April  2022, 16(2): 283-304. doi: 10.3934/ipi.2021050

Variational source conditions for inverse Robin and flux problems by partial measurements

1. 

School of Mathematics and Statistics, Central China Normal University, Wuhan 430079, China

2. 

Universität Duisburg-Essen, Fakultät für Mathematik, Thea-Leymann-Str. 9, D-45127 Essen, Germany

3. 

Department of Mathematics, The Chinese University of Hong Kong, Shatin, Hong Kong, China

* Corresponding author: Jun Zou

Received  February 2021 Revised  May 2021 Published  April 2022 Early access  July 2021

Fund Project: The first author is supported by National Natural Science Foundation of China (Nos. 11701205 and 11871240). The second author is supported by National Natural Science Foundation of China (No. 11871240), NSFC-RGC (China-Hong Kong, No. 11661161017) and self-determined research funds of CCNU from the colleges' basic research and operation of MOE (No. CCNU20TS003). The third author is supported by the German Research Foundation (DFG grants YO159/2-2 and YO159/4-1). The fourth author is supported by Hong Kong RGC grant (project 14306719) and the NSFC/Hong Kong RGC Joint Research Scheme 2016/17 (N_CUHK437/16)

This work is devoted to the convergence analysis of the Tikhonov regularization for the inverse Robin and flux problems.Both inverse problems aim at recovering a respective physical quantity on an inaccessible part of the boundary through some measurement on a partial accessible boundary.The convergence and convergence rate in the desirable L2-norm are derived based on two new logarithmic type stabilities (only in some weak norms,e.g.,the negative Sobolev norms), which enable us to construct and rigorously verify the required variational source conditions.

 

Addendum: The abstract is added since it was missing when the article was published online. To reflect this addition an Addendum is published in this same IPI 16-2 April 2022 regular issue. We apologize for any inconvenience this may cause.

Citation: De-Han Chen, Daijun Jiang, Irwin Yousept, Jun Zou. Variational source conditions for inverse Robin and flux problems by partial measurements. Inverse Problems and Imaging, 2022, 16 (2) : 283-304. doi: 10.3934/ipi.2021050
References:
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G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004. doi: 10.1088/0266-5611/25/12/123004.

[2]

O. M. Alifanov, Inverse Heat Transfer Problems, Springer, Berlin, 1994. doi: 10.1007/978-3-642-76436-3.

[3]

S. W. Anzengruber, B. Hofmann and R. Ramlau, On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization, Inverse Problems, 29 (2013), 125002. doi: 10.1088/0266-5611/29/12/125002.

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R. I. Boţ and B. Hofmann, An extension of the variational inequality approach for nonlinear ill-posed problems, J. Intergral Equ. Appl., 22 (2010), 369-392. doi: 10.1216/JIE-2010-22-3-369.

[5]

L. Bourgeois, About stability and regularization of ill-posed Cauchy problems: the case of $C^{1, 1}$ domain, Model. Math. Anal. Numer., 44 (2010), 715-735.  doi: 10.1051/m2an/2010016.

[6]

M. Burger, J. Flemming and B. Hofmann, Convergence rates in $\ell^1$-regularization if the sparsity assumption fails, Inverse Problems, 29 (2013), 025013. doi: 10.1088/0266-5611/29/2/025013.

[7]

D.-H. Chen, B. Hofmann and I. Yousept, Oversmoothing Tikhonov regularization in Banach spaces, Inverse Problems, 37 (2021), 085007. doi: 10.1088/1361-6420/abcea0.

[8]

D.-H. Chen, B. Hofmann and J. Zou, Elastic-net regularization versus $\ell^1$-regularization for linear inverse problems with quasi-sparse solutions, Inverse Problem, 33 (2017), 015004. doi: 10.1088/1361-6420/33/1/015004.

[9]

D.-H. Chen, D. J. Jiang and J. Zou, Convergence rates of Tikhonov regularizations for elliptic and parabolic inverse radiativity problems, Inverse Problem, 36 (2020), 075001, 21 pp. doi: 10.1088/1361-6420/ab8449.

[10]

D.-H. Chen and I. Yousept, Variational source condition for ill-posed backward nonlinear Maxwell's equations, Inverse Problem, 35 (2019), 025001. doi: 10.1088/1361-6420/aaeebe.

