THERMAL SCIENCE

International Scientific Journal

Jun-Feng Lai

,

Zai-Zai YanORCID

,

Ji-Huan HeORCID

BAYESIAN INFERENCE FOR SOLVING A CLASS OF HEAT CONDUCTION PROBLEMS

ABSTRACT
This paper considers a heat conduction problem of a common continuum-type stochastic mathematical model in an engineering field. The approximate solution is calculated with the Markov chain Monte-Carlo algorithm for the heat conduction problem. Three examples are given to illustrate the solution process of the method.
KEYWORDS
Bayesian inference, heat conduction problem, Markov Chain Monte Carlo
PAPER SUBMITTED: 2019-12-26
PAPER REVISED: 2020-05-10
PAPER ACCEPTED: 2020-05-10
PUBLISHED ONLINE: 2021-03-27
DOI REFERENCE: https://doi.org/10.2298/TSCI191226098L
CITATION EXPORT: view in browser or download as text file
THERMAL SCIENCE YEAR 2021, VOLUME 25, ISSUE No. 3, PAGES [2135 - 2142]
REFERENCES
[1] Milstein, G. N., Approximate Integration of Stochastic Differential Equations. Theory of Probability & Its Applications, 19 (1975), 3, pp. 557-562, 10.1137/1119062
[2] Higham, D. J., A-Stability and Stochastic Mean-Square Stability, Bit: Numerical Mathematics, 40 (2000), 2, pp. 404-409
[3] Higham, D. J., et al., Strong Convergence of Euler-Type Methods for Nonlinear Stochastic Differential Equations, SIAM Journal on Numerical Analysis, 40 (2002), 3, pp. 1041-1063, 10.1023/a:1022386822865
[4] Hutzenthaler, M., et al., Strong Convergence of an Explicit Numerical Method for Sdes With Nonglobally Lipschitz Continuous Coefficients, Annals of Applied Probability, 22 (2012), 4, pp. 1611-1641, 10.1214/11-aap803
[5] Wang, X. J., Gan, S. Q., The Tamed Milstein Method for Commutative Stochastic Differential Equations with Non-Globally Lipschitz Continuous Coefficients, Journal of Difference Equations and Applications, 19 (2013), 3, pp. 466-490, 10.1080/10236198.2012.656617
[6] Geman, S., Geman, D., Stochastic Relaxation, Gibbs Distributions, and the Bayesian Restoration of Images, IEEE transactions on pattern analysis and machine intelligence, 6 (1984), 6, pp. 721-741, 10.1109/tpami.1984.4767596
[7] Gordon, N. J., et al., Novel Approach to Nonlinear/Non-Gaussian Bayesian State Estimation, IEE Proceedings F-Radar and Signal Processing, 140 (1993), 2, pp. 107-113, 10.1049/ip-f-2.1993.0015
[8] Eraker, B., MCMC Analysis of Diffusion Models with Application to Finance, Journal of Business and Economic Statistics, 19 (2001), 2, pp. 177-191, 10.1198/073500101316970403
[9] Golightly, A., Wilkinson, D. J., Bayesian Inference for Nonlinear Multivariate Diffusion Models Ob-served with Error, Computational Statistics and Data Analysis, 52 (2008), 3, pp. 1674-1693, 10.1016/j.csda.2007.05.019
[10] Miguez, J., et al., On the Convergence of Two Sequential Monte-Carlo Methods for Maximum a Posteriori Sequence Estimation and Stochastic Global Optimization, Statistics and Computing, 23 (2013), 1, pp. 91-107, 10.1007/s11222-011-9294-4
[11] Martin-Fernandez, L., Lanzarone, E., A Particle-Filtering Approach for Real-Time Estimation of Thermal Conductivity and Temperature Tracking in Homogeneous Masses, Numerical Heat Transfer, Part B: Fundamentals, 67 (2015), 6, pp. 507-530, 10.1080/10407790.2014.992060
[12] Tifkitsis, K. I., Skordos, A. A., Real-Time Inverse Solution of the Composites' Cure Heat Transfer Problem Under Uncertainty, Inverse Problems in Science and Engineering, 28 (2020), 7, pp. 1011-1030, 10.1080/17415977.2019.1700242
