A parallel tempering algorithm for probabilistic sampling and multimodal optimization

Geophysical Journal International

Volumn 196, Issue 1, 2013, Pages 357-374

  Sambridge, Malcolm ^a  

^a AUSTRALIAN NATIONAL UNIVERSITY   (Australia)

Author keywords

Inverse theory; Numerical solutions

Indexed keywords

COMPUTATIONAL STATISTICS; DIRECT SEARCH OPTIMIZATION; INVERSE THEORY; MARKOV CHAIN MONTE CARLO ALGORITHMS; MULTI-MODAL OPTIMIZATION; NON-LINEAR INVERSE PROBLEM; NUMERICAL SOLUTION; PARALLEL TEMPERING ALGORITHMS;

ALGORITHMS; GLOBAL OPTIMIZATION; INVERSE PROBLEMS; SIMULATED ANNEALING;

TEMPERING;

ALGORITHM; INVERSE ANALYSIS; MARKOV CHAIN; MONTE CARLO ANALYSIS; NONLINEARITY; NUMERICAL MODEL; OPTIMIZATION; SIMULATED ANNEALING;

EID: 84890879984     PISSN: 0956540X     EISSN: 1365246X     Source Type: Journal    
DOI: 10.1093/gji/ggt342     Document Type: Article

References (60)

