Abstract
This study deals with the numerical implementation of a formula in the enclosure method as applied to a prototype inverse initial boundary value problem for thermal imaging in a one-space dimension. A precise error estimate of the formula is given and the effect on the discretization of the used integral of the measured data in the formula is studied. The formula requires a large frequency to converge; however, the number of time interval divisions grows exponentially as the frequency increases. Therefore, for a given number of divisions, we fixed the trusted frequency region of convergence with some given error bound. The trusted frequency region is computed theoretically using the theorems provided in this paper and is numerically implemented for various cases.
Funding source: Japan Society for the Promotion of Science
Award Identifier / Grant number: Grant-in-Aid for Scientific Research (C) (No. 25400155)
Funding source: National Research Foundation of Korea
Award Identifier / Grant number: NRF-2013R1A1A2010624
Funding statement: This work was partially supported by Grant-in-Aid for Scientific Research (C) (No. 25400155) of Japan Society for the Promotion of Science. Kiwoon Kwon was also supported by the National Research Foundation of Korea (NRF) funded by the Ministry of Education (NRF-2013R1A1A2010624).
References
[1] Atkinson K. E., An Introduction to Numerical Analysis, 2nd ed., John Wiley & Sons, New York, 1989. Search in Google Scholar
[2] Banks H. T. and Kojima F., Boundary shape identification problems in two-dimensional domains related to thermal testing of materials, Quart. Appl. Math. 47 (1989), 273–293. 10.1090/qam/998101Search in Google Scholar
[3] Banks H. T., Kojima F. and Winfree W. P., Boundary estimation problems arising in thermal tomography, Inverse Problems 6 (1990), 897–922. 10.1088/0266-5611/6/6/003Search in Google Scholar
[4] Bryan K. and Caudill, Jr. L. F., An inverse problem in thermal imaging, SIAM J. Appl. Math. 56 (1996), 715–735. 10.1007/978-1-4612-0321-6_5Search in Google Scholar
[5] Bryan K. and Caudill, Jr. L. F., Uniqueness for a boundary identification problem in thermal imaging, Electron. J. Differential Equations 1997 (1997), no. 1, 23–39. Search in Google Scholar
[6] Bryan K. and Caudill L., Reconstruction of an unknown boundary portion from Cauchy data in n dimensions, Inverse Problems 21 (2005), 239–255. 10.1088/0266-5611/21/1/015Search in Google Scholar
[7] Chapko R., Kress R. and Yoon J.-R., An inverse boundary value problem for the heat equation: The Neumann condition, Inverse Problems 15 (1999), 1033–1049. 10.1088/0266-5611/15/4/313Search in Google Scholar
[8] Dautray R. and Lions J.-L., Mathematical Analysis and Numerical Methods for Sciences and Technology. Vol. 5: Evolution Problems I, Springer, Berlin, 1992. Search in Google Scholar
[9] Ikehata M., Extracting discontinuity in a heat conductive body. One-space dimensional case, Appl. Anal. 86 (2007), no. 8, 963–1005. 10.1080/00036810701460834Search in Google Scholar
[10] Ikehata M., The framework of the enclosure method with dynamical data and its applications, Inverse Problems 27 (2011), Article ID 065005. 10.1088/0266-5611/27/6/065005Search in Google Scholar
[11] Ikehata M. and Kawashita M., The enclosure method for the heat equation, Inverse Problems 25 (2009), Article ID 075005. 10.1088/0266-5611/25/7/075005Search in Google Scholar
[12] Ikehata M. and Kawashita M., On the reconstruction of inclusions in a heat conductive body from dynamical boundary data over a finite time interval, Inverse Problems 26 (2010), 095004. 10.1088/0266-5611/26/9/095004Search in Google Scholar
[13] Ikehata M. and Kawashita M., An inverse problem for a three-dimensional heat equation in thermal imaging and the enclosure method, Inverse Probl. Imaging 8 (2014), 1073–1116. 10.3934/ipi.2014.8.1073Search in Google Scholar
[14] Ikehata M. and Ohe T., A numerical method for finding the convex hull of polygonal cavities using enclosure method, Inverse Problems 18 (2002), 111–124. 10.1088/0266-5611/18/1/308Search in Google Scholar
[15] Ikehata M. and Siltanen S., Numerical method for finding the convex hull of an inclusion in conductivity from boundary measurements, Inverse Problems 16 (2000), 1043–1052. 10.1088/0266-5611/16/4/311Search in Google Scholar
[16] Killey A. and Sargent J. P., Analysis of thermal non-destructive testing, J. Phys. D Appl. Phys 22 (1989), no. 1, 81–100. 10.1088/0022-3727/22/1/032Search in Google Scholar
[17] https://en.wikipedia.org/wiki/Riemann_zeta_function. Search in Google Scholar
© 2017 by De Gruyter
Articles in the same Issue
- Frontmatter
- On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid
- Reconstruction of the refractive index from transmission eigenvalues for spherically stratified media
- Error minimizing relaxation strategies in Landweber and Kaczmarz type iterations
- Reciprocity gap method for an interior inverse scattering problem
- Feasibility of parameter estimation in hepatitis C viral dynamics models
- Trusted frequency region of convergence for the enclosure method in thermal imaging
- Sequential subspace optimization for nonlinear inverse problems
- Photoacoustic tomography with spatially varying compressibility and density
Articles in the same Issue
- Frontmatter
- On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid
- Reconstruction of the refractive index from transmission eigenvalues for spherically stratified media
- Error minimizing relaxation strategies in Landweber and Kaczmarz type iterations
- Reciprocity gap method for an interior inverse scattering problem
- Feasibility of parameter estimation in hepatitis C viral dynamics models
- Trusted frequency region of convergence for the enclosure method in thermal imaging
- Sequential subspace optimization for nonlinear inverse problems
- Photoacoustic tomography with spatially varying compressibility and density