Multidimensional Inverse and Ill-Posed Problems for Differential Equations

Anikonov, Yu. E.

doi:10.1515/9783110271478

Cited by 43 publications

(81 citation statements)

References 34 publications

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“…The system of eige~functions and associated ,functions of L is complete in s T) .for Re Pm+l > -k 8 Theorem 7.5. …”

Section: Corollarymentioning

confidence: 97%

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Inverse problems of spectral analysis for differential operators and their applications

Yurko¹

2000

J Math Sci

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“…The system of eige~functions and associated ,functions of L is complete in s T) .for Re Pm+l > -k 8 Theorem 7.5. …”

Section: Corollarymentioning

confidence: 97%

“…In See. 8 an IP for integro-differential operators is studied, and in Sec. 9 we consider an IP for one-dimensioiml perturbations of integral Volterra operators.…”

Section: L(q(x) H H1)mentioning

confidence: 99%

Inverse problems of spectral analysis for differential operators and their applications

Yurko¹

2000

J Math Sci

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“…In this framework, the same questions (injectivity, stability, range characterization, reconstruction algorithms, inverse problems with partial data) as for the straight line case are still under active theoretical study. To the author's knowledge, numerical simulations for these transforms remain to be documented.When the metric is simple, injectivity over functions was proved in [12] and injectivity over solenoidal vector fields was established in [1,2]. Under the same simplicity assumption, the problem was recently proved in [17] to be injective over solenoidal tensors ("s-injective") of any order, and previously in [4] under assumptions on the curvature.…”

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confidence: 99%

Numerical Implementation of Geodesic X-Ray Transforms and Their Inversion

Monard¹

2014

SIAM J. Imaging Sci.

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Abstract. We present a numerical implementation of the geodesic ray transform and its inversion over functions and solenoidal vector fields on two-dimensional Riemannian manifolds. For each problem, inversion formulas previously derived by Pestov and Uhlmann in [Int. Math Research Notices 80 (2004)] then extended by Krishnan in [J. Inv. Ill-Posed Problems 18 (2010)] are implemented in the case of simple and some non-simple metrics. These numerical tools are also used to better understand and gain intuition about non-simple manifolds, for which injectivity and stability of the corresponding integral geometric problems are still under active study.Key words. geodesic ray transform, Radon transform, tensor tomography problem, inverse problems, Riemann surfaces AMS subject classifications. 65R10, 65R32, 53D25, 44A121. Introduction. The present article discusses a numerical implementation in MatLab of geodesic X-Ray transforms of functions and solenoidal (i.e. divergence-free) vector fields and their inversion on two dimensional Riemannian manifolds with boundary.Geodesic X-ray transforms appear in problems of mathematical physics where particles travel along some curves and "gather information" along their path. Probably the best known example of geodesic ray transform is that of the Radon transform in two dimensions (also known as the X-Ray transform, as these two transforms coincide in two dimensions), which is the collection of integrals of a given function over all straight lines in the plane, corresponding to the case of a Euclidean metric. Reconstructing a function from its integrals along lines was first considered and solved in [21] and is now used every day in medical imaging. A thorough account of theoretical and numerical aspects for this transform may be found in [13]. In the Euclidean case, solenoidal tensors of any order were also explicitely reconstructed in [23].When optical rays propagate in a medium with variable index of refraction, their trajectories, no longer straight lines, come as geodesics of some Riemannian metric. In this framework, the same questions (injectivity, stability, range characterization, reconstruction algorithms, inverse problems with partial data) as for the straight line case are still under active theoretical study. To the author's knowledge, numerical simulations for these transforms remain to be documented.When the metric is simple, injectivity over functions was proved in [12] and injectivity over solenoidal vector fields was established in [1,2]. Under the same simplicity assumption, the problem was recently proved in [17] to be injective over solenoidal tensors ("s-injective") of any order, and previously in [4] under assumptions on the curvature. Independently, stability estimates were given in [26] via a microlocal study of the normal operator.

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“…This type of linear inverse problem is discussed in [11]. For solving inverse parabolic problems, the reader can consult a book by Anikonov [2].…”

Section: Introductionmentioning

confidence: 99%

Generic well-posedness of a linear inverse parabolic problem with diffusion parameters

Choulli¹,

Yamamoto²

1999

Jiip

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Multidimensional Inverse and Ill-Posed Problems for Differential Equations

Abstract: All rights reserved. No part of this publication may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner.

Cited by 43 publications

References 34 publications

Inverse problems of spectral analysis for differential operators and their applications

Inverse problems of spectral analysis for differential operators and their applications

Numerical Implementation of Geodesic X-Ray Transforms and Their Inversion

Generic well-posedness of a linear inverse parabolic problem with diffusion parameters

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