Inverse Monte Carlo analysis

Fig. 15. Inverse maximum of the collective structure factor of composition fluctuations, N/S k 0), as a function of the incompatibility, x - Symbols correspond to Monte Carlo simulations of the bond fluctuation model, the dashed curve presents the results of a finite-size scaling analysis of simulation data in the vicinity of the critical point, and the straight, solid line indicates the prediction of the Flory-Huggins theory. The critical incompatibility, XcN = 2 predicted by the Flory-Huggins theory and that obtained from Monte Carlo simulations of the bond fluctuation model M 240, N = 64, p = 1/16 and = 25.12) are indicated by arrows. The left inset compares the phase diagram obtained from simulations with the prediction of the Flory-Huggins theory (c.f. (47)). The right inset depicts the compositions at coexistence such that the mean field theory predicts them to fall onto a straight line. Prom Muller [78]...

Thus, we take advantage of the accuracy, robustness and efficiency of the direct problem solution, to tackle the associated inverse heat transfer problem analysis [26, 27] towards the simultaneous estimation of momentum and thermal accommodation coefficients in micro-channel flows with velocity slip and temperature jump. A Bayesian inference approach is adopted in the solution of the identification problem, based on the Monte Carlo Markov Chain method (MCMC) and the Metropolis-Hastings algorithm [28-30]. Only simulated temperature measurements at the external faces of the channel walls, obtained for instance via infrared thermography [30], are used in the inverse analysis in order to demonstrate the capabilities of the proposed approach. A sensitivity analysis allows for the inspection of the identification problem behavior when the external wall Biot number is also included among the parameters to be estimated. 

The value of S(Q) at zero Q value cannot be determined experimentally on the same instrument that is used to measure diffusivities there are not enough points at small Q in Fig. 8b. However, the S(Q) scale, which is given in Fig. 8b in arbitrary units, can be renormalized. At infinite dilution, S(0) should be equal to one (hke in a gas), and sorption thermodynamics also imply that the thermodynamic correction factor should be equal to one, so that Eq. 38 will be fulfilled. On the other hand, at high concentrations, F increases while S(0) goes down. In Fig. 8b, r is equal to 6.6 for a concentration of 14 CF4 per u.c. so that S(0) should go down to 0.15. A more quantitative analysis has been recently performed for n-hexane and n-heptane in sihcalite [36] where the inverse of the thermodynamic factor, calculated from S(Q) was found to be in good agreement with configurational-bias Monte Carlo (CBMC) simulations. 

Myers et al. (2007, 2009) introduced a Bayesian nonlinear inversion framework to multiple-event analysis (Bayesloc). The nonlinear Markov chain Monte Carlo method enables Bayesloc to simultaneously assess the joint posterior distribution spanning event locations, travel-time corrections, phase names, and arrival-time measurement precision. Myers et al. (2011) demonstrated that Bayesloc can be applied to data sets of tens of thousands of events and millions of arrivals because the solution does not involve direct inversion of a matrix and computation demands grow linearly with the number of arrivals. 

The growing influence of the Bayesian viewpoint, is something to be welcomed, but with caution. The relationship between traditional deterministic solutions of inverse problems, maximum probability solutions and Monte Carlo approximation of output statistics needs further clarification and analysis. Studies of model problems, once again, would help us all to understand the issues better. 

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