The Bayesian framework for inverse problems in heat transfer

Heat Transfer Engineering

Volumn 32, Issue 9, 2011, Pages 718-753

  Kaipio, Jari P ^a,b   Fox, Colin ^c  

^a UNIVERSITY OF AUCKLAND   (New Zealand)

^b UNIVERSITY OF EASTERN FINLAND   (Finland)

^c UNIVERSITY OF OTAGO   (New Zealand)

Author keywords

[No Author keywords available]

Indexed keywords

BAYESIAN; BAYESIAN APPROACHES; BAYESIAN FRAMEWORKS; BAYESIAN INVERSION; INVERSE HEAT TRANSFER PROBLEM; PHYSICAL MEASUREMENT; SINGLE POINT;

BAYESIAN NETWORKS; DIFFERENTIAL EQUATIONS; HEAT TRANSFER; UNCERTAINTY ANALYSIS;

INVERSE PROBLEMS;

EID: 79953706294     PISSN: 01457632     EISSN: 15210537     Source Type: Journal    
DOI: 10.1080/01457632.2011.525137     Document Type: Article

References (166)

1. Hadamard, J., Lectures on Cauchy's Problem in Linear Differential Equations, Yale University Press, New Haven, CT, 1923.
2. Campbell, S.L., and Meyer, C.D., Generalized Inverses of Linear Transformations, Dover, New York, 1991.
3. Groetsch, C.W., Inverse Problems in theMathematical Sciences, Vieweg, Braunschweig-Wiesbaden, 1993.
4. Özişik, M. N., and Orlande, H. R. B., Inverse Heat Transfer, Taylor & Francis, New York, 2000.
5. Beck, J. V., and Arnold, K. J., Parameter Estimation in Engineering and Science, Wiley Interscience, New York, 1977.
6. Gauss, C. F., Theoria Combinationis Observationum Erroribus Minimis Obnoxiae. 2 Teile und Supplement, Dieterich, 1823-1826.
7. Laplace, P. S., Mémoire sur la probabilité des causes par lesévénemens, Mem. Acad. Roy. Sci., vol. 6, pp. 621-656, 1774.
8. Tikhonov, A. N., and Arsenin, V. Y., Solution of Ill-Posed Problems, Winston & Sons, Washington, DC, 1977.
9. Morozov, V. A., Methods for Solving Incorrectly Posed Problems, Springer Verlag, New York, 1984.
10. Phillips, D. L., A Technique for the Numerical Solution of Certain Integral Equations of the First Kind, Journal of the Association for Computting Machinery, vol. 9, pp. 84-97, 1962.
11. Twomey. S., On the Numerical Solution of Fredholm integral equations of the first kind by the inversion of the linear system produced by quadrature. Journal of the Association for Computing Machinery, vol. 10, pp. 97-101, 1963.
12. Engl, H. W., Hanke, M., and Neubauer, A., Regularization of Inverse Problems, Kluwer, Dordrecht, 2000.
13. Hanke, M., Conjugate Gradient Type Methods for Ill-posed Problems, Longman Scientific & Technical, Essex, 1995.
14. Hanke,M., and Hansen, P. C.,RegularizationMethods for Large- Scale Problems, Surveys on Mathematics for Industry, vol. 3, pp. 253-315, 1993.
15. Hansen, P. C., Rank-Deficient and Discrete Ill-Posed Problems. Numerical Aspects of Linear Inversion, SIAM, Philadelphia, 1998.
16. Dorofeev, K. Y., and Yagola, A. G., The Method of Extending Compacts and a Posteriori Error Estimates for Nonlinear Ill-Posed Problems, Inverse Ill-Posed Problems, vol. 12, pp. 627-636, 2004. (Pubitemid 40030444)
17. Kaipio, J. P., and Somersalo, E., Statistical and Computational Inverse Problems, Springer-Verlag, New York, 2005.
18. Nissinen, A., Heikkinen, L. M., and Kaipio, J. P., Approximation Errors in Electrical Impedance Tomography-An Experimental Study, Measurement Science and Technology, vol. 19, pp. doi:10.1088/0957-0233/19/1/015501, 2008. (Pubitemid 351189605)
19. Nissinen,A., Heikkinen, L. M., Kolehmainen, V., and Kaipio, J. P., Compensation of Errors due to Discretization, Domain Truncation and Unknown Contact Impedances in Electrical Impedance Tomography. Measurement Science and Technology, vol. 20, doi:10.1088/0957-0233/20/10/105504, 2009.
20. Vauhkonen, M., Kaipio, J. P., Somersalo, E., and Karjalainen, P. A., Electrical Impedance Tomography With Basis Constraints, Inverse Problems, vol. 13, pp. 523-530, 1997. (Pubitemid 127549980)
