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# Uniqueness and numerical reconstruction for inverse problems dealing with interval size search

• *Corresponding author: Jone Apraiz

• We consider a heat equation and a wave equation in one spatial dimension. This article deals with the inverse problem of determining the size of the spatial interval from some extra boundary information on the solution. Under several different circumstances, we prove uniqueness, non-uniqueness and some size estimates. Moreover, we numerically solve the inverse problems and compute accurate approximations of the size. This is illustrated with several satisfactory numerical experiments.

Mathematics Subject Classification: Primary: 35K05, 35L05, 35R30; Secondary: 65M32.

 Citation: • • Figure 1.  Heat equation, $u_0 = 0$ and $\eta\neq 0$. The computed solution

Figure 2.  Heat equation with $u_0 = 0$ and $\eta\neq 0$

Figure 3.  Heat equation, $u_0\neq 0$ and large $\eta$. The computed solution

Figure 4.  Heat equation, fixed $u_0$ and large $\eta$

Figure 5.  Heat equation, $\eta = 0$ and fixed $u_0(x)$

Figure 6.  Heat equation, $\eta = 0$, fixed $u_0(x)$

Figure 7.  Heat equation, fixed $u_0$ and $\eta = 0$

Figure 8.  Wave equation with $(u_0, u_1) = (0, 0)$ and $\eta \not = 0$. The computed solution

Figure 9.  Wave equation with $(u_0, u_1) = (0, 0)$ and $\eta \not = 0$

Figure 10.  Wave equation with $(u_0, u_1) \not = (0, 0)$ and large $\eta = 0$. The computed solution

Figure 11.  Wave equation with $(u_0, u_1) \not = (0, 0)$ and large $\eta = 0$

Figure 12.  Wave equation, $\eta = 0$, fixed $u_0(x)$

Figure 13.  Wave equation, $\eta = 0$, fixed $u_0(x)$

Figure 14.  Wave equation, fixed $u_0$, $\eta = 0$

Figure 15.  Wave equation, fixed $u_0(x)$, $\eta = 0$

Figure 16.  The non-uniqueness cases for the two equations

Table 1.  Heat equation, $u_0 = 0$ and $\eta\neq 0$. Results with random noise in the target (the desired length is $L_d = 2$)

 % noise Cost Iterates Computed $L_c$ 1% 1.e-4 10 1.997586488 0.1% 1.e-6 9 1.999864829 0.01% 1.e-9 8 2.000017283 0.001% 1.e-1 9 1.999998535 0% 1.e-16 9 1.999999991

Table 2.  Heat equation, fixed $u_0$ and large $\eta$. Results with random noise in the target (the desired length is $L_d = 2$)

 % noise Cost Iterates Computed $L_c$ 1% 1.e-3 8 1.999952948 0.1% 1.e-5 11 2.000008948 0.01% 1.e-8 9 2.000003174 0.001% 1.e-10 7 1.999999932 0% 1.e-12 13 1.999999964

Table 3.  Wave equation, $(u_0, u_1) = (0, 0)$ and $\eta \not = 0$. Results with random noise in the target {(the desired length is $L_d = 2$)

 % noise Cost Iterates Computed $L_c$ 1% 1.e-4 12 2.004287790 0.1% 1.e-6 8 2.000029532 0.01% 1.e-8 8 2.000018464 0.001% 1.e-10 9 1.999995613 0% 1.e-17 8 1.999999994

Table 4.  Wave equation, $(u_0, u_1) \not = (0, 0)$ and large $\eta = 0$. Cost and $L_c$ with random noise in the target {(desired length is $L_d = 2$)

 % noise Cost Iterates Computed $L_c$ 1% 1.e-1 12 2.324511735 0.1% 1.e-6 11 2.000008724 0.01% 1.e-7 9 1.999805367 0.001% 1.e-12 10 2.000000789 0% 1.e-17 14 1.999931083
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Tables(4)

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