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

Jone Apraiz; Jin Cheng; Anna Doubova; Enrique Fernández-Cara; Masahiro Yamamoto

doi:10.3934/ipi.2021062

Article Contents

2022, Volume 16, Issue 3: 569-594. Doi: 10.3934/ipi.2021062

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

1.
Universidad del País Vasco, Dpto. Matemáticas, , Barrio Sarriena s/n, 48940 Leioa (Bizkaia), Spain
2.
Fudan University, School of Mathematical Sciences, Shanghai 200433, China
3.
Universidad de Sevilla, Dpto. EDAN e IMUS, Campus Reina Mercedes, 41012 Sevilla, Spain
4.
The University of Tokyo, Department of Mathematical Sciences, 3-8-1 Komaba, Meguro, Tokyo 153, Japan

^*Corresponding author: Jone Apraiz
^*Corresponding author: Jone Apraiz

Received: January 2021

Revised: July 2021

Published: June 2022

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Abstract

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.

Keywords:

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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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

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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

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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

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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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