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D.-H. Chen and I. Yousept, Variational source conditions in $L^p$-spaces, SIAM J. Math. Anal., 53 (2021), 2963-2889.  doi: 10.1137/20M1334462.

[12]

M. Choulli, An inverse problem in corrosion detection: Stability estimates, J. Inverse Ill-Posed Probl., 12 (2004), 349-367.  doi: 10.1515/1569394042248247.

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M. C. Delfour and J.-P. Zolésio, Shapes and Geometries, SIAM, USA, 2001.

[14]

H. W. EnglK. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5 (1989), 523-540.  doi: 10.1088/0266-5611/5/4/007.

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H. W. Engl and J. Zou, A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction, Inverse Problems, 16 (2000), 1907-1923.  doi: 10.1088/0266-5611/16/6/319.

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J. Flemming, Theory and examples of variational regularization with non-metric fitting functionals, J. Inverse Ill-Posed Probl., 18 (2010), 677-699.  doi: 10.1515/JIIP.2010.031.

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M. Grasmair, Generalized Bregman distances and convergence rates for non-convex regularization methods, Inverse Problems, 26 (2010), 115014. doi: 10.1088/0266-5611/26/11/115014.

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D. N. Hao and T. N. T. Quyen, Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equations, Inverse Problems, 26 (2010), 125014, 20pp. doi: 10.1088/0266-5611/26/12/125014.

[20]

B. HofmannB. KaltenbacherC. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.  doi: 10.1088/0266-5611/23/3/009.

[21]

B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Problems, 28 (2012), 104006. doi: 10.1088/0266-5611/28/10/104006.

[22]

T. Hohage and F. Weilding, Verification of a variational source condition for acoustic inverse medium scattering problems, Inverse Problems, 31 (2015), 075006. doi: 10.1088/0266-5611/31/7/075006.

[23]

T. Hohage and F. Weilding, Characerizations of variational source conditions, converse results, and maxisets of spectral regularization methods, SIAM J. Numer. Anal., 55 (2017), 598-620.  doi: 10.1137/16M1067445.

[24]

T. Hohage and F. Weilding, Variational source condition and stability estimates for inverse electromagnetic medium scattering problems, Inverse Probl. Imaging, 11 (2017), 203-220.  doi: 10.3934/ipi.2017010.

[25]

K. Ito and B. Jin, A new approach to nonlinear constrained Tikhonov regularization, Inverse Problems, 27 (2011), 105005. doi: 10.1088/0266-5611/27/10/105005.

[26]

D. J. Jiang, H. Feng and J. Zou, Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system, Inverse Problems, 28 (2012), 104002. doi: 10.1088/0266-5611/28/10/104002.

[27]

B. Jin and J. Zou, Numerical estimation of piecewise constant Robin coefficient, SIAM J. Control Optim., 48 (2009), 1977-2002.  doi: 10.1137/070710846.

[28]

B. Jin and J. Zou, Numerical estimation of the Robin coefficient in a stationary diffusion equation, IMA J. Numer. Anal., 30 (2010), 677-701.  doi: 10.1093/imanum/drn066.

[29]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-585.  doi: 10.1002/cpa.3160130402.

[30]

A. J. KassabE. Divo and J. S. Kapat, Multi-dimensional heat flux reconstruction using narrow-band thermochromic liquid crystal thermography, Inverse Probl. Sci. Eng., 9 (2001), 537-559.  doi: 10.1080/174159701088027780.

[31]

P. K$\ddot{u}$gler, Identification of a temperature dependent heat conductivity from single boundary measurements, SIAM J. Numer. Anal., 41 (2003), 1543-1563.  doi: 10.1137/S0036142902415900.

[32]

K. F. Lam and I. Yousept, Consistency of a phase field regularisation for an inverse problem governed by a quasilinear Maxwell system, Inverse Problems, 36 (2020), 045011. doi: 10.1088/1361-6420/ab6f9f.

[33]

D. Le and H. Smith, Strong positivity of solutions to parabolic and elliptic equations on nonsmooth domains, J. Math. Anal. Appl., 275 (2002), 208-221.  doi: 10.1016/S0022-247X(02)00314-1.

[34]

J. Li, J. Xie and J. Zou, An adaptive finite element reconstruction of distributed fluxes, Inverse Problems, 27 (2011), 075009. doi: 10.1088/0266-5611/27/7/075009.

[35]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer Science & Business Media, 2012.

