publications
search by categories: Calderón problem, Inverse Scattering, Born approximation
2024
- arXivStable factorization of the Calderón problem via the Born approximation (Preprint)Thierry Daudé, Fabricio Macià, Cristóbal J. Meroño, and François Nicoleau
In this article we prove the existence of the Born approximation in the context of the radial Calderón problem for Schrödinger operators. This is the inverse problem of recovering a radial potential on the unit ball from the knowledge of the Dirichlet-to-Neumann map (DtN map from now on) of the corresponding Schrödinger operator. The Born approximation naturally appears as the linear component of a factorization of the Calderón problem; we show that the non-linear part, obtaining the potential from the Born approximation, enjoys several interesting properties. First, this map is local, in the sense that knowledge of the Born approximation in a neighborhood of the boundary is equivalent to knowledge of the potential in the same neighborhood, and, second, it is Hölder stable. This shows in particular that the ill-posedness of the Calderón problem arises solely from the linear step, which consists in computing the Born approximation from the DtN map by solving a Hausdorff moment problem. Moreover, we present an effective algorithm to compute the potential from the Born approximation and show a result on reconstruction of singularities. Finally, we use the Born approximation to obtain a partial characterization of the set of DtN maps for radial potentials. The proofs of these results do not make use of Complex Geometric Optics solutions or its analogues; they are based on results on inverse spectral theory for Schrödinger operators on the half-line, in particular on the concept of A-amplitude introduced by Barry Simon.
@misc{Radial_Born, author = {Daudé, Thierry and Macià, Fabricio and Meroño, Cristóbal J. and Nicoleau, François}, title = {Stable factorization of the Calderón problem via the Born approximation (Preprint)}, year = {2024}, eprint = {arXiv:2402.06321}, } - IPIThe Born approximation in the three-dimensional Calderón problem II: Numerical reconstruction in the radial caseJuan A. Barceló, Carlos Castro, Fabricio Macià, and Cristóbal J. Meroño
In this work we illustrate a number of properties of the Born approximation in the three-dimensional Calderón inverse conductivity problem by numerical experiments. The results are based on an explicit representation formula for the Born approximation recently introduced by the authors. We focus on the particular case of radial conductivities in the ball B_R ⊂\mathbbR^3 of radius R , in which the linearization of the Calderón problem is equivalent to a Hausdorff moment problem. We give numerical evidences that the Born approximation is well defined for L^∞ conductivities, and present a novel numerical algorithm to reconstruct a radial conductivity from the Born approximation under a suitable smallness assumption. We also show that the Born approximation has depth-dependent uniqueness and approximation capabilities depending on the distance (depth) to the boundary ∂B_R . We then investigate how increasing the radius R affects the quality of the Born approximation, and the existence of a scattering limit as R\to ∞. Similar properties are also illustrated in the inverse boundary problem for the Schrödinger operator -∆+q , and strong recovery of singularity results are observed in this case.
@article{MR4685806, author = {Barcel\'o, Juan A. and Castro, Carlos and Maci\`a, Fabricio and Mero\~no, Crist\'obal J.}, title = {The {B}orn approximation in the three-dimensional {C}alder\'on problem {II}: {N}umerical reconstruction in the radial case}, journal = {Inverse Probl. Imaging}, fjournal = {Inverse Problems and Imaging}, volume = {18}, year = {2024}, number = {1}, pages = {183--207}, issn = {1930-8337,1930-8345}, mrclass = {35R30 (35P25 65N21)}, mrnumber = {4685806}, doi = {10.3934/ipi.2023029}, url = {https://doi.org/10.3934/ipi.2023029}, keywords = {Calderón problem, Born approximation, inverse problems, numerical reconstruction, Dirichlet to Neumann map} }
2022
- JFAThe Born approximation in the three-dimensional Calderón problemJuan A. Barceló, Carlos Castro, Fabricio Macià, and Cristóbal J. Meroño
Uniqueness and reconstruction in the three-dimensional Calderón inverse conductivity problem can be reduced to the study of the inverse boundary problem for Schrödinger operators −Δ+q. We study the Born approximation of q in the ball, which amounts to studying the linearization of the inverse problem. We first analyze this approximation for real and radial potentials in any dimension d≥3. We show that this approximation satisfies a closed formula that only involves the spectrum of the Dirichlet-to-Neumann map associated to −Δ+q, which is closely related to a particular moment problem. We then turn to general real and essentially bounded potentials in three dimensions and introduce the notion of averaged Born approximation, which captures the exact invariance properties of the inverse problem. We obtain explicit formulas for the averaged Born approximation in terms of the matrix elements of the Dirichlet to Neumann map in the basis of spherical harmonics. To show that the averaged Born approximation does not destroy information on the potential, we also study the high-energy behavior of the matrix elements of the Dirichlet to Neumann map.
