Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results

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Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Stochastic extension of the boundary map

August 2015, 9(3): 749-766. doi: 10.3934/ipi.2015.9.749

Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results

Daniela Calvetti ^1, , Paul J. Hadwin ^2, , Janne M. J. Huttunen ^3, , David Isaacson ^4, , Jari P. Kaipio ^2, , Debra McGivney ^5, , Erkki Somersalo ^1, and Joseph Volzer ^1,

1.	Case Western Reserve University, Department of Mathematics, Applied Mathematics, and Statistics, Cleveland, OH 44106, United States, United States, United States
2.	University of Auckland, Department of Mathematics, Auckland, New Zealand, New Zealand
3.	University of Eastern Finland, Department of Applied Physics, Kuopio, Finland
4.	Rensselaer Polytechnic Institute, Department of Mathematics, Troy, NY 12180, United States
5.	Case Western Reserve University, Department of Radiology, Cleveland, OH 44106, United States

Received April 2014 Revised January 2015 Published July 2015

Abstract
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Artificial boundary conditions have long been an active research topic in numerical approximation of scattering waves: The truncation of the computational domain and the assignment of the conditions along the fictitious boundary must be done so that no spurious reflections occur. In inverse boundary value problems, a similar problem appears when the estimation of the unknowns is restricted to a domain that represents the whole domain of the solutions of a partial differential equation with unknown coefficient. This problem is significantly more challenging than general scattering problems, because the coefficients representing the unknown material parameter of interest are not known in the truncated portion and assigning suitable condition on the fictitious boundary is part of the problem also. The problem is addressed by defining a Dirichlet-to-Neumann map, or Steklov-Poincaré map, on the boundary of the domain truncation. In this paper we describe the procedure, provide a theoretical justification and illustrate with computed examples the limitations of imposing fixed boundary condition. Extensions of the proposed approach will be presented in a sequel article.

Keywords: Domain decomposition, Dirichlet-to-Neumann, coercivity, Steklov-Poincaré, electrode model..

Mathematics Subject Classification: Primary: 35J20, 35R30, 65M32, 65M5.

Citation: Daniela Calvetti, Paul J. Hadwin, Janne M. J. Huttunen, David Isaacson, Jari P. Kaipio, Debra McGivney, Erkki Somersalo, Joseph Volzer. Artificial boundary conditions and domain truncation in electrical impedance tomography. Part I: Theory and preliminary results. Inverse Problems & Imaging, 2015, 9 (3) : 749-766. doi: 10.3934/ipi.2015.9.749

References:

[1]	D. Calvetti, J. P. Kaipio and E. Somersalo, Aristotelian prior boundary conditions,, Int. J. Math. Comp. Sci., 1 (2006), 63.
[2]	D. Calvetti and E. Somersalo, Statistical compensation of boundary clutter in image deblurring,, Inverse Problems, 21 (2005), 1697. doi: 10.1088/0266-5611/21/5/012.
[3]	D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing - Ten Lectures on Subjective Computing,, Springer Verlag, (2007).
[4]	D. Calvetti, P. J, Hadwin, J. M. J. Huttunen, J. P. Kaipio and E. Somersalo, Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Computational results,, Inv. Probl. Imaging., 12 (2015).
[5]	D. Givoli, Numerical Methods for Problems in Infinite Domains,, Elsevier, (1992).
[6]	D. Givoli, Recent advances in the DtN FE method,, Arch. Comput. Meth. Engin., 6 (1999), 71. doi: 10.1007/BF02736182.
[7]	P. Grisvard, Elliptic Boundary Value Problems in Non-Smooth Domains,, SIAM, (2011).
[8]	M. J. Grote and J. B. Keller, On nonreflecting boundary conditions,, J. Comp. Phys., 122* (1995), 231. doi: 10.1006/jcph.1995.1210.*
[9]	M. J. Grote and C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems,, J. Comp. Phys., 201* (2004), 630. doi: 10.1016/j.jcp.2004.06.012.*
[10]	E. Jonsson, Partial Dirichlet to Neumann Maps in the Approximate Reconstruction of Conductivity Distribution,, PhD Thesis, (1997).
[11]	E. Jonsson, Electrical conductivity reconstruction using nonlocal boundary conditions,, SIAM J. Appl. Math., 59 (1999), 1582. doi: 10.1137/S0036139997327770.
[12]	T.-J. Kao, G. J. Saulnier, H. Xia, C. Tamma, J. C. Newell and D. Isaacson, A compensated radiolucent electrode array for combined EIT and mammography,, Physiol. Meas., 28 (2007). doi: 10.1088/0967-3334/28/7/S22.
[13]	B. S. Kim, G. Boverman, J. C. Newell, G. J. Saulnier and D. Isaacson, The complete electrode model for EIT in a mammography geometry,, Physiol. Meas., 28 (2007). doi: 10.1088/0967-3334/28/7/S05.
[14]	J. B. Keller and D. Givoli, Exact non-reflecting boundary conditions,, J. Comp. Phys., 82 (1989), 172. doi: 10.1016/0021-9991(89)90041-7.
[15]	M. H. Loke and R. D. Barker, Practical techniques for 3D resistivity surveys and data inversion,, Geophys. Prospecting, 44 (1996), 499.
[16]	D. Calvetti, D. McGivney and E. Somersalo, Left and right preconditioning for electrical impedance tomography with structural information,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/5/055015.
[17]	D. McGivney, D. Calvetti and E. Somersalo, Quantitative imaging with electrical impedance spectroscopy,, Phys. Med. Biol., 57 (2012). doi: 10.1088/0031-9155/57/22/7289.
[18]	D. McGivney, Statistical Preconditioners and Quantitative Imaging in Electrical Impedance Tomography,, PhD Thesis, (2013).
[19]	A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations,, Oxford University Press, (1999).
[20]	E. Somersalo, D. Isaacson and M. Cheney, Existence and uniqueness for electrode models for electric current computed tomography,, SIAM J. Appl. Math. 52 (1992), 52 (1992), 1023. doi: 10.1137/0152060.
[21]	H. Triebel, Interpolation Theory, Function Spaces Differential Operators,, 2nd ed. Barth, (1995).
[22]	Y. Zou and Z. Guo, A review of electrical impedance techniques for breast cancer detection,, Med. Eng. Phys., 25 (2003), 79. doi: 10.1016/S1350-4533(02)00194-7.