21. Vauhkonen, M., Vadász, D., Karjalainen, P. A., Somersalo, E. and Kaipio, J. P., Tikhonov Regularization and Prior Information in Electrical Impedance Tomography. IEEE Transactions in Medical Imaging, vol. 17, pp. 285-293, 1998. (Pubitemid 128747681)
22. Jeffreys, H., Scientific Inference, Cambridge University Press, Cambridge, 1931.
23. Gull, S. F., and Daniell, G. J., Image Reconstruction From Incomplete and Noisy Data. Nature, vol. 272, pp. 686-690, 1978.
24. Tarantola, A., and Valette, B., Inverse Problems = Quest for Information. Journal of Geophysics, vol., pp. 159-170, 1982.
25. Jaynes, E. T., Prior Information and Ambiguity in Inverse Problems, In Inverse Problems: SIAM-AMS Proceedings, ed. D. W. Mclaughlin, vol. 14, American Mathematical Society, pp. 151-166, 1984. (Pubitemid 15498695)
26. Berger, J. O., Statistical Decision Theory and Bayesian Analysis, Springer-Verlag, New York, 1980.
27. Beck, J. V., Blackwell, B., and St. Clair, C. R., Inverse Heat Conduction: Ill-Posed Problems,Wiley Interscience, New York, 1985.
28. Alifanov, O. M., Inverse Heat Transfer Problems, Springer- Verlag, New York, 1994.
29. Alifanov, O., Artyukhin, E., and Rumyantsev, A., Extreme Methods for Solving Ill-Posed Problems with Applications to Inverse Heat Transfer Problems, Begell House, New York, 1995.
30. Kurpisz, K., and Nowak, A. J., Inverse Thermal Problems. WIT Press, Southampton, UK, 1995.
31. Stakgold, I., Boundary Value Problems of Mathematical Physics (Classics in Applied Mathematics, 29), 2 vol., Society for ndustrial Mathematics, 1987.
32. Catteneo,C.,AForm of Heat Conduction EquationWhich Eliminates the Paradox of Instantaneous Propagation, Compte Rendus, vol. 247, pp. 431-433, 1958.
33. Vernotte, P., Some Possible Complications in the Phenomenon of Thermal Conduction, Compte Rendus, vol. 252, pp. 2190-191, 1961.
34. Fox, C., Nicholls, G., and Palm, M., Efficient Solution of Boundary-Value Problems for Image Reconstruction via Sampling, Journal of Electronic Imaging, vol. 9, pp. 251-259, July 2000.
35. Orlande,H.R.B., Dulikravich,G. S., andColaço,M. J., Bayesian Estimation of the Thermal Conducitivity Components of rthotropic Solids, In Anais do V Congresso Nacional de Engenharia Mecânica, Salvador, Bahia, Brasil, 2008.
36. Wang, J., and Zabaras, N., Hierarchical Bayesian Models for Inverse Problems in Heat Conduction, Inverse Problems, vol. 21, pp. 183-206, 2005. (Pubitemid 40238422)
37. Emery, A. F., Valenti, E., and Bardot, D., Using Bayesian Inference for Parameter Estimation When the System Response and Experimental Conditions are Measured with Error and Some Variables Are Considered as Nuisance Variables. Measurement Science and Technology, vol. 18, pp. 19-29, 2007. (Pubitemid 46485517)
38. Liu, C.-S., Identification of Temperature-Dependent Thermophysical Properties in a Partial Differential Equation Subject to Extra Final Measurement Data. Numerical Methods and Partial Differential Equation, vol. 23, pp. 1083-1109, 2007. (Pubitemid 47381160)
39. Al-Khairy, R. T., and Al-Ofey, Z. M., Analytical Solution of the Hyperbolic Heat Conduction Equation for Moving Semi-Infinite Medium Under the Effect ofTime-Dependent Laser Heat Source, Journal of Applied Mathematics, vol. 2009, pp. 18, 2009.
40. Ç engel, Y. A., and Ghajar, A., Heat and mass Transfer: Fundamentals and Applications, McGraw-Hill, New York, 4th ed., 2010.
41. Howard, J. R., An Experimental Study of Heat Transfer Through Periodically Contacting Surfaces, International Journal of Heat and Mass Transfer, vol. 19, pp. 367-372, 1976.
