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Convergence analysis of the adjoint ensemble method in inverse source problems for advection-diffusion-reaction models with image-type measurements
Stability estimates in a partial data inverse boundary value problem for biharmonic operators at high frequencies
Department of Mathmatics, University of California, Irvine, Irvine, CA 92697, USA |
We study the inverse boundary value problems of determining a potential in the Helmholtz type equation for the perturbed biharmonic operator from the knowledge of the partial Cauchy data set. Our geometric setting is that of a domain whose inaccessible portion of the boundary is contained in a hyperplane, and we are given the Cauchy data set on the complement. The uniqueness and logarithmic stability for this problem were established in [
References:
[1] |
M. Agranovich, Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains, Springer Monographs in Mathematics, Springer, Cham, 2015.
doi: 10.1007/978-3-319-14648-5. |
[2] |
G. Alessandrini,
Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.
doi: 10.1080/00036818808839730. |
[3] |
Y. Assylbekov and K. Iyer,
Determining rough first order perturbations of the polyharmonic operator, Inverse Problems and Imaging, 13 (2019), 1045-1066.
doi: 10.3934/ipi.2019047. |
[4] |
Y. Assylbekov and Y. Yang,
Determining the first order perturbation of a polyharmonic operator on admissible manifolds, J. Differential Equations, 262 (2017), 590-614.
doi: 10.1016/j.jde.2016.09.039. |
[5] |
S. Bhattacharyya and T. Ghosh,
Inverse boundary value problem of determining up to a second order tensor appear in the lower order perturbation of a polyharmonic operator, J. Fourier Anal. Appl., 25 (2019), 661-683.
doi: 10.1007/s00041-018-9625-3. |
[6] |
P. Caro and K. Marinov,
Stability of inverse problems in an infinite slab with partial data, Comm. Partial Differential Equations, 41 (2016), 683-704.
doi: 10.1080/03605302.2015.1127967. |
[7] |
A. Choudhury and H. Heck,
Stability of the inverse boundary value problem for the biharmonic operator: Logarithmic estimates, J. Inverse Ill-Posed Probl., 25 (2017), 251-263.
doi: 10.1515/jiip-2016-0019. |
[8] |
A. Choudhury and H. Heck,
Increasing stability for the inverse problem for the Schrödinger equation, Math. Methods Appl. Sci., 41 (2018), 606-614.
doi: 10.1002/mma.4632. |
[9] |
A. Choudhury and V. Krishnan,
Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials, J. Math. Anal. Appl., 431 (2015), 300-316.
doi: 10.1016/j.jmaa.2015.05.054. |
[10] |
D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105–S137.
doi: 10.1088/0266-5611/19/6/057. |
[11] |
D. Faraco and K. M. Rogers,
The Sobolev norm of characteristic functions with applications to the Calderón inverse problem, Quart. J. Math., 64 (2013), 133-147.
doi: 10.1093/qmath/har039. |
[12] |
J. Feldman, M. Salo and G. Uhlmann, The Calderón problem: An Introduction to Inverse Problems, Preliminary notes on the book in preparation, 2019. |
[13] |
F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-12245-3. |
[14] |
T. Ghosh and V. Krishnan,
Determination of lower order perturbations of the polyharmonic operator from partial boundary data, Appl. Anal., 95 (2016), 2444-2463.
doi: 10.1080/00036811.2015.1092522. |
[15] |
G. Grubb, Distributions and Operators, Volume 252 of Graduate Texts in Mathematics. Springer, New York, 2009. |
[16] |
H. Heck and J. Wang,
Optimal stability estimate of the inverse boundary value problem by partial measurements, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 369-383.
doi: 10.13137/2464-8728/13164. |
[17] |
P. Hähner,
A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300-308.
doi: 10.1006/jdeq.1996.0096. |
[18] |
T. Hrycak and V. Isakov,
Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.