@article{MR4477941, author = {Barcel\'o, Juan A. and Castro, Carlos and Maci\`a, Fabricio and Mero\~no, Crist\'obal J.}, title = {The {B}orn approximation in the three-dimensional {C}alder\'on problem}, journal = {J. Funct. Anal.}, fjournal = {Journal of Functional Analysis}, volume = {283}, year = {2022}, number = {12}, pages = {Paper No. 109681, 43}, issn = {0022-1236,1096-0783}, mrclass = {35Q60 (42B10 43A90)}, mrnumber = {4477941}, doi = {10.1016/j.jfa.2022.109681}, url = {https://doi.org/10.1016/j.jfa.2022.109681}, keywords = {Inverse problems, Calderón problem, Eigenvalues, Born approximation} } - JDERotational smoothingPedro Caro, Cristóbal J. Meroño, and Ioannis Parissis
Rotational smoothing is a phenomenon consisting in a gain of regularity by means of averaging over rotations. This phenomenon is present in operators that regularize only in certain directions, in contrast to operators regularizing in all directions. The gain of regularity is the result of rotating the directions where the corresponding operator performs the smoothing effect. In this paper we carry out a systematic study of the rotational smoothing for a class of operators that includes k-vector-space Riesz potentials in Rn with k<n, and the convolution with fundamental solutions of elliptic constant-coefficient differential operators acting on k-dimensional linear subspaces. Examples of the latter type of operators are the planar Cauchy transform in Rn, or a solution operator for the transport equation in Rn. The analysis of rotational smoothing is motivated by the resolution of some inverse problems under low-regularity assumptions.
@article{MR4332040, author = {Caro, Pedro and Mero\~no, Crist\'obal J. and Parissis, Ioannis}, title = {Rotational smoothing}, journal = {J. Differential Equations}, fjournal = {Journal of Differential Equations}, volume = {306}, year = {2022}, pages = {101--151}, issn = {0022-0396,1090-2732}, mrclass = {35Q49}, mrnumber = {4332040}, doi = {10.1016/j.jde.2021.10.018}, url = {https://doi.org/10.1016/j.jde.2021.10.018}, keywords = {Rotational smoothing, Multipliers with singular symbols, Riesz potentials, Cauchy transform, Special orthogonal group} }
2021
- AHPThe fixed angle scattering problem with a first-order perturbationCristóbal J. Meroño, Leyter Potenciano-Machado, and Mikko Salo
We study the inverse scattering problem of determining a magnetic field and electric potential from scattering measurements corresponding to finitely many plane waves. The main result shows that the coefficients are uniquely determined by 2n measurements up to a natural gauge. We also show that one can recover the full first-order term for a related equation having no gauge invariance, and that it is possible to reduce the number of measurements if the coefficients have certain symmetries. This work extends the fixed angle scattering results of Rakesh and Salo (SIAM J Math Anal 52(6):5467–5499, 2020) and (Inverse Probl 36(3):035005, 2020) to Hamiltonians with first-order perturbations, and it is based on wave equation methods and Carleman estimates.
