Application of an adaptive MCMC method for the heat flux estimation

Figures & data

Figure 1. Sketch of heat flux inversion.

Figure 2. The heat flux estimated with Piecewise function.

Figure 3. The heat flux estimated with Fourier series.

Figure 4. Fourier coefficients estimated by different methods.

Figure 5. Convergence process of different methods.

Figure 6. The triangle heat flux estimated with noises.

Figure 7. The variances of posterior distribution of heat flux.

Figure 8. The variances of posterior distribution of Fourier coefficients.

Figure 9. The heat flux estimated with greater noises.

Figure 10. The variances of posterior distribution of heat flux with greater noises.

Figure 11. The estimation errors of different methods.

Figure 12. The standard deviation of the estimation heat flux in frequency domain.

Table 1. Cross-correlation of coefficients of the sensitivity.

Piecewise linear function	x1	x2	x3	x4	x5
x1	1.000	0.552	0.184	−0.083	−0.395
x2	0.552	1.000	0.842	0.580	0.355
x3	0.184	0.842	1.000	0.856	0.623
x4	−0.083	0.580	0.856	1.000	0.832
x5	−0.395	0.355	0.623	0.832	1.000
Fourier series	x1	x2	x3	x4	x5
x1	1.000	−0.771	0.066	−0.278	−0.018
x2	−0.771	1.000	−0.026	−0.106	0.046
x3	0.066	−0.026	1.000	0.051	−0.001
x4	−0.278	−0.106	0.051	1.000	0.004
x5	−0.018	0.046	−0.001	0.004	1.000

Table 2. Normalized average squared jumping distances.

Piecewise linear function	Aver sq Dist/Var(xi)	Acceptance rate
x1	6.26E−03	0.449
x2	2.55E−03	0.446
x3	4.25E−03	0.449
x4	4.70E−03	0.448
x5	4.67E−03	0.450
Fourier series	Aver sq Dist/Var(xi)	Acceptance rate
x1	1.46E−02	0.449
x2	8.39E−02	0.453
x3	0.244	0.453
x4	0.258	0.450
x5	0.243	0.447

Figure 13. Efficiency of the first component of piecewise function.

Figure 14. Efficiency of the first component of Fourier series.

Figure 15. The comparison of experiment temperature and estimation result.

Figure 16. The comparison of the test nominal value and the heat flux estimated by CGM and SCMH.