doi: 10.3934/ipi.2021050
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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 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)

Citation: De-Han Chen, Daijun Jiang, Irwin Yousept, Jun Zou. Variational source conditions for inverse Robin and flux problems by partial measurements. Inverse Problems & Imaging, doi: 10.3934/ipi.2021050
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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[45]

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

[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.  Google Scholar

[47]

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

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

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

[50]

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

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.  Google Scholar

[2]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

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

[13]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[18]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[35]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[43]

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

[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.  Google Scholar

[45]

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

[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.  Google Scholar

[47]

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

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

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

[50]

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

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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