[13] Zeng, Y., et al., A novel adaptive approximate Bayesian computation method for inverse heat conduction problem, International Journal of Heat and Mass Transfer, 134 (2019), May, pp. 185-197, 10.1016/j.ijheatmasstransfer.2019.01.002
[14] Ko, Y., et al., Markov Chain Monte-Carlo Simulation of a Bayesian Mixture Model for Gene Network Inference, Genes & Genomics, 41 (2019), 5, pp. 547-555, 10.1007/s13258-019-00789-8
[15] Hemantha, S. B. H., Postauditing and Cost Estimation Applications: An Illustration of MCMC Simulation for Bayesian Regression Analysis, Engineering Economist, 64 (2019), 1, pp. 40-67, 10.1080/0013791x.2018.1498961
[16] Yousaf, R. et al., Bayesian Estimation of the Transmuted Fréchet Distribution, Iranian Journal of Science & Technology Transactions A:Science, 43 (2019), 4, pp. 1629-1641, 10.1007/s40995-018-0581-1
[17] Begona, C., et al., Estimation of Parameters in a Structured SIR Model, Advances in Difference Equations，33 (2017), 1, pp. 1-13, 10.1186/s13662-017-1078-5
[18] Chandra, N. K., Bhattacharya, S., Non-Marginal Decisions: A Novel Bayesian Multiple Testing Procedure, Electronic Journal of Statistics, 13 (2017), 1, pp. 489-535
[19] Vevoda, P., et al., Bayesian Online Regression for Adaptive Direct Illumination Sampling, ACM Trans-actions on Graphics, 37 (2018), 4, pp. 1-12, 10.1145/3197517.3201340
[20] Palacio, A. R., Leisen, F., Bayesian Nonparametric Estimation of Survival Functionswith Multiple-Samples Information, Electronic Journal of Statistics, 12 (2018), 1, pp. 1330-1357
[21] Liang, X., Liu G. N., Application of a New Single Staggered Grid Method, Thermal Science, 23 (2019), Suppl. 3, pp. S631-S637
[22] Zhou, J., Numerical Simulations of the Energy-Stable Scheme for Swift-Hohenberg Equation, Thermal Science, 23 (2019), Suppl. 3, pp. S669-S676, 10.2298/tsci180515080z
[23] Zuo, Y. H., Wang, J. Q., Application of Monte-Carlo Method to Solving Boundary Value Problem of Differential Equations (in Chinese), High Power Laser and Particle Beams, 24 (2012), 12, pp. 3023-3027, 10.3788/hplpb20122412.3023
[24] Gu, S. T., et al., An Energy-Stable Finite-Difference Scheme for the Binary Fluid-Surfactant System, Journal of Computational Physics, 270 (2014), 3, pp. 416-431, 10.1016/j.jcp.2014.03.060
[25] Li, Y. B., et al., An Unconditionally Energy-Stable Second-Order Time-Accurate Schemefor the Cahn-Hilliard Equation on Surfaces, Communications in Nonlinear Science and Numerical Simulation, 53 (2017), Dec., pp. 213-227
[26] Kessler, M., Simple and Explicit Estimating Functions for a Discretely Observed Diffusion Process, Scandinavian Journal of Statistics, 27 (2000), 1, pp. 65-82, 10.1111/1467-9469.00179
[27] He, J.-H., Taylor Series Solution for a Third Order Boundary Value Problem Arising in Architectural Engineering, Ain Shams Engineering Journal, 11 (2020), 4, pp. 1411-1414, 10.1016/j.asej.2020.01.016
[28] He, C. H., et al. Taylor Series Solution for Fractal Bratu-Type Equation Arising in Electrospinning Pro-cess, Fractals, 28 (2020), 1, ID 2050011, 10.1142/s0218348x20500115
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Bayesian inference for solving a class of heat conduction problems

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