Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers

被引：11

作者：

Barcelo, J. A. ^{[2
]}

Faraco, D. ^{[1
]}

Ruiz, A. ^{[1
]}

Vargas, A. ^{[1
]}

机构：

[1] Univ Autonoma Madrid, Dept Matemat, E-28049 Madrid, Spain

[2] Univ Politecn Madrid, ETSI Caminos, E-28040 Madrid, Spain

来源：

MATHEMATISCHE ANNALEN | 2010年 / 346卷 / 03期

关键词：

INVERSE BACKSCATTERING; RECOVERING SINGULARITIES; SCHRODINGER-EQUATION; UNIQUE CONTINUATION; DIMENSIONS; OPERATORS; INTEGRALS; CONVERGENCE; COMMUTATORS; SERIES;

D O I：

10.1007/s00208-009-0398-5

中图分类号：

O1 [数学];

学科分类号：

0701 ; 070101 ;

摘要：

We prove that the singularities of a complex valued potential q in the Schrodinger hamiltonian Delta + q can be reconstructed from the linear Born approximation for full scattering data by averaging in the extra variables. We prove that, with this procedure, the accuracy in the reconstruction improves the previously known accuracy obtained from fixed angle or backscattering data. In particular, for q is an element of W-alpha,W- 2 for alpha >= 0, in 2D we recover the main singularity of q with an accuracy of one derivative; in 3D the accuracy is epsilon > 1/2, increasing with alpha. This gives a mathematical basis for diffraction tomography. The proof is based on some new estimates for multidimensional bilinear Fourier multipliers of independent interest.

引用

页码：505 / 544

页数：40

共 50 条

[1] Reconstruction of singularities from full scattering data by new estimates of bilinear Fourier multipliers
J. A. Barceló
D. Faraco
A. Ruiz
A. Vargas
[J]. Mathematische Annalen, 2010, 346 : 505 - 544
[2] RECOVERING SINGULARITIES OF A POTENTIAL FROM SINGULARITIES OF SCATTERING DATA
GREENLEAF, A
UHLMANN, G
[J]. COMMUNICATIONS IN MATHEMATICAL PHYSICS, 1993, 157 (03) : 549 - 572
[3] Stability estimates for reconstruction from the Fourier transform on the ball
Isaev, Mikhail
Novikov, Roman G.
[J]. JOURNAL OF INVERSE AND ILL-POSED PROBLEMS, 2021, 29 (03): : 421 - 433
[4] Nonnegative image reconstruction from sparse Fourier data: a new deconvolution algorithm
Bonettini, S.
Prato, M.
[J]. INVERSE PROBLEMS, 2010, 26 (09)
[5] Image reconstruction from limited Fourier data
Shieh, Hsin M.
Byrne, Charles L.
[J]. JOURNAL OF THE OPTICAL SOCIETY OF AMERICA A-OPTICS IMAGE SCIENCE AND VISION, 2006, 23 (11) : 2732 - 2736
[6] RECONSTRUCTION OF THE POTENTIAL FROM SCATTERING DATA
KUKULIN, VI
POMERANTSEV, VN
HORACEK, J
[J]. PHYSICAL REVIEW A, 1990, 42 (05): : 2719 - 2727
[7] Recovery of the singularities of a potential from fixed angle scattering data
Ruiz, A
[J]. COMMUNICATIONS IN PARTIAL DIFFERENTIAL EQUATIONS, 2001, 26 (9-10) : 1721 - 1738
[8] Applications of a New Program for the Reconstruction of Protein Envelopes from Solution Scattering Data
Badger, John
[J]. ACTA CRYSTALLOGRAPHICA A-FOUNDATION AND ADVANCES, 2019, 75 : A1 - A1
[9] Optical imaging from scattering data: imaging from Fourier intensity data
Noushin, AJ
Fiddy, MA
Pommet, DA
[J]. COMPUTATIONAL, EXPERIMENTAL, AND NUMERICAL METHODS FOR SOLVING ILL-POSED INVERSE IMAGING PROBLEMS: MEDICAL AND NONMEDICAL APPLICATIONS, 1997, 3171 : 96 - 103
[10] Image reconstruction from real scattering data
Rieger, W
Buchau, A
Haas, M
Huber, C
Lehner, G
Rucker, WM
[J]. COMPEL-THE INTERNATIONAL JOURNAL FOR COMPUTATION AND MATHEMATICS IN ELECTRICAL AND ELECTRONIC ENGINEERING, 1999, 18 (03) : 372 - 384

← 1 2 3 4 5 →