doi: 10.1088/0266-5611/20/3/004. |
[19] |
M. Ikehata,
A special Green's function for the biharmonic operator and its application to an inverse boundary value problem, Comput. Math. Appl, 22 (1991), 53-66.
doi: 10.1016/0898-1221(91)90131-M. |
[20] |
V. Isakov,
Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations, 92 (1991), 305-316.
doi: 10.1016/0022-0396(91)90051-A. |
[21] |
V. Isakov,
On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105.
doi: 10.3934/ipi.2007.1.95. |
[22] |
V. Isakov,
Increased stability in the continuation for the Helmholtz equation with variable coefficient, Control Methods in PDE-dynamical systems, Contemp. Math., 426 (2007), 255-267.
doi: 10.1090/conm/426/08192. |
[23] |
V. Isakov,
Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, DCDS-S, 4 (2011), 631-640.
doi: 10.3934/dcdss.2011.4.631. |
[24] |
V. Isakov, R. Lai and J. Wang,
Increasing stability for the conductivity and attenuation coefficients, SIAM J. Math. Anal., 48 (2016), 569-594.
doi: 10.1137/15M1019052. |
[25] |
V. Isakov, S. Nagayasu, G. Uhlmann and J. Wang,
Increasing stability of the inverse boundary value problem for the Schrödinger equation, Contemp. Math., 615 (2014), 131-141.
doi: 10.1090/conm/615/12268. |
[26] |
V. Isakov and J. Wang,
Increasing stability for determining the potential in the Schr${\rm{\ddot d}}$inger equation with attenuation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 8 (2014), 1139-1150.
doi: 10.3934/ipi.2014.8.1139. |
[27] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Determining a first order perturbation of the biharmonic operator by partial boundary measurements, J. Funct. Anal., 262 (2012), 1781-1801.
doi: 10.1016/j.jfa.2011.11.021. |
[28] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Inverse boundary value problems for the perturbed polyharmonic operator, Trans. Amer. Math. Soc., 366 (2014), 95-112.
doi: 10.1090/S0002-9947-2013-05713-3. |
[29] |
K. Krupchyk and G. Uhlmann,
Inverse boundary problems for polyharmonic operators with unbounded potentials, J. Spectr. Theory, 6 (2016), 145-183.
doi: 10.4171/JST/122. |
[30] |
K. Krupchyk and G. Uhlmann,
Inverse problems for advection diffusion equations in admissible geometries, Comm. Partial Differential Equations, 43 (2018), 585-615.
doi: 10.1080/03605302.2018.1446163. |
[31] |
K. Krupchyk and G. Uhlmann,
Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, J. Math. Pures Appl., 126 (2019), 273-291.
doi: 10.1016/j.matpur.2019.02.017. |
[32] |
L. Liang,
Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data, Inverse Probl. Imaging, 9 (2015), 469-478.
doi: 10.3934/ipi.2015.9.469. |
[33] |
N. Mandache,
Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.
doi: 10.1088/0266-5611/17/5/313. |
[34] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
![]() ![]() |
[35] |
V. Serov, Borg-Levinson theorem for perturbations of the bi-harmonic operator, Inverse Problems, 32 (2016), 045002, 19 pp.
doi: 10.1088/0266-5611/32/4/045002. |
[36] |
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. |
[37] |
Y. Yang,
Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data, J. Differential Equations, 257 (2014), 3607-3639.
doi: 10.1016/j.jde.2014.07.003. |
show all references
References:
[1] |
M. Agranovich, Sobolev Spaces, Their Generalizations and Elliptic Problems in Smooth and Lipschitz Domains, Springer Monographs in Mathematics, Springer, Cham, 2015.
doi: 10.1007/978-3-319-14648-5. |
[2] |
G. Alessandrini,
Stable determination of conductivity by boundary measurements, Appl. Anal., 27 (1988), 153-172.
doi: 10.1080/00036818808839730. |
[3] |
Y. Assylbekov and K. Iyer,
Determining rough first order perturbations of the polyharmonic operator, Inverse Problems and Imaging, 13 (2019), 1045-1066.
