A computational method to estimate the unknown coefficient in a wave equation using boundary measurements

Figures & data

Table 1. Maximal error ‖u − u_exact‖_∞ between the exact solution and the numerical solution by the second-order scheme defined in Equation (15).

(h, Δt)	(1/8, 1/10)	(1/16, 1/20)	(1/32, 1/40)	(1/64, 1/80)	(1/128, 1/160)
E(h, Δt)	1.2138E−04	3.0503E−05	7.6392E−06	1.9108E−06	4.7773E−07
	–	3.9793	3.9930	3.9979	3.9997
	–	1.9925	1.9975	1.9993	1.9999

Table 2. Maximal error ‖u − u_exact‖_∞ between the exact solution and the numerical solution by the fourth-order scheme defined in Equation (23).

(h, Δt)	(1/8, 1/10)	(1/16, 1/20)	(1/32, 1/40)	(1/64, 1/80)	(1/128, 1/160)
E(h, Δt)	7.8152E−06	4.6310E−07	2.6581E−08	1.5215E−09	8.7092E−11
	–	16.8758	17.42258	17.4699	17.4702
	–	4.0769	4.1229	4.1268	4.1268

Table 3. Comparison of derivatives generated by various methods.

xi	FD (ε = 0.01)	FD (ε = 0.001)	FD (ε = 0.0001)	TAMC	TAPENADE
0.1	0.1547459996	0.1547356869	0.1547355778	0.1547355843	0.1548949188
0.2	0.2193119526	0.2192973117	0.2192971754	0.2192971651	0.2192654826
0.3	0.2851436441	0.2851247931	0.2851245907	0.2851246039	0.2849388791
0.4	0.3465546654	0.3465319277	0.3465316861	0.3465316996	0.3462826013
0.5	0.4160232908	0.4159961009	0.4159958051	0.4159958286	0.4156669989
0.6	0.4372516947	0.4372231528	0.4372228554	0.4372228672	0.4362583627
0.7	0.4669640295	0.4669334619	0.4669331458	0.4669331575	0.4659774826
0.8	0.4764092251	0.4763779912	0.4763776626	0.4763776802	0.4751535480
0.9	0.4485708252	0.4485416621	0.4485413880	0.4485413683	0.4462902136

Figure 1. Comparison of the results by the second-order and fourth-order methods for Example 2, with h = 1/40 and Δt = 1/80.

Figure 2. Comparison of the results with various regularizers.

Figure 3. Comparison of the results by different optimization algorithms.

Table 4. Computational cost of various algorithms.

Number of iterations	Number of function evaluations
	L-BFGS	Fletcher–Reeves	Polak–Ribiere
20	23	47	44
35	41	78	75
50	57	109	107
	CPU time (s)
	L-BFGS	Fletcher–Reeves	Polak–Ribiere
20	4.64	8.96	8.45
35	7.94	14.62	14.10
50	11.04	20.31	20.06

Figure 4. Comparison of the results by using different number of boundary measurements.

Figure 5. Comparison of the results by using partial and complete boundary measurements, λ = 1 × 10⁻³.

Figure 6. Comparison of the results by using partial and complete boundary measurements, λ = 1 × 10⁻⁴.

Figure 7. Comparison of the results by different levels of perturbations.

Figure 8. Numerical result for problem with non-smooth c(x), Δt = 1/80, h = 1/40.