[36]

T. J. Martin and G. S. Dulikravich, Inverse determination of steady heat convection coefficient distribution, J. Heat Transfer., 120 (1998), 328-334.  doi: 10.1115/1.2824251.

[37]

A. M. Osman and J. V. Beck, Nonlinear inverse problem for the estimation of time-and-space dependent heat transfer coefficients, J. Thérmophys, 3 (1989), 146-152.  doi: 10.2514/3.141.

[38]

L. E. Payne, On a priori bounds in the Cauchy problem for elliptic equations, SIAM J. Math. Anal., 1 (1970), 82-89.  doi: 10.1137/0501008.

[39]

C. Pechstein, Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems, Lecture Notes in Computational Science and Engineering, 90. Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-23588-7.

[40]

A. J. Pryde, Second order elliptic equations with mixed boundary conditions, J. Math. Anal. Appl., 80 (1981), 203-244.  doi: 10.1016/0022-247X(81)90102-5.

[41]

G. Savaré, Regularity and perturbation results for mixed second order elliptic problems, Commun. Partial. Differ. Equ., 22 (1997), 869-899.  doi: 10.1080/03605309708821287.

[42]

S. A. Sauter and C. Schwab, Boundary Element Methods, volume 39 of Springer Series in Compuational Mathematics, Berlin: Springer, 2011. doi: 10.1007/978-3-540-68093-2.

[43]

W. Sickel, Superposition of functions in Sobolev spaces of fractional order, Partial Dif Equ., 27 (1992), 481-497. 

[44]

J. Tambača, Estimates of the Sobolev norm of a product of two functions, J. Math. Anal. Appl., 255 (2001), 137-146.  doi: 10.1006/jmaa.2000.7209.

[45]

M. E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981.

[46]

F. Werner and B. Hofmann, Convergence analysis of (statistical) inverse problems under conditional stability estimates, Inverse Problems, 36 (2020), 015004. doi: 10.1088/1361-6420/ab4cd7.

[47]

F. M. White, Heat and Mass Transfer, Addison-Wesley, Reading, MA, 1988.

[48] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.
[49]

J. Xie and J. Zou, Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal., 43 (2005), 1504-1535.  doi: 10.1137/030602551.

[50]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.

show all references

References:
[1]

G. Alessandrini, L. Rondi, E. Rosset and S. Vessella, The stability for the Cauchy problem for elliptic equations, Inverse Problems, 25 (2009), 123004. doi: 10.1088/0266-5611/25/12/123004.

[2]

O. M. Alifanov, Inverse Heat Transfer Problems, Springer, Berlin, 1994. doi: 10.1007/978-3-642-76436-3.

[3]

S. W. Anzengruber, B. Hofmann and R. Ramlau, On the interplay of basis smoothness and specific range conditions occurring in sparsity regularization, Inverse Problems, 29 (2013), 125002. doi: 10.1088/0266-5611/29/12/125002.

[4]

R. I. Boţ and B. Hofmann, An extension of the variational inequality approach for nonlinear ill-posed problems, J. Intergral Equ. Appl., 22 (2010), 369-392. doi: 10.1216/JIE-2010-22-3-369.

[5]

L. Bourgeois, About stability and regularization of ill-posed Cauchy problems: the case of $C^{1, 1}$ domain, Model. Math. Anal. Numer., 44 (2010), 715-735.  doi: 10.1051/m2an/2010016.

[6]

M. Burger, J. Flemming and B. Hofmann, Convergence rates in $\ell^1$-regularization if the sparsity assumption fails, Inverse Problems, 29 (2013), 025013. doi: 10.1088/0266-5611/29/2/025013.

[7]

D.-H. Chen, B. Hofmann and I. Yousept, Oversmoothing Tikhonov regularization in Banach spaces, Inverse Problems, 37 (2021), 085007. doi: 10.1088/1361-6420/abcea0.

[8]

D.-H. Chen, B. Hofmann and J. Zou, Elastic-net regularization versus $\ell^1$-regularization for linear inverse problems with quasi-sparse solutions, Inverse Problem, 33 (2017), 015004. doi: 10.1088/1361-6420/33/1/015004.

[9]

D.-H. Chen, D. J. Jiang and J. Zou, Convergence rates of Tikhonov regularizations for elliptic and parabolic inverse radiativity problems, Inverse Problem, 36 (2020), 075001, 21 pp. doi: 10.1088/1361-6420/ab8449.