doi: 10.3934/ipi.2019047. |
[4] |
Y. Assylbekov and Y. Yang,
Determining the first order perturbation of a polyharmonic operator on admissible manifolds, J. Differential Equations, 262 (2017), 590-614.
doi: 10.1016/j.jde.2016.09.039. |
[5] |
S. Bhattacharyya and T. Ghosh,
Inverse boundary value problem of determining up to a second order tensor appear in the lower order perturbation of a polyharmonic operator, J. Fourier Anal. Appl., 25 (2019), 661-683.
doi: 10.1007/s00041-018-9625-3. |
[6] |
P. Caro and K. Marinov,
Stability of inverse problems in an infinite slab with partial data, Comm. Partial Differential Equations, 41 (2016), 683-704.
doi: 10.1080/03605302.2015.1127967. |
[7] |
A. Choudhury and H. Heck,
Stability of the inverse boundary value problem for the biharmonic operator: Logarithmic estimates, J. Inverse Ill-Posed Probl., 25 (2017), 251-263.
doi: 10.1515/jiip-2016-0019. |
[8] |
A. Choudhury and H. Heck,
Increasing stability for the inverse problem for the Schrödinger equation, Math. Methods Appl. Sci., 41 (2018), 606-614.
doi: 10.1002/mma.4632. |
[9] |
A. Choudhury and V. Krishnan,
Stability estimates for the inverse boundary value problem for the biharmonic operator with bounded potentials, J. Math. Anal. Appl., 431 (2015), 300-316.
doi: 10.1016/j.jmaa.2015.05.054. |
[10] |
D. Colton, H. Haddar and M. Piana, The linear sampling method in inverse electromagnetic scattering theory, Inverse Problems, 19 (2003), S105–S137.
doi: 10.1088/0266-5611/19/6/057. |
[11] |
D. Faraco and K. M. Rogers,
The Sobolev norm of characteristic functions with applications to the Calderón inverse problem, Quart. J. Math., 64 (2013), 133-147.
doi: 10.1093/qmath/har039. |
[12] |
J. Feldman, M. Salo and G. Uhlmann, The Calderón problem: An Introduction to Inverse Problems, Preliminary notes on the book in preparation, 2019. |
[13] |
F. Gazzola, H.-C. Grunau and G. Sweers, Polyharmonic Boundary Value Problems, Springer-Verlag, Berlin, 2010.
doi: 10.1007/978-3-642-12245-3. |
[14] |
T. Ghosh and V. Krishnan,
Determination of lower order perturbations of the polyharmonic operator from partial boundary data, Appl. Anal., 95 (2016), 2444-2463.
doi: 10.1080/00036811.2015.1092522. |
[15] |
G. Grubb, Distributions and Operators, Volume 252 of Graduate Texts in Mathematics. Springer, New York, 2009. |
[16] |
H. Heck and J. Wang,
Optimal stability estimate of the inverse boundary value problem by partial measurements, Rend. Istit. Mat. Univ. Trieste, 48 (2016), 369-383.
doi: 10.13137/2464-8728/13164. |
[17] |
P. Hähner,
A periodic Faddeev-type solution operator, J. Differential Equations, 128 (1996), 300-308.
doi: 10.1006/jdeq.1996.0096. |
[18] |
T. Hrycak and V. Isakov,
Increased stability in the continuation of solutions to the Helmholtz equation, Inverse Problems, 20 (2004), 697-712.
doi: 10.1088/0266-5611/20/3/004. |
[19] |
M. Ikehata,
A special Green's function for the biharmonic operator and its application to an inverse boundary value problem, Comput. Math. Appl, 22 (1991), 53-66.
doi: 10.1016/0898-1221(91)90131-M. |
[20] |
V. Isakov,
Completeness of products of solutions and some inverse problems for PDE, J. Differential Equations, 92 (1991), 305-316.