show all references

References:

[1]	D. Calvetti, J. P. Kaipio and E. Somersalo, Aristotelian prior boundary conditions,, Int. J. Math. Comp. Sci., 1 (2006), 63.
[2]	D. Calvetti and E. Somersalo, Statistical compensation of boundary clutter in image deblurring,, Inverse Problems, 21 (2005), 1697. doi: 10.1088/0266-5611/21/5/012.
[3]	D. Calvetti and E. Somersalo, Introduction to Bayesian Scientific Computing - Ten Lectures on Subjective Computing,, Springer Verlag, (2007).
[4]	D. Calvetti, P. J, Hadwin, J. M. J. Huttunen, J. P. Kaipio and E. Somersalo, Artificial boundary conditions and domain truncation in electrical impedance tomography. Part II: Computational results,, Inv. Probl. Imaging., 12 (2015).
[5]	D. Givoli, Numerical Methods for Problems in Infinite Domains,, Elsevier, (1992).
[6]	D. Givoli, Recent advances in the DtN FE method,, Arch. Comput. Meth. Engin., 6 (1999), 71. doi: 10.1007/BF02736182.
[7]	P. Grisvard, Elliptic Boundary Value Problems in Non-Smooth Domains,, SIAM, (2011).
[8]	M. J. Grote and J. B. Keller, On nonreflecting boundary conditions,, J. Comp. Phys., 122* (1995), 231. doi: 10.1006/jcph.1995.1210.*
[9]	M. J. Grote and C. Kirsch, Dirichlet-to-Neumann boundary conditions for multiple scattering problems,, J. Comp. Phys., 201* (2004), 630. doi: 10.1016/j.jcp.2004.06.012.*
[10]	E. Jonsson, Partial Dirichlet to Neumann Maps in the Approximate Reconstruction of Conductivity Distribution,, PhD Thesis, (1997).
[11]	E. Jonsson, Electrical conductivity reconstruction using nonlocal boundary conditions,, SIAM J. Appl. Math., 59 (1999), 1582. doi: 10.1137/S0036139997327770.
[12]	T.-J. Kao, G. J. Saulnier, H. Xia, C. Tamma, J. C. Newell and D. Isaacson, A compensated radiolucent electrode array for combined EIT and mammography,, Physiol. Meas., 28 (2007). doi: 10.1088/0967-3334/28/7/S22.
[13]	B. S. Kim, G. Boverman, J. C. Newell, G. J. Saulnier and D. Isaacson, The complete electrode model for EIT in a mammography geometry,, Physiol. Meas., 28 (2007). doi: 10.1088/0967-3334/28/7/S05.
[14]	J. B. Keller and D. Givoli, Exact non-reflecting boundary conditions,, J. Comp. Phys., 82 (1989), 172. doi: 10.1016/0021-9991(89)90041-7.
[15]	M. H. Loke and R. D. Barker, Practical techniques for 3D resistivity surveys and data inversion,, Geophys. Prospecting, 44 (1996), 499.
[16]	D. Calvetti, D. McGivney and E. Somersalo, Left and right preconditioning for electrical impedance tomography with structural information,, Inverse Problems, 28 (2012). doi: 10.1088/0266-5611/28/5/055015.
[17]	D. McGivney, D. Calvetti and E. Somersalo, Quantitative imaging with electrical impedance spectroscopy,, Phys. Med. Biol., 57 (2012). doi: 10.1088/0031-9155/57/22/7289.
[18]	D. McGivney, Statistical Preconditioners and Quantitative Imaging in Electrical Impedance Tomography,, PhD Thesis, (2013).
[19]	A. Quarteroni and A. Valli, Domain Decomposition Methods for Partial Differential Equations,, Oxford University Press, (1999).
[20]	E. Somersalo, D. Isaacson and M. Cheney, Existence and uniqueness for electrode models for electric current computed tomography,, SIAM J. Appl. Math. 52 (1992), 52 (1992), 1023. doi: 10.1137/0152060.
[21]	H. Triebel, Interpolation Theory, Function Spaces Differential Operators,, 2nd ed. Barth, (1995).
[22]	Y. Zou and Z. Guo, A review of electrical impedance techniques for breast cancer detection,, Med. Eng. Phys., 25 (2003), 79. doi: 10.1016/S1350-4533(02)00194-7.

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