@article{MR4325875, author = {Mero\~no, Crist\'obal J. and Potenciano-Machado, Leyter and Salo, Mikko}, title = {The fixed angle scattering problem with a first-order perturbation}, journal = {Ann. Henri Poincar\'e}, fjournal = {Annales Henri Poincar\'e. A Journal of Theoretical and Mathematical Physics}, volume = {22}, year = {2021}, number = {11}, pages = {3699--3746}, issn = {1424-0637,1424-0661}, mrclass = {35R30 (81U40)}, mrnumber = {4325875}, doi = {10.1007/s00023-021-01081-w}, url = {https://doi.org/10.1007/s00023-021-01081-w}, keywords = {Inverse Scattering, Inverse Problems} } - RMIThe double dispersion operator in backscattering: Hölder estimates and optimal Sobolev estimates for radial potentialsCristóbal J. Meroño
@article{MR4236805, author = {Mero\~no, Crist\'obal J.}, title = {The double dispersion operator in backscattering: {H}\"older estimates and optimal {S}obolev estimates for radial potentials}, journal = {Rev. Mat. Iberoam.}, fjournal = {Revista Matem\'atica Iberoamericana}, volume = {37}, year = {2021}, number = {3}, pages = {1175--1205}, issn = {0213-2230,2235-0616}, mrclass = {35R30 (42B20 81U40)}, mrnumber = {4236805}, doi = {10.4171/rmi/1223}, url = {https://doi.org/10.4171/rmi/1223}, keywords = {Schrödinger equation, Inverse Scattering, Inverse Problems} }
2020
- SIAM JMAThe observational limit of wave packets with noisy measurementsPedro Caro, and Cristóbal J. Meroño
The authors consider the problem of recovering an observable from certain measurements containing random errors. The observable is given by a pseudodifferential operator while the random errors are generated by a Gaussian white noise. The authors show how wave packets can be used to partially recover the observable from the measurements almost surely. Furthermore, they point out the limitation of wave packets to recover the remaining part of the observable and show how the errors hide the signal coming from the observable. The recovery results are based on an ergodicity property of the errors produced by wave packets.
@article{MR4165940, author = {Caro, Pedro and Mero\~no, Crist\'obal J.}, title = {The observational limit of wave packets with noisy measurements}, journal = {SIAM J. Math. Anal.}, fjournal = {SIAM Journal on Mathematical Analysis}, volume = {52}, year = {2020}, number = {5}, pages = {5196--5212}, issn = {0036-1410,1095-7154}, mrclass = {35S05 (35R30 60H40)}, mrnumber = {4165940}, mrreviewer = {Akhilesh\ Prasad}, doi = {10.1137/20M1324946}, url = {https://doi.org/10.1137/20M1324946}, keywords = {Wave Packets, Inverse Problems} } - Uniqueness for the inverse fixed angle scattering problemJ.A. Barceló, C. Castro, T. Luque, C. J. Meroño, A. Ruiz, and M. C. Vilela
@article{MR4129366, author = {Barcel\'o, J.A. and Castro, C. and Luque, T. and Mero\~no, C. J. and Ruiz, A. and Vilela, M. C.}, title = {Uniqueness for the inverse fixed angle scattering problem}, journal = {J. Inverse Ill-Posed Probl.}, fjournal = {Journal of Inverse and Ill-Posed Problems}, volume = {28}, year = {2020}, number = {4}, pages = {465--470}, issn = {0928-0219,1569-3945}, mrclass = {35P25 (35J05 35R30 81U40)}, mrnumber = {4129366}, doi = {10.1515/jiip-2019-0019}, url = {https://doi.org/10.1515/jiip-2019-0019}, keywords = {Inverse Scattering, Inverse Problems} } - RMCResolvent estimates for the magnetic Schrödinger operator in dimensions ≥2Cristóbal J. Meroño, Leyter Potenciano-Machado, and Mikko Salo
It is well known that the resolvent of the free Schrödinger operator on weighted $L^2 spaces has norm decaying like λ^-\frac12 at energy λ. There are several works proving analogous high frequency estimates for magnetic Schrödinger operators, with large long or short range potentials, in dimensions n \ge 3. We prove that the same estimates remain valid in all dimensions n \ge 2$.