42. Yan, L., Yang, F., and Fu, C., A Bayesian Inference Approach to Identify a Robin Coefficient in One-Dimensional Parabolic Problems, Journal of Computational and Applied Mathematics, vol. 231, pp. 840-850, 2009.
43. Lesnic, D., Elliot, L., and Ingham, D. B., Application of the Boundary Element Method to Inverse Heat Conduction Problems, International Journal of Heat and Mass Transfer, vol. 39, pp. 1503-1517, 1996. (Pubitemid 126379612)
44. Wrobel, L. C., The Boundary Element Method, Wiley, Chichester, 2002.
45. Langdon, S., A Boundary Integral EquationMethod for the Heat Equation. In Proc. 5th Int. Conf. on Integral Methods in Science and Engineering (IMSE98), eds. A. Struthers and B. Bertram, CRC Press, pp. 211-216, Boca Raton, FL, pp. 211-216, 2000.
46. Johansson, B. T., and Lesnic, D., A Method of Fundamental Solutions for Transient Heat Conduction in Layered Materials. Engineering Analysis with Boundary Elements, vol. 33, 2009.
47. Davies, A. J., The Finite Element Method: A First Approach, Clarendon Press, New York, 1980.
48. Ern, A., and Guermond, J. L., Theory and Practice of Finite Elements, Springer-Verlag, New York, 2004.
49. Butcher, J. C., Numerical Methods for Ordinary Differential Equations, Wiley, New York, 2008.
50. Ascher, U., Mattheij, R., and Russell, R., Numerical Solution of Boundary Value Problems for Ordinary Differential Equations, SIAM, Philadelphia, 1995.
51. Tausch, J. A., Fast Method for Solving the Heat Equation by Layer Potentials, Journal of Computational Physics, vol. 224, pp. 956-969, 2007.
52. Brenner, S. C., and Scott, L. R., The Mathematical Theory of Finite Element Methods, Springer-Verlag, New York, 1994.
53. Solin, P., Segeth, K., and Dolezel, I., Higher Order Finite Element Methods, Chapmann and Hall, Suffolk, 2004.
54. Sinclair, E. Volatility Trading, Wiley, New York, 2008.
55. Abramowitz, M., and Stegun, I. A., Handbook of Mathematical Functions, Dover Publications, New York, 1965.
56. Springer, M. D.,The Algebra of Random Variables, John Wiley & Sons, New York, 1979.
57. Berger, J. O., De Oliveira, V., and Sanso, B., Objective Bayesian Analysis of Spatially Correlated Data, Journal of the American Statistical Association, vol. 96, pp. 1361-1374, 2001. (Pubitemid 33695586)
58. Hartigan, J. A., Bayes Theory, Springer-Verlag, New York, 1983.
59. Robert, C. P., The Bayesian Choice: From Decision-Theoretic Foundations to Computational Implementation. SpringerVerlag, New York, 2nd ed., 2001.
60. Nicholls, G. K., and Fox, C., Prior Modeling and Posterior Sampling in Impedance Imaging, in Proc SPIE, vol. 3459, Bayesian Inference for Inverse Problems, ed. A. Mohammad- Djafari, SPIE, Bellinghan WA, pp. 116-127, 1998.
61. Robert, C. P., and Casella, G., Monte Carlo Statistical Methods, Springer, New York, 2004.
62. Lee, P. M., Bayesian Statistics: An Introduction, Wiley, Chichester, 3rd ed., 2009.
63. Carlin, B. P., and Louis, T. A., Bayesian Methods for Data Analysis, Chapman & Hall/CRC, Boca Raton, FL, 3rd ed., 2008.
64. Tarantola, A., Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM, Philadelphia, 2004.
65. Calvetti, D., and Somersalo, E., An Introduction to Bayesian Scientific Computing-Ten Lectures on Subjective Computing, Springer-Verlag, New York, 2007.
66. Horn, R. A., and Johnson, C. R., Matrix Analysis, Cambridge University Press, Cambridge, 1985.
67. Gilks,W. R.,Richardson, S., and Spiegelhalter, D. J., Introducing Markov Chain Monte Carlo, in Markov Chain Monte Carlo in Practice, eds.W. R. Gilks, S. Richardson, and D. J. Spiegehalter, Chapman & Hall, Suffolk, pp. 1-19, 1996.
68. Kipnis, C., and Varadhan, S. R. S., Central Limit Theorem for Additive Functionals of Reversible Markov Processes and pplications to Simple Exclusions. Communications in Mathematical Physics, vol. 104, pp. 1-19, 1986.