1. Aarts, E. & Korst, J., 1989. Simulated Annealing and Boltzmann Machines, John Wiley & Sons.
2. Ammon, C.J., Randall, G.E. & Zandt, G., 1990. On the nonuniqueness of receiver function inversions, J. geophys. Res., 95, 15 303-15 318.
3. Aster, R., Borchers, B. & Thurber, C.H., 2012. Parameter Estimation and Inverse Problems, 2nd edn, Elsevier Academic Press.
4. Atchad ́e, Y.F., Roberts, G.O. & Rosenthal, J.S., 2011. Towards optimal scaling of metropolis-coupled Markov chain Monte Carlo, Stat. Comput., 21(4), 555-568.
5. Bai, Y., Craiu, R.V. & Di Narrzo, F., 2011. Divide and conquer: a mixturebased approach to regional adaptation for MCMC, J. Comp. Graph. Stat., 20(1), 63-79.
6. Beck, J.L. & Au, S.-K., 2002. Bayesian updating of structural models and reliability using Markov chain Monte Carlo simulation, J. Eng. Mech., 128, 380-391.
7. Bina, C.R., 1998. Free energy minimization by simulated annealing with applications to lithospheric slabs and mantle plumes, Pure appl.Geophys., 151, 605-618.
8. Bodin, T., Sambridge, M., Rawlinson, N. & Arroucau, P., 2012a. Transdimensional tomography with unknown data noise, Geophys. J. Int., 189, 1536-1556.
9. Bodin, T., Sambridge, M., Tkalčci ́c, H., Arroucau, P., Gallagher, K. & Rawlinson, N., 2012b. Transdimensional inversion of receiver functions and surface wave dispersion, J. geophys. Res., 117, B02301, doi:10.1029/2011JB008560.
10. Brooks, S., Gelman, A., Jones, G.L. & Meng, X.E., 2011. Handbook of Markov Chain Monte Carlo, Chapman & Hall/CRC.
11. Ching, J. & Chen, Y.-C., 2007. Transitional Markov chain Monte Carlo method for Bayesian model updating, model class selection, and model averaging, J. Eng. Mech. 133(7), 816-832.
12. Cornish, N.J., 2012. Gravitational wave astronomy: needle in a haystack, Phil. Trans. R. Soc. Lond., A, 371(1984), doi: 10.1098/rsta.2011.0540.
13. Craiu, R.V., Rosenthal, J. & Yang, C., 2009. Learn from thy neighbor: parallel-chain and regional adaptive McMC, J. Am. Stat. Assoc., 104(488), 1454-1466.
14. Curtis,A.&Wood, R., 2004. Optimal elicitation of probabilistic information from experts, in Geological Prior Information, Publication 239, pp. 1-14, edsCurtis, A. & Wood,, R., The Geological Society of London.
15. Dettmer, J. & Dosso, S.E., 2012. Trans-dimensional matched-field geoacoustic inversion with hierarchical error models and interacting Markov chains, J. acoust. Soc. Am., 132, 2239-2250.
16. Dettmer, J. & Dosso, S.E., 2013. Probabilistic two dimensional joint watercolumn and seabed inversion, J. acoust. Soc. Am., 133, 2612-2623.
17. Dosso, S.E. &Wilmut,M.J., 2006. Data uncertainty estimation inMatched- Field geoacoustic inversion, IEEE J. Ocean. Eng., 31, 470-479.
18. Dosso, S.E., Holland, C.W. & Sambridge, M., 2012. Parallel tempering in strongly nonlinear geoacoustic inversion, J. acoust. Soc. Am., 132(5), 3030-3040.
19. Earl, D.J. & Deem, M.W., 2005. Parallel tempering: theory, applications, and new perspectives, Phys. Chem. Chem. Phys., 7, 3910-3916.
20. Falcioni,M. & Deem,M.W., 1999. A biased Monte Carlo scheme for zeolite structure solution, J. Chem. Phys., 110, 1754-1766.
21. Gallagher, K., Charvin, K., Nielsen, S., Sambridge, M. & Stephenson, J., 2009. Markov chain Monte Carlo (McMC) sampling methods to determine optimal models,model resolution and model choice for earth science problems, J. Mar. Petrol. Geol., 26, 525-535.
22. Gallagher, K., Bodin, T., Sambridge, M., Weiss, D., Kylander, M. & Large, D., 2011. Inference of abrupt changes in noisy geochemical records using transdimensional change point models, Earth planet. Sci. Lett., 311, 182-194.
23. Garcia, R.H., Tkalčci ́c, H. & Chevrot, S., 2006. A new global PKP data set to study earth's core and deep mantle, Phys. Earth planet. Inter., 159, 15-31.
24. Geyer, C.J., 1991. Markov Chain Monte Carlo maximum likelihood, in Computing Science and Statistics: Proceedings of the 23rd Symposium on the Interface, American Statistical Association, New York, pp. 156-163.
25. Geyer, C.J., 2011. Importance sampling, simulated tempering and umbrella sampling, in Handbook of Markov Chain Monte Carlo, pp. 295-311, edsBrooks, S., Gelman, A., Jones, G.L. & Meng, X., Chapman & Hall/CRC.
26. Geyer, C.J.&Thompson, E.A., 1995. AnnealingMarkov chain Monte Carlo with applications to ancestral inference, J. Am. Stat. Assoc., 90, 909-920.
27. Haario, H., Laine, M., Mira, A. & Saksman, E., 2006. DRAM: efficient adaptive MCMC, Stat. Comput., 16(4), 339-354.
28. Habeck, M., Nilges, M. & Rieping, W., 2005. Replica-exchange Monte Carlo scheme for Bayesian data analysis, Phys. Rev. Lett., 94, 018105-1-018105-4.
