A wavelet multiscale-adaptive homotopy method for the inverse problem of nonlinear diffusion equation

Figures & data

Figure 1. Simulation geometry with the locations of producer points and injector points.

Figure 2. The errors of numerical solution and ground truth with respect to the iteration numbers with different initial values in Example 5.1: (a) the initial value is 0.17; (b) the initial value is 2.45; (c) the initial value is 4.15; (d) the initial value is 6.58.

Figure 3. The numerical solutions with respect to the iteration numbers with the initial value 0.17 at some grid points in Example 5.1: (a) the grid point is (1/8, 1/8); (b) the grid point is (1/8, 7/8); (c) the grid point is (7/8, 1/8); (d) the grid point is (7/8, 7/8).

Figure 4. The numerical solutions with respect to the iteration numbers with the initial value 2.45 at some grid points in Example 5.1: (a) the grid point is (1/8, 1/8); (b) the grid point is (1/8, 7/8); (c) the grid point is (7/8, 1/8); (d) the grid point is (7/8, 7/8).

Figure 5. The true model based on (2.22.2 $$\begin{array}{c}\hfill u_t-\mathrm{\nabla }·(q(\mathbf{x})N(\mathrm{\nabla }u)\mathrm{\nabla }u)=s(\mathbf{x},t),(\mathbf{x},t)\in \mathrm{\Omega }\times (0,T),\end{array}$$2.2 ) and inversion results $q^{\ast }$ with the noise level $\mathit{ϵ}=0.05$ in Example 5.2: (a) the true model based on (2.22.2 $$\begin{array}{c}\hfill u_t-\mathrm{\nabla }·(q(\mathbf{x})N(\mathrm{\nabla }u)\mathrm{\nabla }u)=s(\mathbf{x},t),(\mathbf{x},t)\in \mathrm{\Omega }\times (0,T),\end{array}$$2.2 ); (b) the inversion result $q^{\ast }$ for $N(\mathrm{\nabla }u)=1+{0.1|\mathrm{\nabla }u|}^2$; (c) the inversion result $q^{\ast }$ for $N(\mathrm{\nabla }u)=1/(1-0.1|\mathrm{\nabla }u{|}^2)$.

Figure 6. The true model based on (2.12.1 $$\begin{array}{c}\hfill u_t-\mathrm{\nabla }·(q(\mathbf{x})N(u)\mathrm{\nabla }u)=s(\mathbf{x},t),(\mathbf{x},t)\in \mathrm{\Omega }\times (0,T),\end{array}$$2.1 ) and inversion results $q^{\ast }$ with the noise level $\mathit{ϵ}=0.05$ in Example 5.3: (a) the true model based on (2.12.1 $$\begin{array}{c}\hfill u_t-\mathrm{\nabla }·(q(\mathbf{x})N(u)\mathrm{\nabla }u)=s(\mathbf{x},t),(\mathbf{x},t)\in \mathrm{\Omega }\times (0,T),\end{array}$$2.1 ); (b) the inversion result $q^{\ast }$ for $N(u)=u^4-u^3+u^2-u+1$; (c) the inversion result $q^{\ast }$ for $N(u)=u^4+u^3+u^2+u+1$.

Figure 7. The true model based on (2.12.1 $$\begin{array}{c}\hfill u_t-\mathrm{\nabla }·(q(\mathbf{x})N(u)\mathrm{\nabla }u)=s(\mathbf{x},t),(\mathbf{x},t)\in \mathrm{\Omega }\times (0,T),\end{array}$$2.1 ) in Example 5.4.

Figure 8. The inversion result $q^{\ast }$ at the original scale with the noise level $\mathit{ϵ}=0.01$ in Example 5.4.

Figure 9. The true model based on (2.22.2 $$\begin{array}{c}\hfill u_t-\mathrm{\nabla }·(q(\mathbf{x})N(\mathrm{\nabla }u)\mathrm{\nabla }u)=s(\mathbf{x},t),(\mathbf{x},t)\in \mathrm{\Omega }\times (0,T),\end{array}$$2.2 ) in Example 5.5.

Figure 10. The inversion result $q^{\ast }$ at the original scale with the noise level $\mathit{ϵ}=0.01$ in Example 5.5.

Figure 11. The true model based on (2.12.1 $$\begin{array}{c}\hfill u_t-\mathrm{\nabla }·(q(\mathbf{x})N(u)\mathrm{\nabla }u)=s(\mathbf{x},t),(\mathbf{x},t)\in \mathrm{\Omega }\times (0,T),\end{array}$$2.1 ) in Example 5.6.

Figure 12. The inversion results $q^{\ast }$ with the noise level $\mathit{ϵ}=0.04$ in Example 5.6: (a) the inversion result $q^{\ast }$ at scale 1; (b) the inversion result $q^{\ast }$ at scale 0.

Table 1. The errors of numerical solution and ground truth by WAH, WM and AH with the initial value 0.17 in Example 5.1.