69. Geyer, C. J., Practical Markov Chain Monte Carlo, Statistical Science, vol. 7, pp. 473-483, 1992.
70. Liu, J. S., Monte Carlo Strategies in Scientific Computing, Springer-Verlag, New York, 2005.
71. Metropolis, N., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. H., and Teller, E., Equation of State Calculations by Fast Computing Machines, Journal of Chemical Physics, vol. 21, pp. 1087-1092, 1953.
72. Hastings,W. K., Monte Carlo SamplingMethods UsingMarkov Chains and Their Applications, Biometrika, vol. 57, pp. 97-109, 1970.
73. Green, P. J., Reversible Jump Markov Chain Monte Carlo Computation and Bayesian Model Determination, Biometrika, vol. 82, pp. 711-732, 1995.
74. Andersen, K. E., Brooks, S. P., and Hansen, M. B., Bayesian Inversion of Geoelectrical Resistivity Data. Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 65, pp. 619-642, 2003. (Pubitemid 36931859)
75. Brooks, S. P., Giudici, P., and Roberts, G. O., Efficient Construction of Reversible Jump Markov Chain Monte Carlo Proposal Distributions, Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 65, pp. 3-39, 2003. (Pubitemid 37155334)
76. Teirney, L., Introduction to General State-Space Markov Chain Theory, in Markov Chain Monte Carlo in Practice, eds., W. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Chapman & Hall, Suffolk, pp. 59-74. 1996.
77. Siltanen, S.,Kolehmainen,V., Järvenpää, S., Kaipio, J. P.,Koistinen, P., Lassas, M., Pirttilä, J., and Somersalo, E., Statistical Inversion for Medical X-Ray Tomography with Few Radiographs I: General Theory. Physicss in Medicine and Biology, vol. 48, pp. 1437-1463, 2003. (Pubitemid 36700475)
78. Kolehmainen,V., Siltanen, S., Järvenpää, S., Kaipio, J. P.,Koistinen, P. Lassas, M., Pirttilä, J., and Somersalo, E., Statistical nversion for Medical X-Ray Tomography with Few Radiographs I: Application to Dental Radiology. Physics in Medicine and Biology, vol. 48, pp. 1465-1480, 2003.
79. Watzenig, D., and Fox, C., A Review of Statistical Modeling and Inference for Electrical Capacitance Tomography, Measurement in Science and Technology, vol. 20, p. 22 2009.
80. Carlin, B. P., and Chib, S., Bayesian Model Choice via Markov Chain Monte Carlo Methods, vol. 57, pp. 473-484, 1995.
81. Kaipio, J., and Somersalo, E., Statistical Inverse Problems: Discretization, Model Reduction and Inverse Crimes, Journal of Computational and Applied Mathematics, vol. 198, pp. 493-504, 2007. (Pubitemid 44402950)
82. Banerjee, S. Carlin, B. P., and Gelfand, A. E., Hierachical Modeling and Analysis for SpatialData, Chapman & Hall/CRC, New York, 2004.
83. Higdon, D., A Primer on Space-Time Modeling from a Bayesian Perspective, in Statistics of Spatio-Temporal Systems, eds. B. Finkenstadt, L. Held, and V. Isham, Chapman & Hall/CRC, New York, pp. 217-279, 2006.
84. Hurn, M. A., Husby, O., and Rue, H., Advances in Bayesian Image Analysis, in Highly Structured Stochastic Systems, eds. Green, P. J., N. Hjort, and S. Richardson, Oxford: Oxford University Press, Oxford, pp. 302-322, 2003.
85. Green, P. J., MCMC in Image Analysis, in Markov chain Monte Carlo in Practice, eds. Gilks,W. R., S. Richardson, and piegelhalter, D. J., Chapman & Hall, Suffolk, pp. 381-399. 1996.
86. Lindgren. F., Reconstruction of flames, in The Art and Science of Bayesian Image Analysis, eds. Mardia, K. V., Gill, C. A., and Aykroyd, R. G., editors, Leeds University Press, Leeds, pp. 52-59. 1997.
87. Hanson, K. M., Cunningham, G. S., and McKee, R. J., Uncertainty Assessment for Reconstructions Based on Deformable Geometry, International Journal of Imaging System Technology, vol. 8, pp. 506-512, 1997. (Pubitemid 127549488)