[10]

D.-H. Chen and I. Yousept, Variational source condition for ill-posed backward nonlinear Maxwell's equations, Inverse Problem, 35 (2019), 025001. doi: 10.1088/1361-6420/aaeebe.

[11]

D.-H. Chen and I. Yousept, Variational source conditions in $L^p$-spaces, SIAM J. Math. Anal., 53 (2021), 2963-2889.  doi: 10.1137/20M1334462.

[12]

M. Choulli, An inverse problem in corrosion detection: Stability estimates, J. Inverse Ill-Posed Probl., 12 (2004), 349-367.  doi: 10.1515/1569394042248247.

[13]

M. C. Delfour and J.-P. Zolésio, Shapes and Geometries, SIAM, USA, 2001.

[14]

H. W. EnglK. Kunisch and A. Neubauer, Convergence rates for Tikhonov regularization of nonlinear ill-posed problems, Inverse Problems, 5 (1989), 523-540.  doi: 10.1088/0266-5611/5/4/007.

[15]

H. W. Engl and J. Zou, A new approach to convergence rate analysis of Tikhonov regularization for parameter identification in heat conduction, Inverse Problems, 16 (2000), 1907-1923.  doi: 10.1088/0266-5611/16/6/319.

[16]

J. Flemming, Theory and examples of variational regularization with non-metric fitting functionals, J. Inverse Ill-Posed Probl., 18 (2010), 677-699.  doi: 10.1515/JIIP.2010.031.

[17]

M. Grasmair, Generalized Bregman distances and convergence rates for non-convex regularization methods, Inverse Problems, 26 (2010), 115014. doi: 10.1088/0266-5611/26/11/115014.

[18]

J. Hadamard, Lectures on Cauchy's Problem in Linear Partial Differential Equations, Dover Publications, New York, 1953.

[19]

D. N. Hao and T. N. T. Quyen, Convergence rates for Tikhonov regularization of coefficient identification problems in Laplace-type equations, Inverse Problems, 26 (2010), 125014, 20pp. doi: 10.1088/0266-5611/26/12/125014.

[20]

B. HofmannB. KaltenbacherC. Pöschl and O. Scherzer, A convergence rates result for Tikhonov regularization in Banach spaces with non-smooth operators, Inverse Problems, 23 (2007), 987-1010.  doi: 10.1088/0266-5611/23/3/009.

[21]

B. Hofmann and P. Mathé, Parameter choice in Banach space regularization under variational inequalities, Inverse Problems, 28 (2012), 104006. doi: 10.1088/0266-5611/28/10/104006.

[22]

T. Hohage and F. Weilding, Verification of a variational source condition for acoustic inverse medium scattering problems, Inverse Problems, 31 (2015), 075006. doi: 10.1088/0266-5611/31/7/075006.

[23]

T. Hohage and F. Weilding, Characerizations of variational source conditions, converse results, and maxisets of spectral regularization methods, SIAM J. Numer. Anal., 55 (2017), 598-620.  doi: 10.1137/16M1067445.

[24]

T. Hohage and F. Weilding, Variational source condition and stability estimates for inverse electromagnetic medium scattering problems, Inverse Probl. Imaging, 11 (2017), 203-220.  doi: 10.3934/ipi.2017010.

[25]

K. Ito and B. Jin, A new approach to nonlinear constrained Tikhonov regularization, Inverse Problems, 27 (2011), 105005. doi: 10.1088/0266-5611/27/10/105005.

[26]

D. J. Jiang, H. Feng and J. Zou, Convergence rates of Tikhonov regularizations for parameter identification in a parabolic-elliptic system, Inverse Problems, 28 (2012), 104002. doi: 10.1088/0266-5611/28/10/104002.

[27]

B. Jin and J. Zou, Numerical estimation of piecewise constant Robin coefficient, SIAM J. Control Optim., 48 (2009), 1977-2002.  doi: 10.1137/070710846.

[28]

B. Jin and J. Zou, Numerical estimation of the Robin coefficient in a stationary diffusion equation, IMA J. Numer. Anal., 30 (2010), 677-701.  doi: 10.1093/imanum/drn066.

[29]

F. John, Continuous dependence on data for solutions of partial differential equations with a prescribed bound, Comm. Pure Appl. Math., 13 (1960), 551-585.  doi: 10.1002/cpa.3160130402.

