A variational MAX ensemble numerical algorism for a transient heat model with random inputs

Tingfu Yao; Changlun Ye; Xianbing Luo; Shuwen Xiang; Tingfu Yao; Changlun Ye; Xianbing Luo; Shuwen Xiang

doi:10.3934/nhm.2024045

Networks and Heterogeneous Media

2024, Volume 19, Issue 3: 1013-1037. doi: 10.3934/nhm.2024045

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A variational MAX ensemble numerical algorism for a transient heat model with random inputs

1.
School of Mathematics and Statistics, Guizhou University, Guiyang 550025, China
2.
College of Science, Guiyang University, Guiyang 550005, China
3.
School of Mathematical Sciences, Guizhou Normal University, Guiyang 550001, China

Received: 22 August 2024 Revised: 18 September 2024 Accepted: 23 September 2024 Published: 27 September 2024

A variational MAX ensemble-based time-stepping numerical method is proposed to simulate a transient heat equation with uncertain Robin boundary and diffusion coefficients. Instead of employing ensemble means for Robin coefficients as well as diffusion coefficients, the maximums of these coefficients are utilized at per time step. This is a new variational ensemble Monte Carlo (MC) numerical method, which we call the variational MAX ensemble Monte Carlo (VMEMC) method. In contrast with related methodologies, the novelty of this algorithm is that it is unconditionally stable. And also, the error estimates are proved. Numerical tests illustrate the theoretical properties for the VMEMC method.
- transient heat equation,
- Robin coefficient,
- diffusion coefficient,
- uncertain inputs,
- variational MAX ensemble,
- MC sampling
Citation: Tingfu Yao, Changlun Ye, Xianbing Luo, Shuwen Xiang. A variational MAX ensemble numerical algorism for a transient heat model with random inputs[J]. Networks and Heterogeneous Media, 2024, 19(3): 1013-1037. doi: 10.3934/nhm.2024045

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Abstract

A variational MAX ensemble-based time-stepping numerical method is proposed to simulate a transient heat equation with uncertain Robin boundary and diffusion coefficients. Instead of employing ensemble means for Robin coefficients as well as diffusion coefficients, the maximums of these coefficients are utilized at per time step. This is a new variational ensemble Monte Carlo (MC) numerical method, which we call the variational MAX ensemble Monte Carlo (VMEMC) method. In contrast with related methodologies, the novelty of this algorithm is that it is unconditionally stable. And also, the error estimates are proved. Numerical tests illustrate the theoretical properties for the VMEMC method.