29. Hopcroft, P., Gallagher, K. & Pain, C., 2007. Inference of past climate from borehole temperature data using Bayesian reversible jump Markov chain Monte Carlo, Geophys. J. Int., 171(3), 1430-1439.
30. Kirkpatrick, S., Gelatt, C.D. Jr & Vecchi, M.P., 1983 Optimization by simulated annealing, Science, 220, 671-680.
31. Kofke,D.A., 2004. Comment on the incomplete beta function lawfor parallel tempering sampling of classical canonical systems [J. Chem. Phys. 120, 4119 (2004)], J. Chem. Phys., 121, 1167, doi:10.1063/1.1758211.
32. Kone, A. & Kofke, D.A., 2005. Selection of temperature intervals for parallel-tempering simulations, J. Chem. Phys., 122, 206101-1- 206101-2.
33. Laloy, E.&Vrugt, J.A., 2012. High-dimensional posterior exploration of hydrologic models using multiple-try DREAM and high performance computing, Water Resour. Res., 48, W01526, doi: 10.1029/2011WR010608.
34. Li, Y., Protopopescu, V.A. & Gorin, A., 2004. Accelerated simulated tempering, Phys. Lett. A, 328(45), 274-283.
35. Malinverno, A., 2002. Parsimonious Bayesian Markov chain Monte Carlo inversion in a nonlinear geophysical problem, Geophys. J. Int., 151, 675-688.
36. Marinari, E. & Parisi, G., 1992. Simulated tempering: a new Monte Carlo scheme, Europhys. Lett., 19(6), 451-458.
37. Metropolis, N. & Ulam, S.M., 1949. The Monte Carlo method, J. Am. Stat. Assoc., 44, 335-341.
38. Minsley,B., 2011.Atrans-dimensionalBayesian Markov chain MonteCarlo algorithm for model assessment using frequency-domain electromagnetic data, Geophys. J. Int., 187, 252-272.
39. Minson, S.E., Simons, M. & Beck, J.L., 2013. Bayesian inversion for finite fault earthquake source models: I - theory and algorithm, Geophys. J. Int., 194, 1701-1726.
40. Mira, A., 2001. On Metropolis-Hastings algorithms with delayed rejection, Metron, 59(3-4), 231-241.
41. Mosegaard, K. & Sambridge, M., 2002. Monte Carlo analysis of inverse problems, Inverse Probl., 18, R29-R54.
42. Mosegaard, K. & Tarantola, A., 1995. Monte Carlo sampling of solutions to inverse problems, J. geophys. Res., 100, 12 431-12 447.
43. Parker, R.L., 1994. Geophysical Inverse Theory, Princeton University Press.
44. Pedrescu, C., Pedrescu, M. & Ciobanu, C.V., 2004. The incomplete beta function law for parallel tempering sampling of classical canonical systems, J. Chem. Phys, 120, 4119-4128.
45. Press, W.H., Tuekolsky, S.A., Vetterling, W.T. & Flannery, B.P., 1992. Numerical Recipes in FORTRAN, Cambridge University Press.
46. Rosenthal, J., 2000. Parallel computing and Monte Carlo algorithmas, Far East J. Theor. Stat., 4(2), 207-236.
47. Rothman,D.H., 1985. Nonlinear inversion statistical mechanics and residual statics corrections, Geophysics, 50, 2784-2796.
48. Rothman, D.H., 1986. Automatic estimation of large residual statics corrections, Geophysics, 51, 332-346.
49. Sambridge, M., 1999. Geophysical inversion with a neighbourhood algorithm-I. Searching a parameter space, Geophys. J. Int., 138, 479-494.
50. Sambridge, M., Gallagher, K., Jackson, A. & Rickwood, P., 2006. Transdimensional inverse problems, model comparison and the evidence, Geophys. J. Int., 167, 528-542.
51. Sen, M.K. & Stoffa, P.L., 2013. Global Optimization Methods in Geophysical Inversion, 2nd edn, Cambridge University Press.
52. Shibutani, T., Sambridge, M. & Kennett, B.L.N., 1996. Genetic algorithm inversion for receiver functions with application to crust and uppermost mantle structure beneath eastern australia, Geophys. Res. Lett., 23, 1829-1832.
53. Steininger, G.A.M.W., Dettmer, J., Holland, C.W. & Dosso, S.E., 2013. Trans-dimensional joint inversion of seabed scattering and reflection data, J. acoust. Soc. Am., 133, 1347-1357.
54. Sugita, Y. & Okamoto, Y., 1999. Replica-exchange molecular dynamics method for protein folding, Chem. Phys. Lett., 314(12), 141-151.
55. Swendsen, R.H. & Wang, J.S., 1986. Replica Monte Carlo simulation of spin glasses, Phys. Rev. Lett., 57, 2607-2609.
56. Tarantola, A., 2005. Inverse Problem Theory and Methods for Model Parameter Estimation, SIAM.
57. Kofke,D.A., 2002. On the acceptance probability of replica-exchange Monte Carlo trials, J. Chem. Phys., 117, 6911-6914.
58. Tarantola, A.&Valette,B., 1982a. Inverse problems=quest for information, J. Geophys., 50, 159-170.
59. Tarantola, A. & Valette, B., 1982b. Generalized nonlinear inverse problems solved using the least squares criterion, Rev. Geophys. Space Phys., 20, 219-232.
60. Tierney, L. & Mira, A., 1999. Some adaptive Monte Carlo methods for Bayesian inference, Stat. Med., 18(1718), 2507-2515.