88. Stawinski, G., Doucet, A., and Duvaut, P., Reversible Jump Markov ChainMonte Carlo for Bayesian Deconvolution of Point Sources, In Proc SPIE, vol. 3459, "Bayesian Inference for Inverse Problems", ed. A. Mohammad-Djafari, SPIE, Bellinghan WA, pp. 179-190. 1998.
89. Jaynes, E. T., Prior Probabilities, IEEE Transactions on Systems Science and Cybernetics, vol. 4, pp. 227-241, 1968.
90. Rue.,H. Fast Sampling of Gaussian Markov Random Fields, Journal of the Royal Statistical Society: Series B, vol. 63, pp. 325-338, 2001. (Pubitemid 33267689)
91. Geman, S., and Geman, D., Stochastic Relaxation, Gibbs Distributions and the Bayesian Restoration of Images, IEEE Transactions of Pattern Analysis and Machine Intelligence, vol. 6, pp. 721-741, 1984. (Pubitemid 15453722)
92. Dobson, D., and Santosa, F., Recovery of Blocky Images from Noisy and Blurred Data. SIAM Journal of Applied Mathematics, vol., pp. 1181-1198, 1996. (Pubitemid 126551909)
93. Nicholls, G. K., Bayesian Image Analysis With Markov Chain Monte Carlo and Coloured Continuum Triangulation Models, Journal of the Royal Statistical Society: Series B (Statistical Methodology), vol. 60, pp. 325-338, 1998.
94. Voutilainen, A., Stratmann, F., and Kaipio, J. P., A Non- Homogeneous RegularizationMethod for Estimation of Narrow Aerosol Size Distributions, Journal of Aerosol Science, vol. 31, pp. 1433-1445, 2000.
95. Lehikoinen, A., Finsterle, S., Voutilainen, Heikkinen, L. M., Vauhkonen, M., and Kaipio, J. P., Approximation Errors and Truncation of Computational Domains with Application to Geophysical Tomography, Inverse Problem Imaging, vol. 1, pp. 371-389, 2007.
96. Lassas, M., and Siltanen S., Can One Use Total Variation Prior for Edge Preserving Bayesian inversion. Inverse Problems, vol. 20, pp. 1537-1564, 2004.
97. Kolehmainen, V., Arridge , S. R., Lionheart, W. R. B., Vauhkonen, M., and Kaipio, J. P., Recovery of Region Boundaries of Piecewise Constant Coefficients of an Elliptic PDE from Boundary Data, Inverse Problems, vol. 15, pp. 1375-1391, 1999. (Pubitemid 129618024)
98. Calvetti, D., and Somersalo, E., Image Inpainting with Structural Bootstrap Priors, Image Vision Computing, vol. 24, pp. 782-93, 2006. (Pubitemid 44163390)
99. Berger, J. O., The Case for Objective Bayesian Analysis, Bayesian Analysis, vol. 1, pp. 1-17, 2004.
100. Kaipio, J. P., Seppänen, A., Somersalo, E., and Haario, H., Posterior Covariance Related Optimal Current Patterns in Electrical Impedance Tomography, Inverse Problems, vol. 20, pp. 919-936, 2004.
101. Bardot, D., and Emery, A. F., Combining Discrepancy Models with Hierarchical Bayesian Inference for Parameter Estimation of Very Ill Posed Thermal Problems, Journal of Physics: Conference Series, vol. 135, pp. 012013, 2008.
102. Arridge, S. R., Kaipio, J. P., Kolehmainen, V., Schweiger, M., Somersalo, E., Tarvainen, T., and Vauhkonen, M., pproximation Errors and Model Reduction with an Application in Optical Diffusion Tomography, Inverse Problems, vol. 22, pp. 175-195, 2006. (Pubitemid 43185096)
103. Kolehmainen, V., Tarvainen, T., Arridge, S. R., and Kaipio, J. P., Marginalization of Uninteresting Distributed Parameters in Inverse Problems - Application to Optical Tomography, International Journal of Uncertainty Quantification, vol., 2009. in review.
104. Arridge, S. R., Kaipio, J. P., Kolehmainen, V., Schweiger, M., Somersalo, E., Tarvainen, T., and Vauhkonen. M., pproximation Errors andModel ReductionWith an Application in Optical Diffusion Tomography, Inverse Problems, vol. 22, pp. 175-195, 2006. (Pubitemid 43185096)
105. Heino, J., and Somersalo, E., A Modeling Error Approach for the Estimation of Optical Absorption in the Presence of Anisotropies, Physics Medicine and Biology, vol. 49, pp. 4785-4798, 2004. (Pubitemid 39433506)