References

[1]	I. Babu$\check{s}$ka, F. Nobile, R. Tempone, A stochastic collocation method for elliptic partial differential equations with random input data, SIAM J Numer Anal, 52 (2010), 317–355. https://doi.org/10.1137/050645142 doi: 10.1137/050645142
[2]	S. C. Brenner, L. R. Scott, The mathematical theory of finite element methods, 3rd edn., Springer, New York, 2008. https://doi.org/10.1007/978-0-387-75934-0
[3]	R. Chiba, Stochastic analysis of heat conduction and thermal stresses in solids: a review, Chapter 9 in Heat Transfer Phenomena and Applications, IntechOpen, London, 2012. https://doi.org/10.5772/50994
[4]	R. Chiba, Stochastic heat conduction analysis of a functionally graded annular disc with spatially random heat transfer coefficients, Appl. Math. Model., 33 (2009), 507–523. https://doi.org/10.1016/j.apm.2007.11.014 doi: 10.1016/j.apm.2007.11.014
[5]	A. Ern, J. L. Guermond, Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004. https://doi.org/10.1007/978-1-4757-4355-5
[6]	J. A. Fiordilino, M. Winger, Unconditionally energy stable and first-order accurate numerical schemes for the heat equation with uncertain temperature-dependent conductivity, Int. J. Num. Ana. Model., 20 (2023), 805–831. https://doi.org/10.4208/ijnam2023-1035 doi: 10.4208/ijnam2023-1035
[7]	X. Feng, Y. Luo, L. Vo, Z. Wang, An efficient iterative method for solving parameter-dependent and random convection-diffusion problems, J. Sci. Comput., 90 (2022), 72. https://doi.org/10.1007/s10915-021-01737-z doi: 10.1007/s10915-021-01737-z
[8]	G. Fishman., Monte Carlo: Concepts, Algorithms, and Applications, Springer, New York, 1996. https://doi.org/10.1007/978-1-4757-2553-7
[9]	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
[10]	M. B. Giles, Multilevel Monte Carlo methods, Acta Numer., 24 (2015), 259–328. https://doi.org/10.1017/S09624929 doi: 10.1017/S09624929
[11]	M. D. 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
[12]	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
[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]	B. Jin, J. Zou, Numerical estimation of the Robin coefficient in a stationary diffusion equation, IMA J. Numer. Anal., 30 (2010), 677–701. https://doi.org/10.1093/imanum/drn066 doi: 10.1093/imanum/drn066
[15]	B. Jin, J. Zou, Numerical identification of a Robin coefficient in parabolic problems, Math. Comp., 81 (2012), 1369–1398. https://doi.org/10.1090/S0025-5718-2012-02559-2 doi: 10.1090/S0025-5718-2012-02559-2
[16]	L. Ju, W. Leng, Z. Wang, S. Yuan, Numerical investigation of ensemble methods with block iterative solvers for evolution problems, Discrete Contin. Dyn. Syst. Ser. B., 25 (2020), 4905–4923. https://doi.org/10.3934/dcdsb.2020132 doi: 10.3934/dcdsb.2020132
[17]	M. Li, X. Luo, An EMC-HDG scheme for the convection-diffusion equation with random diffusivity, Numer Algor, 90 (2022), 1755–1776. https://doi.org/10.1007/s11075-021-01250-2 doi: 10.1007/s11075-021-01250-2
[18]	M. Li, X. Luo, An MLMCE-HDG method for the convection diffusion equation with random diffusivity, Comput. Math. Appl., 127 (2022), 127–143. https://doi.org/10.1007/s11075-021-01250-2 doi: 10.1007/s11075-021-01250-2
[19]	M. Li, X. Luo, An ensemble Monte Carlo HDG method for parabolic PDEs with random coefficients, Int. J. Comput. Math., 100 (2022), 405–421. https://doi.org/10.1016/j.camwa.2022.10.002 doi: 10.1016/j.camwa.2022.10.002
[20]	M. Li, X. Luo, A multilevel Monte Carlo ensemble and hybridizable discontinuous Galerkin method for a stochastic parabolic problem, Numer Methods Partial Differ Equ, 39 (2023), 2840–2864. https://doi.org/10.1002/num.22990 doi: 10.1002/num.22990
[21]	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
[22]	G. J. Lord, C. E. Powell, T. Shardlow, An introduction to computational stochastic PDEs, Cambridge University Press, New York, 2014. https://doi.org/10.1017/CBO9781139017329
[23]	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
[24]	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
[25]	J. Martínez-Frutos, M. Kessler, A. Münch, 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
[26]	J. Martínez-Frutos, F. P. Esparza, Optimal control of PDEs under uncertainty. An introduction with application to optimal shape design of structures, Springer, Cham, 2018. https://doi.org/10.1007/978-3-319-98210-6
[27]	L. Mathelin, M. Y. Hussaini, T. A. Zang, Stochastic approaches to uncertainty quantification in CFD simulations, Numer Algor, 38 (2005), 209–236. https://doi.org/10.1007/BF02810624 doi: 10.1007/BF02810624
[28]	J. Meng, P. Y. Zhu, H. B. Li, A block method for linear systems with multiple right-hand sides, J. Comput. Appl. Math., 255 (2014), 544–554. https://doi.org/10.1016/j.cam.2013.06.014 doi: 10.1016/j.cam.2013.06.014
[29]	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
[30]	T. F. Yao, C. L. Ye, X. B. Luo, S. W. Xiang, An ensemble scheme for the numerical solution of a random transient heat equation with uncertain inputs, Numer Algor, 94 (2023), 643–668. https://doi.org/10.1007/s11075-023-01514-z doi: 10.1007/s11075-023-01514-z
[31]	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

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