doi: 10.1016/0022-0396(91)90051-A. |
[21] |
V. Isakov,
On uniqueness in the inverse conductivity problem with local data, Inverse Probl. Imaging, 1 (2007), 95-105.
doi: 10.3934/ipi.2007.1.95. |
[22] |
V. Isakov,
Increased stability in the continuation for the Helmholtz equation with variable coefficient, Control Methods in PDE-dynamical systems, Contemp. Math., 426 (2007), 255-267.
doi: 10.1090/conm/426/08192. |
[23] |
V. Isakov,
Increasing stability for the Schrödinger potential from the Dirichlet-to-Neumann map, DCDS-S, 4 (2011), 631-640.
doi: 10.3934/dcdss.2011.4.631. |
[24] |
V. Isakov, R. Lai and J. Wang,
Increasing stability for the conductivity and attenuation coefficients, SIAM J. Math. Anal., 48 (2016), 569-594.
doi: 10.1137/15M1019052. |
[25] |
V. Isakov, S. Nagayasu, G. Uhlmann and J. Wang,
Increasing stability of the inverse boundary value problem for the Schrödinger equation, Contemp. Math., 615 (2014), 131-141.
doi: 10.1090/conm/615/12268. |
[26] |
V. Isakov and J. Wang,
Increasing stability for determining the potential in the Schr${\rm{\ddot d}}$inger equation with attenuation from the Dirichlet-to-Neumann map, Inverse Probl. Imaging, 8 (2014), 1139-1150.
doi: 10.3934/ipi.2014.8.1139. |
[27] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Determining a first order perturbation of the biharmonic operator by partial boundary measurements, J. Funct. Anal., 262 (2012), 1781-1801.
doi: 10.1016/j.jfa.2011.11.021. |
[28] |
K. Krupchyk, M. Lassas and G. Uhlmann,
Inverse boundary value problems for the perturbed polyharmonic operator, Trans. Amer. Math. Soc., 366 (2014), 95-112.
doi: 10.1090/S0002-9947-2013-05713-3. |
[29] |
K. Krupchyk and G. Uhlmann,
Inverse boundary problems for polyharmonic operators with unbounded potentials, J. Spectr. Theory, 6 (2016), 145-183.
doi: 10.4171/JST/122. |
[30] |
K. Krupchyk and G. Uhlmann,
Inverse problems for advection diffusion equations in admissible geometries, Comm. Partial Differential Equations, 43 (2018), 585-615.
doi: 10.1080/03605302.2018.1446163. |
[31] |
K. Krupchyk and G. Uhlmann,
Stability estimates for partial data inverse problems for Schrödinger operators in the high frequency limit, J. Math. Pures Appl., 126 (2019), 273-291.
doi: 10.1016/j.matpur.2019.02.017. |
[32] |
L. Liang,
Increasing stability for the inverse problem of the Schrödinger equation with the partial Cauchy data, Inverse Probl. Imaging, 9 (2015), 469-478.
doi: 10.3934/ipi.2015.9.469. |
[33] |
N. Mandache,
Exponential instability in an inverse problem for the Schrödinger equation, Inverse Problems, 17 (2001), 1435-1444.
doi: 10.1088/0266-5611/17/5/313. |
[34] |
W. McLean, Strongly Elliptic Systems and Boundary Integral Equations, Cambridge University Press, Cambridge, 2000.
![]() ![]() |
[35] |
V. Serov, Borg-Levinson theorem for perturbations of the bi-harmonic operator, Inverse Problems, 32 (2016), 045002, 19 pp.
doi: 10.1088/0266-5611/32/4/045002. |
[36] |
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. |
[37] |
Y. Yang,
Determining the first order perturbation of a bi-harmonic operator on bounded and unbounded domains from partial data, J. Differential Equations, 257 (2014), 3607-3639.
doi: 10.1016/j.jde.2014.07.003. |
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