106. Heino, J., Somersalo, and Kaipio, J. P., Compensation for Geometric Mismodelling by Anisotropies in Optical Tomography, Optics Express, vol. 13, pp. 296-308, 2005. (Pubitemid 40224979)
107. Calvetti, D., Kaipio, J. P., and Somersalo, E., Aristotelian Prior Boundary Conditions, International Journal of Mathematics, vol. 1, pp. 63-81, 2006.
108. Nissinen, A., Heikkinen, L. M., Kolehmainen, V., and Kaipio, J. P., Compensation of Modeling Errors Due to Unknown Domain Boundary in Electrical Impedance Tomography, Measurement in Science and Technology, vol., 2009.
109. Tarvainen, T., Kolehmainen, V., Pulkkinen, A., Vauhkonen, M., Schweiger, M., Arridge, S. R., and Kaipio, J. P., Approximation Error Approach for CompensatingModeling Errors Between the Radiative Transfer Equation and the Diffusion Approximation in Diffuse Optical Tomography, Inverse Problems, vol. 26, pp. doi:10.1088/0266-5611/26/1/015005, 2010.
110. Huttunen, J. M. J., and Kaipio, J. P., Approximation Error Analysis in Nonlinear State EstimationWith An Application to Statespace Identification, Inverse Problems, vol. 23, pp. 2141-2157, 2007. (Pubitemid 351453862)
111. Kaipio, J. P., and Somersalo, E., Nonstationary Inverse Problems and State Estimation, Journal of Inverse Ill-Posed Problems, vol. 7, pp. 273-282, 1999.
112. Doucet, A. de Freitas, N., and Gordon, N., Sequential Monte Carlo Methods in Practice, Springer, 2001.
113. Kalman, R. E., and Bucy, R. S., New Results in Linear Filtering and Prediction Theory, Trans. ASME Journal of Basic Engineering, vol. 83, pp. 95-108, 1961.
114. Anderson, B. D. O., and Moore, J. B., Optimal Filtering, Prentice Hall, Englewood Cliffs, NJ, 1979.
115. Kaipio, J. P.,Duncan, S., Seppanen, A., Somersalo, E., andVoutilainen, A., State estimation. in Handbook of Process Imaging for Automatic Control, eds. D. Scott and H. McCann, CRC Press, pp. 207-235, 2005.
116. Sinha, N. K., and Kuszta, B., Modeling and Identification of Dynamic Systems, Van Nostrand Reinhold, New York, 1983.
117. Durbin, J., and Koopman, J., Time Series Analysis by State Space Methods, Oxford University Press, Oxford, 2001.
118. Risken, H., The Fokker-Planck Equation, Springer-Verlag, New York, 1989.
119. Brockwell, P. J., and Davis, R. A., Time Series: Theory and Methods, Springer-Verlag, New York, 1991.
120. Lehikoinen, A., Huttunen, J. M. J., Voutilainen. A., Finsterle, S., Kowalsky, M. B., and Kaipio, J. P., A New Dynamic Inversion Approach for Hydrological Process Monitoring. Water Resources Research, vol., 2009.
121. Bonnans, J. F., Glbert, J. C., Lemarechal, and Sagastizabal, C. A., Numerical Optimization. Theoretical and Practical Aspects, Springer-Verlag, New York, 2003.
122. Shumway, R. H., Applied Statistical Time Series Analysis, Prentice-Hall, Englewood Cliffs, NJ, 1988.
123. Huttunen, J. M. J., and Kaipio, J. P., Approximation Errors in Nostationary Inverse Problems, Inverse Problem and Imaging, vol. 1, pp. 77-93, 2007.
124. Huttunen J. M. J., and Kaipio, J. P., Model Reduction in State Identification ProblemsWith an Application to Determination of Thermal Parameters. Applied Numerical Mathematics, vol. 59, pp. 877-890, 2009.
125. Huttunen, J. M. J., Lehikoinen, A., Hämäläinen, J., and Kaipio, J. P., Importance Filtering Approach for the Nonstationary Approximation Error Method. Inverse Problems, vol., 2009.
126. Antoulas, A. C., Approximation of Large-Scale Dynamical Systems, SIAM, Philadelphia, 2005.
127. Moulton, J. D., Fox, C., and Svyatskiy, D., Multilevel Approximations in Sample-Based Inversion from the Dirichletto- Neumann Map. Journal of Physics, Conf. Ser., vol. 124, 012035+, 2008.
