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Capped $\ell_p$ approximations for the composite $\ell_0$ regularization problem
Stability estimates in tensor tomography
| 1. | Department of Mathematics, Stockholm University, 106 91 Stockholm, Sweden |
| 2. | Sobolev Institute of Mathematics, 4 Koptyug Avenue, Novosibirsk, 630090, Russia |
| 3. | Novosibirsk State University, 2 Pirogov street, Novosibirsk, 630090, Russia |
We study the X-ray transform $I$ of symmetric tensor fields on a smooth convex bounded domain $Ω\subset{\mathbb R}^n$. The main result is the stability estimate $\|^{s}f\|_{L^2}≤ C\|If\|_{H^{1/2}}$, where $^{s}f$ is the solenoidal part of the tensor field $f$. The proof is based on a comparison of the Dirichlet integrals for the exterior and interior Dirichlet problems and on a generalization of the Korn inequality to symmetric tensor fields of arbitrary rank.
References:
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G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972. |
| [2] |
G. Fichera, Existence Theorems in Elasticity, Springer, 1972. |
| [3] |
K. O. Friedrichs,
On the boundary value problems of the theory of elasticity and Korn's inequality, Ann. Math., 48 (1947), 441-471.
doi: 10.2307/1969180. |
| [4] |
J. Gobert,
Une inégalité fondamentale de la théorie de l'élasticité, Bull. de la Roy. des Sci. de Liège, 31 (1962), 182-191.
|
| [5] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1, Springer, 1983.
doi: 10.1007/978-3-642-96750-4. |
| [6] |
V. A. Kondrat'ev and O. A. Oleinik,
Boundary value problems for the system of elasticity theory in unbounded domains. Korn's inequalities, Russian Mathematical Surveys, 43 (1988), 65-120.
doi: 10.1070/RM1988v043n05ABEH001945. |
| [7] |
A. Korn, Solution générale du problème d'équilibre dans la théorie de l'élasticité dans le cas où les efforts sont donnés à la surface, in Ann. Fac. Sci. Univ. Toulouse, 10 (1908), 165–269.
doi: 10.5802/afst.251. |
| [8] |
W. Lionheart and V. Sharafutdinov, Reconstruction algorithm for the linearized polarization tomography problem with incomplete data, in Imaging Microstructures: Mathematical and Computational Challenges, (eds. H. Ammari and Hyeonbae Kang), Contemporary Mathematics, 494 (2009), 137–159.
doi: 10.1090/conm/494/09648. |
| [9] |
S. G. Mikhlin, Variational Methods in Mathematical Physics, Oxford, Pergamon Press, 1964. |
| [10] |
F. Natterer, The Mathematics of Computerized Tomography, John Willey & Sons, 1986. |
| [11] |
L. E. Payne and H. F. Weinberger,
On Korn's inequality, Arch. Rat. Mech. Anal., 8 (1961), 89-98.
doi: 10.1007/BF00277432. |
| [12] |
L. N. Pestov and V. A. Sharafutdinov,
Integral geometry of tensor fields on a manifold of negative curvature, Siberian Math. J., 29 (1988), 427-441.
doi: 10.1007/BF00969652. |
| [13] |
Yu. G. Reshetnyak,
Estimates for certain differential operators with finite-dimensional kernel, Siberian Math. J., 11 (1970), 315-326.
|
| [14] |
V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994.
doi: 10.1515/9783110900095. |
| [15] |
V. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002 (20 pp).
doi: 10.1088/1361-6420/33/2/025002. |
| [16] |
V. Sharafutdinov, M. Skokan and G. Uhlmann,
Regularity of ghosts in tensor tomography, J. Geom. Anal., 15 (2005), 499-542.