[30]

A. J. KassabE. Divo and J. S. Kapat, Multi-dimensional heat flux reconstruction using narrow-band thermochromic liquid crystal thermography, Inverse Probl. Sci. Eng., 9 (2001), 537-559.  doi: 10.1080/174159701088027780.

[31]

P. K$\ddot{u}$gler, Identification of a temperature dependent heat conductivity from single boundary measurements, SIAM J. Numer. Anal., 41 (2003), 1543-1563.  doi: 10.1137/S0036142902415900.

[32]

K. F. Lam and I. Yousept, Consistency of a phase field regularisation for an inverse problem governed by a quasilinear Maxwell system, Inverse Problems, 36 (2020), 045011. doi: 10.1088/1361-6420/ab6f9f.

[33]

D. Le and H. Smith, Strong positivity of solutions to parabolic and elliptic equations on nonsmooth domains, J. Math. Anal. Appl., 275 (2002), 208-221.  doi: 10.1016/S0022-247X(02)00314-1.

[34]

J. Li, J. Xie and J. Zou, An adaptive finite element reconstruction of distributed fluxes, Inverse Problems, 27 (2011), 075009. doi: 10.1088/0266-5611/27/7/075009.

[35]

J. L. Lions and E. Magenes, Non-Homogeneous Boundary Value Problems and Applications, Springer Science & Business Media, 2012.

[36]

T. J. Martin and G. S. Dulikravich, Inverse determination of steady heat convection coefficient distribution, J. Heat Transfer., 120 (1998), 328-334.  doi: 10.1115/1.2824251.

[37]

A. M. Osman and J. V. Beck, Nonlinear inverse problem for the estimation of time-and-space dependent heat transfer coefficients, J. Thérmophys, 3 (1989), 146-152.  doi: 10.2514/3.141.

[38]

L. E. Payne, On a priori bounds in the Cauchy problem for elliptic equations, SIAM J. Math. Anal., 1 (1970), 82-89.  doi: 10.1137/0501008.

[39]

C. Pechstein, Finite and Boundary Element Tearing and Interconnecting Solvers for Multiscale Problems, Lecture Notes in Computational Science and Engineering, 90. Springer, Heidelberg, 2013. doi: 10.1007/978-3-642-23588-7.

[40]

A. J. Pryde, Second order elliptic equations with mixed boundary conditions, J. Math. Anal. Appl., 80 (1981), 203-244.  doi: 10.1016/0022-247X(81)90102-5.

[41]

G. Savaré, Regularity and perturbation results for mixed second order elliptic problems, Commun. Partial. Differ. Equ., 22 (1997), 869-899.  doi: 10.1080/03605309708821287.

[42]

S. A. Sauter and C. Schwab, Boundary Element Methods, volume 39 of Springer Series in Compuational Mathematics, Berlin: Springer, 2011. doi: 10.1007/978-3-540-68093-2.

[43]

W. Sickel, Superposition of functions in Sobolev spaces of fractional order, Partial Dif Equ., 27 (1992), 481-497. 

[44]

J. Tambača, Estimates of the Sobolev norm of a product of two functions, J. Math. Anal. Appl., 255 (2001), 137-146.  doi: 10.1006/jmaa.2000.7209.

[45]

M. E. Taylor, Pseudodifferential Operators, Princeton University Press, Princeton, NJ, 1981.

[46]

F. Werner and B. Hofmann, Convergence analysis of (statistical) inverse problems under conditional stability estimates, Inverse Problems, 36 (2020), 015004. doi: 10.1088/1361-6420/ab4cd7.

[47]

F. M. White, Heat and Mass Transfer, Addison-Wesley, Reading, MA, 1988.

[48] J. Wloka, Partial Differential Equations, Cambridge University Press, Cambridge, 1987.  doi: 10.1017/CBO9781139171755.
[49]

J. Xie and J. Zou, Numerical reconstruction of heat fluxes, SIAM J. Numer. Anal., 43 (2005), 1504-1535.  doi: 10.1137/030602551.

[50]

A. Yagi, Abstract Parabolic Evolution Equations and Their Applications, Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04631-5.

Figure 1.  An example of $ \Omega $ in $ {{\mathbb R}}^2 $ and $ \Gamma_a $ (resp. $ \Gamma_b $) is an open part of $ \Gamma_1 $ (resp. $ \Gamma_2 $)
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