collocation method for computation of statistical moments, J. Comput. Phys., 341 (2017), 386–396. https://doi.org/10.1016/j.jcp.2017.04.022 doi: 10.1016/j.jcp.2017.04.022
[8]	L. Mathelin, M. Y. Hussaini, T. A. Zang, Stochastic approaches to uncertainty quantification in CFD simulations, Numer. Algorithms., 38 (2005), 209–236. https://doi.org/10.1007/BF02810624 doi: 10.1007/BF02810624
[9]	G. Fishman, Monte Carlo: Concepts, Algorithms, and Applications, New York: Springer, 1996. https://doi.org/10.1007/978-1-4757-2553-7
[10]	M. B. Giles, Multilevel monte carlo methods, Acta Numer., 24 (2015), 259–328. https://doi.org/10.1017/S096249291500001X doi: 10.1017/S096249291500001X
[11]	J. C. Helton, F. J. Davis, Latin hypercube sampling and the propagation of uncertainty in analyses of complex systems, Reliab. Eng. Syst. Safe., 81 (2003), 23–69. https://doi.org/10.1016/S0951-8320(03)00058-9 doi: 10.1016/S0951-8320(03)00058-9
[12]	Y. Luo, Z. Wang, A multilevel Monte Carlo ensemble scheme for solving random parabolic PDEs, SIAM J. Sci. Comput., 41 (2019), A622–A642. https://doi.org/10.1137/18M1174635 doi: 10.1137/18M1174635
[13]	N. Jiang, W. Layton, An algorithm for fast calculation of flow ensembles, Int. J. Uncertain. Quan., 4 (2014), 273–301. https://doi.org/10.1615/Int.J.UncertaintyQuantification.2014007691 doi: 10.1615/Int.J.UncertaintyQuantification.2014007691
[14]	N. Jiang, A higher order ensemble simulation algorithm for fluid flows, J. Sci. Comput., 64 (2015), 264–288. https://doi.org/10.1007/s10915-014-9932-z doi: 10.1007/s10915-014-9932-z
[15]	M. Li, X. Luo, An MLMCE-HDG method for the convection diffusion equation with random diffusivity, Comput. Math. with Appl., 127 (2022), 127–143. https://doi.org/10.1016/j.camwa.2022.10.002 doi: 10.1016/j.camwa.2022.10.002
[16]	Y. Luo, Z. Wang, An ensemble algorithm for numerical solutions to deterministic and random parabolic PDEs, SIAM J. Numer. Anal., 56 (2018), 859–876. https://doi.org/10.1137/17M1131489 doi: 10.1137/17M1131489
[17]	T. Yao, C. Ye, X. Luo, S. Xiang, An ensemble scheme for the numerical solution of a random transient heat equation with uncertain inputs, Numer. Algorithms, 94 (2023), 643–668. https://doi.org/10.1007/s11075-023-01514-z doi: 10.1007/s11075-023-01514-z
[18]	C. Ye, T. Yao, H. Bi, X. Luo, A variational Crank-Nicolson ensemble Monte Carlo algorithm for a heat equation under uncertainty, J. Comput. Appl. Math., 451 (2024), 116068. https://doi.org/10.1016/j.cam.2024.116068 doi: 10.1016/j.cam.2024.116068
[19]	J. Yong, C. Ye, X. Luo, S. Sun, Improved error estimates of ensemble Monte Carlo methods for random transient heat equations with uncertain inputs, Comp. Appl. Math, 44 (2025), 58. https://doi.org/10.1007/s40314-024-03022-9 doi: 10.1007/s40314-024-03022-9
[20]	S. Du, F. J. Sayas, An Invitation to the Theory of the Hybridizable Discontinuous Galerkin Method: Projections, Estimates, Tools, Cham: Springer, 2019. https://doi.org/10.1007/978-3-030-27230-2
[21]	B. Cockburn, F. J. Sayas, Divergence-conforming HDG methods for Stokes flows, Math. Comp., 83 (2014), 1571–1598. https://doi.org/10.1090/S0025-5718-2014-02802-0 doi: 10.1090/S0025-5718-2014-02802-0
[22]	S. Rhebergen, B. Cockburn, J. J. W. Van Der Vegt, A space-time discontinuous Galerkin method for the incompressible Navier-Stokes equations, J. Comput. Phys., 233 (2013), 339–358. https://doi.org/10.1016/j.jcp.2012.08.052 doi: 10.1016/j.jcp.2012.08.052
[23]	M. Stanglmeier, N. C. Nguyen, J. Peraire, B. Cockburn, An explicit hybridizable discontinuous Galerkin method for the acoustic wave equation, Comput. Methods Appl. Mech. Eng., 300 (2016), 748–769. https://doi.org/10.1016/j.cma.2015.12.003 doi: 10.1016/j.cma.2015.12.003
[24]	S. C. Brenner, L. R. Scott, The Mathematical Theory of Finite Element Methods, New York: Springer, 2008. https://doi.org/10.1007/978-0-387-75934-0
[25]	Y. H. Hao, Q. M. Huang, C. Wang, A third order BDF energy stable linear scheme for the no-slope-selection thin film model, Commun. Comput. Phys., 29 (2021), 905–929. https://doi.org/10.4208/cicp.OA-2020-0074 doi: 10.4208/cicp.OA-2020-0074
[26]	Z. Y. Li, H. L. Liao, Stability of variable-step BDF2 and BDF3 methods, SIAM J. Numer. Anal., 60 (2022), 2253–2272. https://doi.org/10.1137/21M1462398 doi: 10.1137/21M1462398
[27]	K. L. Zheng, C. Wang, S. M. Wise, Y. Wu, A third order accurate in time, BDF-type energy stable scheme for the Cahn-Hilliard equation, Numer. Math. Theor. Meth. Appl., 15 (2022), 279–303. https://doi.org/10.4208/nmtma.OA-2021-0165 doi: 10.4208/nmtma.OA-2021-0165
[28]	B. Cockburn, J. Gopalakrishnan, F. J. Sayas, A projection-based error analysis of HDG methods, Math. Comp., 79 (2010), 1351–1367. https://doi.org/10.1090/S0025-5718-10-02334-3 doi: 10.1090/S0025-5718-10-02334-3
[29]	J. Schöberl, C++ 11 Implementation of Finite Elements in NGSolve, Institute for Analysis and Scientific Computing, Vienna University of Technology, 2014.
[30]	J. Martinez-Frutos, M. Kessler, A. Mnch, F. Periago, Robust optimal Robin boundary control for the transient heat equation with random input data, Int. J. Numer. Methods Eng., 108 (2016), 116–135. https://doi.org/10.1002/nme.5210 doi: 10.1002/nme.5210
[31]	J. Martinez-Frutos, F. P. Esparza, Optimal Control of PDEs Under Uncertainty: An introduction with Application to Optimal Shape Design of Structures, Cham: Springer, 2018. https://doi.org/10.1007/978-3-319-98210-6

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