@article{MR4082592, author = {Mero\~no, Crist\'obal J. and Potenciano-Machado, Leyter and Salo, Mikko}, title = {Resolvent estimates for the magnetic {S}chr\"odinger operator in dimensions {$\geq 2$}}, journal = {Rev. Mat. Complut.}, fjournal = {Revista Matem\'atica Complutense}, volume = {33}, year = {2020}, number = {2}, pages = {619--641}, issn = {1139-1138,1988-2807}, mrclass = {35J10 (35P20 47A10 58J50)}, mrnumber = {4082592}, mrreviewer = {L.\ V.\ Kritskov}, doi = {10.1007/s13163-019-00316-z}, url = {https://doi.org/10.1007/s13163-019-00316-z}, keywords = {Inverse Scattering, Resolvent estimates} }
2019
- JDERecovery of the singularities of a potential from backscattering data in general dimensionCristóbal J. Meroño
We prove that in dimension n≥2 the main singularities of a complex potential q having a certain a priori regularity are contained in the Born approximation qB constructed from backscattering data. This is archived using a new explicit formula for the multiple dispersion operators in the Fourier transform side. We also show that q−qB can be up to one derivative more regular than q in the Sobolev scale. On the other hand, we construct counterexamples showing that in general it is not possible to have more than one derivative gain, sometimes even strictly less, depending on the a priori regularity of q.
@article{MR3926070, author = {Mero\~no, Crist\'obal J.}, title = {Recovery of the singularities of a potential from backscattering data in general dimension}, journal = {J. Differential Equations}, fjournal = {Journal of Differential Equations}, volume = {266}, year = {2019}, number = {10}, pages = {6307--6345}, issn = {0022-0396,1090-2732}, mrclass = {35J10 (35P25 35R30 81U40)}, mrnumber = {3926070}, mrreviewer = {Tomio\ Umeda}, doi = {10.1016/j.jde.2018.11.003}, url = {https://doi.org/10.1016/j.jde.2018.11.003}, keywords = {Schrödinger equation, Inverse scattering, Inverse Problems} }
2018
- SIAM JMAFixed angle scattering: recovery of singularities and its limitationsCristóbal J. Meroño
We prove that in dimension \n \ge 2 the main singularities of a complex potential \q having a certain a priori regularity are contained in the Born approximation \q_θconstructed from fixed angle scattering data. Moreover, {q-q_θ can be up to one derivative more regular than \q in the Sobolev scale. In fact, this result is optimal. We construct a family of compactly supported and radial potentials for which it is not possible to have more than one derivative gain. Also, these functions show that for \n>3 the maximum derivative gain can be very small for potentials in the Sobolev scale not having a certain a priori level of regularity which grows with the dimension.
@article{MR3867616, author = {Mero\~no, Crist\'obal J.}, title = {Fixed angle scattering: recovery of singularities and its limitations}, journal = {SIAM J. Math. Anal.}, fjournal = {SIAM Journal on Mathematical Analysis}, volume = {50}, year = {2018}, number = {5}, pages = {5616--5636}, issn = {0036-1410,1095-7154}, mrclass = {35R30 (35J05 35P25 42B20 81U40)}, mrnumber = {3867616}, mrreviewer = {Drossos\ Gintides}, doi = {10.1137/18M1164871}, url = {https://doi.org/10.1137/18M1164871}, keywords = {Schrödinger equation, Inverse scattering, Inverse Problems} }