June  2017, 11(3): 553-575. doi: 10.3934/ipi.2017026

Probabilistic interpretation of the Calderón problem

1. 

Department of Mathematics and Statistics, University of Helsinki, FI-00014 Helsinki, Finland

2. 

Institute of Mathematics, Johannes Gutenberg University, 55128 Mainz, Germany

* Corresponding author

Received  March 2015 Revised  January 2017 Published  April 2017

In this paper, we use the theory of symmetric Dirichlet forms to give a probabilistic interpretation of Calderón's inverse conductivity problem in terms of reflecting diffusion processes and their corresponding boundary trace processes. This probabilistic interpretation comes in three equivalent formulations which open up novel perspectives on the classical question of unique determinability of conductivities from boundary data. We aim to make this work accessible to both readers with a background in stochastic process theory as well as researchers working on deterministic methods in inverse problems.

Citation: Petteri Piiroinen, Martin Simon. Probabilistic interpretation of the Calderón problem. Inverse Problems and Imaging, 2017, 11 (3) : 553-575. doi: 10.3934/ipi.2017026
References:
[1]

M. Aizenman and B. Simon, Brownian Motion and Harnack inequality for Schrödinger Operator, Comm. Pure Appl. Math., 35 (1982), 209-273.  doi: 10.1002/cpa.3160350206.

[2]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Analysis, 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[3]

K. AstalaM. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207-224.  doi: 10.1081/PDE-200044485.

[4]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane., Ann. of Math.(20), 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.

[5]

A. Benchérif-Madani and É. Pardoux, A probabilistic formula for a Poisson equation with Neumann boundary condition, Stoch. Anal. Appl., 27 (2009), 739-746.  doi: 10.1080/07362990902976520.

[6]

P. Caro and K. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum Math. Pi, 4 (2016), e2, 28 pp.  doi: 10.1017/fmp.2015.9.

[7]

Z. Q. ChenM. Fukushima and J. Ying, Traces of symmetric Markov processes and their characterizations, Ann. Probab., 34 (2006), 1052-1102.  doi: 10.1214/009117905000000657.

[8]

D. DosSantosFerreiraC. KenigM. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171.  doi: 10.1007/s00222-009-0196-4.

[9]

N. Falkner and G. Teschl, On the substitution rule for Lebesgue-Stieltjes integrals, Exp. Math., 30 (2012), 412-418.  doi: 10.1016/j.exmath.2012.09.002.

[10]

M. Fukushima, Dirichlet spaces and strong Markov processes, Trans. Amer. Math. Soc., 162 (1971), 185-224.  doi: 10.1090/S0002-9947-1971-0295435-0.

[11]

M. Fukushima, On a decomposition of additive functionals in the strict sense for a symmetric Markov process, in Dirichlet forms and stochastic processes (Beijing, 1993), de Gruyter, Berlin, (1995), 155–169.

[12]

M. Fukushima, Y. Ōshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter & Co. , Berlin, 2011. doi: 10.1515/9783110889741.

[13]

M. Fukushima and M. Tomisaki, Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps, Probab. Theory Related Fields, 106 (1996), 521-557.  doi: 10.1007/s004400050074.

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985.

[15]

B. Haberman, Uniqueness in Calder´on's problem for conductivities with unbounded gradient, Commun. Math. Phys, 340 (2015), 639-659.  doi: 10.1007/s00220-015-2460-3.

[16]

B. Haberman and D. Tataru, Uniqueness in Calder´on's problem with Lipschitz conductivities,, Duke Math. J., 162 (2013), 496-516.  doi: 10.1215/00127094-2019591.

[17]

M. HankeN. Hyvönen and S. Reusswig, Convex backscattering support in electric impedance tomography, Numer. Math., 117 (2011), 373-396.  doi: 10.1007/s00211-010-0320-9.

[18]

P. Hsu, Probabilistic approach to the Neumann problem, Comm. Pure Appl. Math., 38 (1985), 445-472.  doi: 10.1002/cpa.3160380406.

[19]

P. Hsu, On the Poisson kernel for the Neumann problem of Schrödinger operators,, J. London Math. Soc.(2), 36 (1987), 370-384.  doi: 10.1112/jlms/s2-36.2.370.

[20]

P. Hsu, On excursions of reflecting Brownian motion, Trans. Amer. Math. Soc., 296 (1986), 239-264.  doi: 10.1090/S0002-9947-1986-0837810-X.

[21]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, NorthHolland Publishing Co. , Amsterdam, 1981.

[22]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.

[23]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289-298.  doi: 10.1002/cpa.3160370302.

[24]

J.-P. Lepeltier and B. Marchal, Problème des martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel, Ann. Inst. H. Poincaré Sect. B., 12 (1976), 43-103. 

[25]

J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Scuola Norm. Sup. Pisa, 16 (1962), 305-326. 

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

P. Piiroinen and M. Simon, From Feynman-Kac Formulae to Numerical Stochastic Homogenization in Electrical Impedance Tomography, Ann. Appl. Probab., 26 (2016), 3001-3043.  doi: 10.1214/15-AAP1168.

[28]

M. Simon, Anomaly Detection in Random Heterogeneous Media, Springer-Verlag, Berlin, 2015. doi: 10.1007/978-3-658-10993-6.

[29]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

show all references

References:
[1]

M. Aizenman and B. Simon, Brownian Motion and Harnack inequality for Schrödinger Operator, Comm. Pure Appl. Math., 35 (1982), 209-273.  doi: 10.1002/cpa.3160350206.

