Application of the generalized extremal optimization algorithm to an inverse radiative transfer problem

Figures & data

Figure 1. Outline of the GEO algorithm as implemented for the solution of the inverse radiative transfer problem.

Figure 2. Average of the best objective function values found in 20 runs of GEO for each value of γ.

Table 1. Exact values of the radiative properties

Radiative property	Case 1	Case 2	Case 3
Optical thickness, τ0	1.0	2.0	0.5
Single scattering albedo, ω	0.5	0.8	0.3
Diffuse reflectivity, ρ1	0.2	0.1	0.1
Diffuse reflectivity, ρ2	0.2	0.8	0.8

Figure 3. Average of the best values of the objective function, as a function of the number of function evaluations for Case 1, without noise.

Figure 4. Average of the best values of the objective function, as a function of the number of function evaluations for Case 1, with noise.

Table 2. Worst, average and best estimates for Case 1

			τ0	ω	ρ1	ρ2	fx^a
Exact value			1.000	0.500	0.200	0.200	–
SA	Without noise	Worst	1.000	0.499	0.198	0.200	6.07 × 10^−8
		Average^b	1.000	0.500	0.199	0.200	1.80 × 10^−8
		Best	1.001	0.500	0.201	0.200	1.84 × 10^−9
	With noise	Worst	1.032	0.525	0.253	0.204	4.03 × 10^−4
		Average^2	1.034	0.526	0.254	0.203	4.03 × 10^−4
		Best	1.034	0.526	0.254	0.203	4.03 × 10^−4
μGA	Without noise	Worst	0.963	0.509	0.251	0.250	2.50 × 10^−4
		Average	1.010	0.508	0.218	0.201	4.93 × 10^−5
		Best	0.987	0.495	0.190	0.205	2.79 × 10^−6
	With noise	Worst	1.540	0.817	0.000	0.000	2.2755
		Average	1.505	0.780	0.092	0.019	1.8205
		Best	1.173	0.589	0.373	0.163	5.58 × 10^−4
GEO	Without noise	Worst	0.943	0.497	0.231	0.253	2.63 × 10^−4
		Average	0.965	0.481	0.157	0.211	1.31 × 10^−4
		Best	1.012	0.502	0.200	0.194	9.00 × 10^−6
	With noise	Worst	0.924	0.503	0.234	0.266	9.46 × 10^−4
		Average	1.030	0.527	0.256	0.212	6.16 × 10^−4
		Best	1.046	0.541	0.295	0.212	4.37 × 10^−4
^afx is the value of the objective function at the end of the run.
^bThe SA stopping criterion was reached for all runs in <25,000 evaluations.

Figure 5. Average of the best values of the objective function, as a function of the number of function evaluations for Case 2, without noise.

Figure 6. Average of the best values of the objective function, as a function of the number of function evaluations for Case 2, with noise.

Figure 7. Average of the best values of the objective function, as a function of the number of function evaluations for Case 3, without noise.

Figure 8. Average of the best values of the objective function, as a function of the number of function evaluations for Case 3, with noise.

Table 3. Worst, average and best estimates for Case 2

			τ0	ω	ρ1	ρ2	fx^a
Exact value			2.000	0.800	0.100	0.800	–
SA	Without noise	Worst	2.003	0.800	0.101	0.800	1.72 × 10^−8
		Average^b	2.000	0.800	0.100	0.800	8.64 × 10^−9
		Best	2.000	0.801	0.102	0.800	0.92 × 10^−9
	With noise	Worst	2.027	0.803	0.109	0.799	7.76 × 10^−5
		Average^b	2.017	0.801	0.104	0.799	7.75 × 10^−5
		Best	2.016	0.801	0.104	0.799	7.75 × 10^−5
μGA	Without noise	Worst	2.845	0.851	0.243	0.732	1.66 × 10^−4
		Average	2.240	0.821	0.169	0.789	7.11 × 10^−5
		Best	1.955	0.790	0.062	0.797	7.63 × 10^−6
	With noise	Worst	3.847	0.894	0.402	0.669	4.01 × 10^−4
		Average	3.751	0.878	0.336	0.602	3.10 × 10^−4
		Best	2.933	0.853	0.249	0.720	2.50 × 10^−4
GEO	Without noise	Worst	4.844	0.891	0.373	0.249	4.28 × 10^−4
		Average	2.507	0.817	0.148	0.718	1.72 × 10^−4
		Best	2.033	0.809	0.138	0.803	1.80 × 10^−5
	With noise	Worst	2.077	0.816	0.164	0.831	5.21 × 10^−4
		Average	2.385	0.824	0.167	0.774	2.62 × 10^−4
		Best	2.160	0.811	0.129	0.787	9.90 × 10^−5
^afx is the value of the objective function at the end of the run.
^bThe SA stopping criterion was reached for all runs in <25,000 evaluations.

Table 4. Worst, average and best estimates for Case 3

			τ0	ω	ρ1	ρ2	fx^a
Exact value			0.500	0.300	0.100	0.800	–
SA	Without noise	Worst	0.500	0.300	0.100	0.800	3.08 × 10^−8
		Average^b	0.500	0.300	0.100	0.800	1.57 × 10^−8
		Best	0.500	0.300	0.100	0.800	8.49 × 10^−10
	With noise	Worst	0.487	0.305	0.147	0.806	1.08 × 10^−4
		Average^b	0.486	0.305	0.149	0.806	1.08 × 10^−4
		Best	0.487	0.305	0.148	0.806	1.08 × 10^−4
μGA	Without noise	Worst	0.430	0.362	0.375	0.841	5.82 × 10^−4
		Average	0.463	0.319	0.226	0.818	2.08 × 10^−4
		Best	0.518	0.299	0.049	0.793	2.52 × 10^−5
	With noise	Worst	0.254	0.343	0.680	0.902	3.22 × 10^−3
		Average	0.431	0.327	0.298	0.833	8.01 × 10^−4
		Best	0.469	0.325	0.233	0.818	1.6 × 10^−4
GEO	Without noise	Worst	0.396	0.210	0.152	0.750	1.91 × 10^−2
		Average	0.501	0.290	0.107	0.786	4.32 × 10^−3
		Best	0.523	0.277	0.003	0.789	2.08 × 10^−4
	With noise	Worst	0.831	0.543	0.094	0.728	3.31 × 10^−2
		Average	0.525	0.378	0.237	0.812	6.82 × 10^−3
		Best	0.386	0.302	0.382	0.844	8.17 × 10^−4
^afx is the value of the objective function at the end of the run.
^bThe SA stopping criterion was reached for all runs in <25,000 evaluations.