doi: 10.1007/BF02930983. |
| [17] |
V. Sharafutdinov and J. Wang J, Tomography of small residual stresses, Inverse Problems, 28 (2012), 065017, 17 pp.
doi: 10.1088/0266-5611/28/6/065017. |
| [18] |
P. Stefanov, A sharp stability estimate in tensor tomography, in J. of Physics: Conference Series, 124 (2008), 012007.
doi: 10.1088/1742-6596/124/1/012007. |
| [19] |
M. E. Taylor, Partial Differential Equations I. Basic Theory, Texts in Applied Mathematics, 23. Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4684-9320-7. |
show all references
References:
| [1] |
G. Duvaut and J.-L. Lions, Les Inéquations en Mécanique et en Physique, Dunod, Paris, 1972. |
| [2] |
G. Fichera, Existence Theorems in Elasticity, Springer, 1972. |
| [3] |
K. O. Friedrichs,
On the boundary value problems of the theory of elasticity and Korn's inequality, Ann. Math., 48 (1947), 441-471.
doi: 10.2307/1969180. |
| [4] |
J. Gobert,
Une inégalité fondamentale de la théorie de l'élasticité, Bull. de la Roy. des Sci. de Liège, 31 (1962), 182-191.
|
| [5] |
L. Hörmander, The Analysis of Linear Partial Differential Operators, Vol. 1, Springer, 1983.
doi: 10.1007/978-3-642-96750-4. |
| [6] |
V. A. Kondrat'ev and O. A. Oleinik,
Boundary value problems for the system of elasticity theory in unbounded domains. Korn's inequalities, Russian Mathematical Surveys, 43 (1988), 65-120.
doi: 10.1070/RM1988v043n05ABEH001945. |
| [7] |
A. Korn, Solution générale du problème d'équilibre dans la théorie de l'élasticité dans le cas où les efforts sont donnés à la surface, in Ann. Fac. Sci. Univ. Toulouse, 10 (1908), 165–269.
doi: 10.5802/afst.251. |
| [8] |
W. Lionheart and V. Sharafutdinov, Reconstruction algorithm for the linearized polarization tomography problem with incomplete data, in Imaging Microstructures: Mathematical and Computational Challenges, (eds. H. Ammari and Hyeonbae Kang), Contemporary Mathematics, 494 (2009), 137–159.
doi: 10.1090/conm/494/09648. |
| [9] |
S. G. Mikhlin, Variational Methods in Mathematical Physics, Oxford, Pergamon Press, 1964. |
| [10] |
F. Natterer, The Mathematics of Computerized Tomography, John Willey & Sons, 1986. |
| [11] |
L. E. Payne and H. F. Weinberger,
On Korn's inequality, Arch. Rat. Mech. Anal., 8 (1961), 89-98.
doi: 10.1007/BF00277432. |
| [12] |
L. N. Pestov and V. A. Sharafutdinov,
Integral geometry of tensor fields on a manifold of negative curvature, Siberian Math. J., 29 (1988), 427-441.
doi: 10.1007/BF00969652. |
| [13] |
Yu. G. Reshetnyak,
Estimates for certain differential operators with finite-dimensional kernel, Siberian Math. J., 11 (1970), 315-326.
|
| [14] |
V. A. Sharafutdinov, Integral Geometry of Tensor Fields, VSP, Utrecht, The Netherlands, 1994.
doi: 10.1515/9783110900095. |
| [15] |
V. Sharafutdinov, The Reshetnyak formula and Natterer stability estimates in tensor tomography, Inverse Problems, 33 (2017), 025002 (20 pp).
doi: 10.1088/1361-6420/33/2/025002. |
| [16] |
V. Sharafutdinov, M. Skokan and G. Uhlmann,
Regularity of ghosts in tensor tomography, J. Geom. Anal., 15 (2005), 499-542.
doi: 10.1007/BF02930983. |
| [17] |
V. Sharafutdinov and J. Wang J, Tomography of small residual stresses, Inverse Problems, 28 (2012), 065017, 17 pp.
doi: 10.1088/0266-5611/28/6/065017. |
| [18] |
P. Stefanov, A sharp stability estimate in tensor tomography, in J. of Physics: Conference Series, 124 (2008), 012007.
doi: 10.1088/1742-6596/124/1/012007. |
| [19] |
M. E. Taylor, Partial Differential Equations I. Basic Theory, Texts in Applied Mathematics, 23. Springer-Verlag, New York, 1996.
doi: 10.1007/978-1-4684-9320-7. |
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