Iteration number	WAH	WM	AH
1	10.0039	9.7011	10.2474
2	9.1854	8.6512	10.1695
3	7.7769	6.9255	10.0695
4	5.6471	4.5503	9.9362
5	3.1213	2.0821	9.7486
6	1.0361	0.4841	9.4676
7	0.1289	0.0296	9.0165
8	0.0021	$1.1705\times {10}^{-4}$	8.2459
9	$2.6095\times {10}^{-7}$	$2.4680\times {10}^{-8}$	6.8729
10	$5.7368\times {10}^{-11}$	$5.6682\times {10}^{-12}$	4.4157
11	$1.3339\times {10}^{-14}$	$7.5187\times {10}^{-16}$	1.9299
12	$1.4239\times {10}^{-15}$	$1.1123\times {10}^{-15}$	0.4075
13	$9.7081\times {10}^{-16}$	$8.8817\times {10}^{-16}$	0.0203
14	$7.2875\times {10}^{-16}$	$6.1328\times {10}^{-16}$	$4.8887\times {10}^{-5}$
15	$7.4638\times {10}^{-16}$	$4.9649\times {10}^{-16}$	$1.0874\times {10}^{-8}$

Table 2. The errors of numerical solution and ground truth by WAH, WM and AH with the initial value 2.45 in Example 5.1.

Iteration number	WAH	WM	AH
1	3.2287	2.0417	3.5632
2	2.5617	1.8221	3.1935
3	1.1558	1.5371	2.8413
4	0.5521	1.1836	2.4964
5	0.2077	0.6964	2.1467
6	0.0039	0.2145	1.7830
7	$3.2263\times {10}^{-6}$	0.0211	1.3972
8	$8.3792\times {10}^{-10}$	$2.4191\times {10}^{-4}$	0.9818
9	$2.4202\times {10}^{-13}$	$1.3159\times {10}^{-7}$	0.5289
10	$1.4201\times {10}^{-15}$	$9.2188\times {10}^{-11}$	0.0319
11	$1.0793\times {10}^{-15}$	$6.4001\times {10}^{-14}$	$1.2034\times {10}^{-4}$
12	$1.3798\times {10}^{-15}$	$2.0261\times {10}^{-15}$	$2.6732\times {10}^{-8}$
13	$1.2429\times {10}^{-15}$	$1.1633\times {10}^{-15}$	$6.9694\times {10}^{-12}$

Table 3. The errors of numerical solution and ground truth by WAH, WM and AH with the initial value 4.15 in Example 5.1.

Iteration number	WAH	WM	AH
1	4.1052	12.9611	5.8474
2	3.2551	$\infty$	4.6877
3	0.7618	$\infty$	3.6352
4	0.6079	$\infty$	2.8049
5	0.2067	$\infty$	2.1178
6	0.0041	$\infty$	1.5461
7	$3.2381\times {10}^{-6}$	$\infty$	1.0642
8	$8.3425\times {10}^{-10}$	$\infty$	0.6526
9	$2.3170\times {10}^{-13}$	$\infty$	0.2967
10	$1.6043\times {10}^{-15}$	$\infty$	0.0153
11	$1.3068\times {10}^{-15}$	$\infty$	$4.2494\times {10}^{-5}$
12	$1.6677\times {10}^{-15}$	$\infty$	$2.6573\times {10}^{-8}$

Table 4. The errors of numerical solution and ground truth by WAH, WM and AH with the initial value 6.58 in Example 5.1.

Iteration number	WAH	WM	AH
1	6.7614	47.6033	11.4152
2	4.9733	$\infty$	9.2323
3	0.9618	$\infty$	6.9785
4	0.5663	$\infty$	5.3391
5	0.2092	$\infty$	4.0014
6	0.0041	$\infty$	2.9099
7	$3.3133\times {10}^{-6}$	$\infty$	1.9987
8	$8.5869\times {10}^{-10}$	$\infty$	1.2243
9	$2.3371\times {10}^{-13}$	$\infty$	0.5554
10	$2.2577\times {10}^{-15}$	$\infty$	0.0307
11	$1.2000\times {10}^{-15}$	$\infty$	$9.9211\times {10}^{-5}$
12	$2.0861\times {10}^{-15}$	$\infty$	$4.7145\times {10}^{-8}$
13	$1.1922\times {10}^{-15}$	$\infty$	$2.6898\times {10}^{-11}$

Table 5. The required iteration numbers by WAH, WM and AH in Example 5.1.

Initial value	WAH	WM	AH
0.17	10	10	15
2.45	8	10	13
4.15	8	No convergence	12
6.58	8	No convergence	13
8.23	10	No convergence	13
9.99	10	No convergence	13

Table 6. The required CPU times (in seconds) by WAH, WM and AH in Example 5.1.

Initial value	WAH	WM	AH
0.17	14.73	14.39	22.17
2.45	11.82	15.02	18.84
4.15	11.78	No convergence	16.93
6.58	11.93	No convergence	18.78
8.23	14.30	No convergence	18.87
9.99	14.51	No convergence	18.86