128. Bui-Thanh, T., Willcox, K., and Ghattas, O., Model Reduction for Large-scale syStems with High-Dimensional Parametric Input Space, SIAM Journal on Scientific Computing, vol. 30, pp. 3270-3288, 2008.
129. Golub, G. H., and van Loan, C. F., Matrix Computations, vol. 3 Johns Hopkins University Press, Baltimore, MD 1996.
130. Beatson, R. K., and Greengard, L., A Short Course on Fast MultipoleMethods; inWavelets, Multilevel Methods and Elliptic PDEs, eds.M.Ainsworth, J. Levesley,W.Light, andM.Marletta, Oxford University Press, Oxford, pp. 1-37. 1997.
131. Christen, J. A., and Fox, C. Markov Chain Monte Carlo Using an Approximation, Journal of Computational and Graphical Statistics, vol. 14, pp. 795-810, 2005. (Pubitemid 43017087)
132. Hettlich F., and Rundell. W. A Second Degree Method for Nonlinear Inverse Problems. SIAM Journal on Numerical Analysis, vol. 37, pp. 587-620, 1999.
133. Leoni, N., and Amon C. H. Bayesian Surrogates for Integrating Numerical, Analytical, and experimental data: application to inverse heat transfer in wearable computers. Components and Packaging Technologies, IEEE Transactions on, vol. 23, pp. 23-32, 2000. (Pubitemid 30593779)
134. Orlande,H. R. B., Colaço,M. J., andDulikravich,G. S., Approximation of the Likelihood Function in the Bayesian Technique for the Solution of Inverse Problems. Inverse Problems in Science and Engineering, vol. 16, pp. 677-692, 2008.
135. Geyer, C. J., Markov chain Monte Carlo Maximum Likelihood Calculations, in Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, Interface Foundation, pp. 156-163, 1991.
136. Gelman A., and Rubin, D. B., Inference From Iterative Simulation Using Multiple Sequences, Statistical Science, vol. 7, pp. 457-472, 1992.
137. Sokal, A. D., Monte Carlo Methods in Statistical Mechanics: Foundations and New Algorithms, in Lectures at the Cargése summer school on "Functional Integration: Basics and Applications," 1996.
138. Roberts, G. O., Markov Chain Concepts Related to Sampling Algorithms, inW. R. Gilks, S. Richardson, and D. J. Spiegelhalter, Markov Chain Monte Carlo in Practice, Chapman & Hall, Suffolk, pp. 45-57, 1996.
139. Roberts, G. O., and Rosenthal, J. S., Optimal Scaling for Various Metropolis-Hastings Algorithms, Statistical Science, vol. 16, pp. 351-367, 2001. (Pubitemid 33636356)
140. Lee, J. E., Sample Based Inference For Inverse Obstacle Scattering, Master's thesis, Department of Mathematics, University of Auckland; Auckland New Zealand, 2005.
141. Fox, C., Recent Advances in Inferential Solutions to Inverse Problems, Inverse Problems in Science Engineering, vol. 16, pp. 797-810, 2008.
142. Marinari, E., and Parisi, G., Simulated Tempering: A New Monte Carlo Scheme, Europhysics Letters, vol. 19, pp. 451-458, 1992.
143. Palm. M., Monte Carlo Methods in Electrical Conductance Imaging, Master's thesis, Department of Mathematics, University of Auckland: Auckland, New Zealand, 1999.
144. Higdon, D., Lee, H., and Holloman, C., Markov Chain Monte Carlo-based Approaches for Inference in Computationally Intensive Inverse Problems, in eds. Bernardo, J. M., Bayarri, M. J., Berger, J. O., Dawid, A. P., D. Heckerman, Smith, A. F. M., and M. West, editors, Bayesian Statistics 7. Oxford University Press, 2003.
145. Liang, F., and Wong, W., Real Parameter Evolutionary Monte Carlo with Applications to Bayesian Mixture Models, Journals of the Amercian Statistics Association, vol. 96, pp. 653-666, 2001.
146. Cui, T., Bayesian Inference for Geothermal Model Calibration, Master's Thesis, Department of Mathematics, University of Auckland, Auckland New Zealand, 2005.
147. Cui, T., Fox, C., O'Sullivan, M., and Nicholls, G. K., Using Parallel MCMC Sampling to Calibrate a Computer Model of a Geothermal Resevoir, manuscript, 2010.