[2]

G. Alessandrini, Stable determination of conductivity by boundary measurements, Appl. Analysis, 27 (1988), 153-172.  doi: 10.1080/00036818808839730.

[3]

K. AstalaM. Lassas and L. Päivärinta, Calderón's inverse problem for anisotropic conductivity in the plane, Comm. Partial Differential Equations, 30 (2005), 207-224.  doi: 10.1081/PDE-200044485.

[4]

K. Astala and L. Päivärinta, Calderón's inverse conductivity problem in the plane., Ann. of Math.(20), 163 (2006), 265-299.  doi: 10.4007/annals.2006.163.265.

[5]

A. Benchérif-Madani and É. Pardoux, A probabilistic formula for a Poisson equation with Neumann boundary condition, Stoch. Anal. Appl., 27 (2009), 739-746.  doi: 10.1080/07362990902976520.

[6]

P. Caro and K. Rogers, Global uniqueness for the Calderón problem with Lipschitz conductivities, Forum Math. Pi, 4 (2016), e2, 28 pp.  doi: 10.1017/fmp.2015.9.

[7]

Z. Q. ChenM. Fukushima and J. Ying, Traces of symmetric Markov processes and their characterizations, Ann. Probab., 34 (2006), 1052-1102.  doi: 10.1214/009117905000000657.

[8]

D. DosSantosFerreiraC. KenigM. Salo and G. Uhlmann, Limiting Carleman weights and anisotropic inverse problems, Invent. Math., 178 (2009), 119-171.  doi: 10.1007/s00222-009-0196-4.

[9]

N. Falkner and G. Teschl, On the substitution rule for Lebesgue-Stieltjes integrals, Exp. Math., 30 (2012), 412-418.  doi: 10.1016/j.exmath.2012.09.002.

[10]

M. Fukushima, Dirichlet spaces and strong Markov processes, Trans. Amer. Math. Soc., 162 (1971), 185-224.  doi: 10.1090/S0002-9947-1971-0295435-0.

[11]

M. Fukushima, On a decomposition of additive functionals in the strict sense for a symmetric Markov process, in Dirichlet forms and stochastic processes (Beijing, 1993), de Gruyter, Berlin, (1995), 155–169.

[12]

M. Fukushima, Y. Ōshima and M. Takeda, Dirichlet Forms and Symmetric Markov Processes, Walter de Gruyter & Co. , Berlin, 2011. doi: 10.1515/9783110889741.

[13]

M. Fukushima and M. Tomisaki, Construction and decomposition of reflecting diffusions on Lipschitz domains with Hölder cusps, Probab. Theory Related Fields, 106 (1996), 521-557.  doi: 10.1007/s004400050074.

[14]

P. Grisvard, Elliptic Problems in Nonsmooth Domains, Pitman, Boston, MA, 1985.

[15]

B. Haberman, Uniqueness in Calder´on's problem for conductivities with unbounded gradient, Commun. Math. Phys, 340 (2015), 639-659.  doi: 10.1007/s00220-015-2460-3.

[16]

B. Haberman and D. Tataru, Uniqueness in Calder´on's problem with Lipschitz conductivities,, Duke Math. J., 162 (2013), 496-516.  doi: 10.1215/00127094-2019591.

[17]

M. HankeN. Hyvönen and S. Reusswig, Convex backscattering support in electric impedance tomography, Numer. Math., 117 (2011), 373-396.  doi: 10.1007/s00211-010-0320-9.

[18]

P. Hsu, Probabilistic approach to the Neumann problem, Comm. Pure Appl. Math., 38 (1985), 445-472.  doi: 10.1002/cpa.3160380406.

[19]

P. Hsu, On the Poisson kernel for the Neumann problem of Schrödinger operators,, J. London Math. Soc.(2), 36 (1987), 370-384.  doi: 10.1112/jlms/s2-36.2.370.

[20]

P. Hsu, On excursions of reflecting Brownian motion, Trans. Amer. Math. Soc., 296 (1986), 239-264.  doi: 10.1090/S0002-9947-1986-0837810-X.

[21]

N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, NorthHolland Publishing Co. , Amsterdam, 1981.

[22]

I. Karatzas and S. E. Shreve, Brownian Motion and Stochastic Calculus, Springer-Verlag, New York, 1991. doi: 10.1007/978-1-4612-0949-2.

[23]

R. Kohn and M. Vogelius, Determining conductivity by boundary measurements, Comm. Pure Appl. Math., 37 (1984), 289-298.  doi: 10.1002/cpa.3160370302.

[24]

J.-P. Lepeltier and B. Marchal, Problème des martingales et équations différentielles stochastiques associées à un opérateur intégro-différentiel, Ann. Inst. H. Poincaré Sect. B., 12 (1976), 43-103. 

[25]

J. Nečas, Sur une méthode pour résoudre les équations aux dérivées partielles du type elliptique, voisine de la variationnelle, Ann. Scuola Norm. Sup. Pisa, 16 (1962), 305-326. 

[26]

A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer-Verlag, New York, 1983. doi: 10.1007/978-1-4612-5561-1.

[27]

P. Piiroinen and M. Simon, From Feynman-Kac Formulae to Numerical Stochastic Homogenization in Electrical Impedance Tomography, Ann. Appl. Probab., 26 (2016), 3001-3043.  doi: 10.1214/15-AAP1168.

[28]

M. Simon, Anomaly Detection in Random Heterogeneous Media, Springer-Verlag, Berlin, 2015. doi: 10.1007/978-3-658-10993-6.

[29]

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem, Ann. of Math., 125 (1987), 153-169.  doi: 10.2307/1971291.

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