| 1. |
- Avdonin, Sergei, et al.
(författare)
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Inverse problems for quantum trees
- 2008
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Ingår i: INVERSE PROBLEMS AND IMAGING. - 1930-8337 .- 1930-8345. ; 2:1, s. 1-21
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Tidskriftsartikel (refereegranskat)abstract
- Three different inverse problems for the Schrodinger operator on a metric tree are considered, so far with standard boundary conditions at the vertices. These inverse problems are connected with the matrix Titchmarsh-Weyl function, response operator ( dynamic Dirichlet-to-Neumann map) and scattering matrix. Our approach is based on the boundary control ( BC) method and in particular on the study of the response operator. It is proven that the response operator determines the quantum tree completely, i.e. its connectivity, lengths of the edges and potentials on them. The same holds if the response operator is known for all but one boundary points, as well as for the Titchmarsh-Weyl function and scattering matrix. If the connectivity of the graph is known, then the lengths of the edges and the corresponding potentials are determined by just the diagonal terms of the data.
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| 2. |
- Avdonin, Sergei, et al.
(författare)
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INVERSE PROBLEMS FOR QUANTUM TREES II : RECOVERING MATCHING CONDITIONS FOR STAR GRAPHS
- 2010
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Ingår i: Inverse Problems and Imaging. - : American Institute of Mathematical Sciences (AIMS). - 1930-8337 .- 1930-8345. ; 4:4, s. 579-598
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Tidskriftsartikel (refereegranskat)abstract
- The inverse problem for the Schrodinger operator on a star graph is investigated. It is proven that such Schrodinger operator, i.e. the graph, the real potential on it and the matching conditions at the central vertex, can be reconstructed from the Titchmarsh-Weyl matrix function associated with the graph boundary. The reconstruction is also unique if the spectral data include not the whole Titchmarsh-Weyl function but its principal block (the matrix reduced by one dimension). The same result holds true if instead of the Titchmarsh-Weyl function the dynamical response operator or just its principal block is known.
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| 3. |
- Beilina, Larisa, 1970, et al.
(författare)
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Optimization approach for the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions from limited observations
- 2015
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Ingår i: Inverse Problems and Imaging. - : American Institute of Mathematical Sciences (AIMS). - 1930-8337 .- 1930-8345. ; 9:1, s. 1-25
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Tidskriftsartikel (refereegranskat)abstract
- We consider the inverse problem of the simultaneous reconstruction of the dielectric permittivity and magnetic permeability functions of the Maxwell's system in 3D with limited boundary observations of the electric field. The theoretical stability for the problem is provided by the Carleman estimates. For the numerical computations the problem is formulated as an optimization problem and hybrid finite element/difference method is used to solve the parameter identification problem.
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| 4. |
- Beltita, Ingrid, et al.
(författare)
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The quadratic contribution to the backscattering transform in the rotation invariant case
- 2010
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Ingår i: Inverse Problems and Imaging. - : American Institute of Mathematical Sciences (AIMS). - 1930-8345 .- 1930-8337. ; 4:4, s. 599-618
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Konferensbidrag (refereegranskat)abstract
- Considerations of the backscattering data for the Schrodinger operator H-v = -Delta + v in R-n, where n >= 3 is odd, give rise to an entire analytic mapping from C-0(infinity)(R-n) to C-0(infinity)(R-n), the backscattering transformation. The aim of this paper is to give formulas for B-2(v, w) where B-2 is the symmetric bilinear operator that corresponds to the quadratic part of the backscattering transformation and v and w are rotation invariant.
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| 5. |
- Boman, Jan, 1933-
(författare)
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A local uniqueness theorem for weighted Radon transforms
- 2010
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Ingår i: Inverse Problems and Imaging. - : American institute of mathematical sciences. - 1930-8337 .- 1930-8345. ; 4:4, s. 631-637
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Tidskriftsartikel (refereegranskat)abstract
- We consider a weighted Radon transform in the plane, , where is a smooth, positive function. Using an extension of an argument of Strichartz we prove a local injectivity theorem for for essentially the same class of that was considered by Gindikin in his article in this issue.
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| 6. |
- Boman, Jan, et al.
(författare)
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STABILITY ESTIMATES IN TENSOR TOMOGRAPHY
- 2018
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Ingår i: Inverse Problems and Imaging. - : American Institute of Mathematical Sciences (AIMS). - 1930-8337 .- 1930-8345. ; 12:5, s. 1245-1262
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Tidskriftsartikel (refereegranskat)abstract
- We study the X-ray transform I of symmetric tensor fields on a smooth convex bounded domain Omega C R-n. The main result is the stability estimate parallel to(s)f parallel to(L2) <= C parallel to If parallel to(H1/2), 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.
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| 7. |
- Boman, Jan, 1933-
(författare)
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Unique continuation of microlocally analytic distributions and injectivity theorems for the ray transform
- 2010
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Ingår i: Inverse Problems and Imaging. - : American Institute of Mathematical Sciences. - 1930-8337 .- 1930-8345. ; 4:4, s. 619-630
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Tidskriftsartikel (refereegranskat)abstract
- Using a vanishing theorem for microlocally real analytic distributions and a theorem on flatness of a distribution vanishing on infinitely many hyperplanes we give a new proof of an injectivity theorem of Bélisle, Massé, and Ransford for the ray transform on R^n. By means of an example we show that this result is sharp. An extension is given where real analyticity is replaced by quasianalyticity.
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| 8. |
- Quinto, Eric Todd, et al.
(författare)
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LOCAL SINGULARITY RECONSTRUCTION FROM INTEGRALS OVER CURVES IN R-3
- 2013
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Ingår i: Inverse Problems and Imaging. - : American Institute of Mathematical Sciences (AIMS). - 1930-8337 .- 1930-8345. ; 7:2, s. 585-609
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Tidskriftsartikel (refereegranskat)abstract
- We define a general curvilinear Radon transform in R-3, and we develop its microlocal properties. We prove that singularities can be added (or masked) in any backprojection reconstruction method for this transform. We use the microlocal properties of the transform to develop a local backprojection reconstruction algorithm that decreases the effect of the added singularities and reconstructs the shape of the object. This work was motivated by new models in electron microscope tomography in which the electrons travel over curves such as helices or spirals, and we provide reconstructions for a specific transform motivated by this electron microscope tomography problem.
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| 9. |
- Rullgård, Hans, et al.
(författare)
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LOCAL SOBOLEV ESTIMATES OF A FUNCTION BY MEANS OF ITS RADON TRANSFORM
- 2010
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Ingår i: Inverse Problems and Imaging. - : American Institute of Mathematical Sciences (AIMS). - 1930-8337 .- 1930-8345. ; 4:4, s. 721-734
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Tidskriftsartikel (refereegranskat)abstract
- In this article, we will define local and microlocal Sobolev seminorms and prove local and microlocal inverse continuity estimates for the Radon hyperplane transform in these seminorms. The relation between the Sobolev wavefront set of a function f and of its Radon transform is well-known [18]. However, Sobolev wavefront is qualitative and therefore the relation in [18] is qualitative. Our results will make the relation between singularities of a function and those of its Radon transform quantitative. This could be important for practical applications, such as tomography, in which the data are smooth but can have large derivatives.
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| 10. |
- Boman, Jan, 1933-
(författare)
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Mathematical reminiscences
- 2010
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Ingår i: Inverse problems and imaging. - : American Institute of Matheamtical Sciences. - 1930-8337. ; 4:4, s. 571-577
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Tidskriftsartikel (refereegranskat)
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