148. Fox, C., and Nicholls, G., Sampling Conductivity Images Via MCMC, in eds, K. V. Mardia, K. V., Gill, C. A., and Aykroyd, R. G., editors, The art and science of Bayesian Image Analysis. Proceedings of the Leeds Annual Statistics Research Workshop, 1-4 July pp. 91-100, 1997. Leeds University Press. Leeds, UK, 1997.
149. Haario, H., Saksman, E., and Tamminen. J., An Adaptive Metropolis Algorithm, Bernoulli, vol. 7, pp. 223-242, 2001. (Pubitemid 33793059)
150. Andrieu, C., and Thoms, J., A Tutorial on Adaptive MCMC, Statistics and Computing, vol. 18, pp. 343-373.
151. Bai, Y., Roberts, G. O., and Rosenthal, J. S., On the Containment Condition for Adaptive Markov ChainMonte Carlo Algorithms, Technical report, University of Toronto, Toranto, 2008.
152. Roberts, G. O., and Rosenthal, J. S., Examples of Adaptive MCMC, Journal of Computational and Graphical Statistics, vol. 18, pp. 349-367, 2009.
153. Box, G. E. P., and Tiao, G. C., Bayesian Inference in Statistical Analysis, Wiley, New York 1992 (1973).
154. Gelman, A.,Carlin, J.B., Stern, H. S., and Rubin, D. B., Bayesian Data Analysis, Chapman & Hall, Suffolk, 1995.
155. Kaipio, J. P., Seppänen, A., Voutilainen, A., and Haario, H., Optimal Current Patterns in Nonstationary Electrical Impedance Tomography, Inverse Problems, vol. 23, pp. 1201-1214, 2007. (Pubitemid 46838849)
156. Ferrero, C., and Gallagher, K., Stochastic Thermal HistoryModelling. 1. Constraining Heat Flow Histories and Their Uncertainty, Marine and Petroleum Geology, vol. 19, pp. 633-648, 2002. (Pubitemid 35274157)
157. Wang, J., and Zabaras, N., A Bayesian Inference Approach to The Inverse Heat Conduction Problem, International Journal of Heat and Mass Transfer, vol. 47, pp. 3927-3941, 2004. (Pubitemid 38790937)
158. Wang, J., and Zabaras, N., Using Bayesian Statistics in the Estimation of Heat Source in Radiation. International Journal of Heat and Mass Transfer, vol. 48, pp. 15-29, 2005. (Pubitemid 39553919)
159. Zabaras, N., Inverse Problems in Heat Transfer, in Handbook of Numerical Heat Transfer, eds. W. J., Minkowycz, E. M., Sparrow, and J. Y., Murthy, Wiley, Chichester, 2nd edition, pp. 525-558, 2006.
160. Kolehmainen, V., Kaipio, J. P., and Orlande, H. R. B., Reconstruction of Thermal Conductivity and Heat Capacity Using a TomographicApproach, International Journal of Heat and Mass Transfer, vol. 51, pp. 1866-1876, 2008. (Pubitemid 351295884)
161. Jin, B., and Zou, J., A Bayesian Inference Approach to the Ill- Posed Cauchy Problem of Steady-State Heat Conduction. International Journal for Numerical Methods in Engineering, vol. 76, pp. 521-544, 2008.
162. Jin, B., Fast Bayesian Approach for Parameter Estimation, International Journal for Numerical Methods in Engineering, vol. 76, pp. 230-252, 2008.
163. Parthasarathy, S., and Balaji, C., Estimation of Parameters in Multi-Mode Heat Transfer Problems Using Bayesian Inference-Effect of Noise and A Priori, International Journal of Heat and Mass Transfer, vol. 51, pp. 2313-2334, 2008. (Pubitemid 351492523)
164. Mota, C. A. A., Orlande, H. R. B., Carvalho, M. O. M., Kolehmainen, V., and Kaipio. J. P., Bayesian Estimation of Temperature-Dependent Thermophysical Properties and Boundary Heat Flux. Heat Transfer Engineering, vol. 31, pp. 570-580, 2010.
165. Mota, C.A.A., Orlande, H. R. B., Wellele,O. J.M., Kolehmainen, V., and Kaipio. J. P., Inverse Problem of Simultaneous Identification of Thermophysical Properties And Boundary Heat Flux. High Temperatures-High Pressure, vol., 2010.
166. Huttunen, J., Malinen, M., Huttunen, T., and Kaipio, J. P., Determination of Heterogeneous Thermal Parameters Using Ultrasound Induced Heating and Mr Thermal Mapping Physics in Medicine and Biology, vol. 51, pp. 1011-1032, 2006.