A Qualitative Approach to Inverse Scattering Theory

Cakoni, Fioralba; Colton, David

doi:10.1007/978-1-4614-8827-9

Cited by 237 publications

(393 citation statements)

References 118 publications

(239 reference statements)

Supporting

Mentioning

383

Contrasting

Unclassified

Order By: Relevance

“…Then there exists a unique solution ψ + ∈ H p−2 σ (R + ,H −1/2 ( + )) of Equation (18). Moreover, if C 2 is the constant defined in Lemma 5, we obtain that…”

Section: Downloaded By [University Of Bath] At 05:40 21 June 2016mentioning

confidence: 88%

“…Moreover, the LSM is relatively fast and easy to implement because of its non-iterative nature. For more details on the frequency domain LSM, we refer to [2,17,18] and the references therein. In recent years, the time domain version of LSM has been successfully applied to the time-dependent inverse scattering problems of recovering bounded objects.…”

Section: Introductionmentioning

confidence: 99%

See 1 more Smart Citation

Time domain scattering and inverse scattering problems in a locally perturbed half-plane

Chen

Guo

2016

Applicable Analysis

View full text Add to dashboard Cite

This paper is concerned with efficient numerical methods for solving the time-dependent scattering and inverse scattering problems of acoustic waves in a locally perturbed half-plane. By symmetric continuation, the scattering problem is reformulated as an equivalent symmetric problem defined in the whole plane. The retarded potential boundary integral equation method is modified to solve the forward problem. Then we consider the inverse scattering problem of determinating the local perturbation from the measured scattered data. The time domain linear sampling method is employed to deal with the inverse problem. The computation schemes proposed in this paper are relatively simple and easy to implement. Several numerical examples are presented to show the effectiveness of the proposed methods. ARTICLE HISTORY

show abstract

“…Then there exists a unique solution ψ + ∈ H p−2 σ (R + ,H −1/2 ( + )) of Equation (18). Moreover, if C 2 is the constant defined in Lemma 5, we obtain that…”

Section: Downloaded By [University Of Bath] At 05:40 21 June 2016mentioning

confidence: 88%

Section: Introductionmentioning

confidence: 99%

Time domain scattering and inverse scattering problems in a locally perturbed half-plane

Chen

Guo

2016

Applicable Analysis

View full text Add to dashboard Cite

show abstract

“…. , 2 6 ) for fixed κ = 0.25. The former case in plot (b) describes the asymptotic behavior of the tested methods as κ = kh 2 → 0.…”

Section: The A-posteriori Numerical Analysismentioning

confidence: 99%

“…Except for eigenvalues, the unique solvability of the variational equation (3.1) is argued by the Fredholm's theorem, see e.g. [6,Section 5.3]. The solution of (3.1) fulfills the homogeneous Helmholtz equation: − u−k 2 u = 0 in .…”

Section: The Helmholtz Problem Formulation and Discretizationmentioning

confidence: 99%

Revisiting generalized FEM: a Petrov–Galerkin enrichment based FEM interpolation for Helmholtz problem

Kovtunenko

Kunisch

2018

Calcolo

View full text Add to dashboard Cite

A concept of Petrov-Galerkin enrichment which is appropriate for highly accurate and stable interpolation of variational solutions is introduced. In the finite element context, the setting refers to standard trial functions for the solution, while the test space will be enriched. The FEM interpolation procedure that we propose will be justified by local wavelets with vanishing moments based on Gegenbauer polynomials. For the reference Helmholtz equation, the continuous piecewise polynomial test functions are enriched using dispersion analysis on uniform meshes in 2d and 3d. From a-priori and a-posteriori numerical analysis it follows that the Petrov-Galerkin based enrichment approximates the exact interpolate solution of the Helmholtz equation with at least seventh order of accuracy.

show abstract

“…Many of the inverse problems that arise in these fields can mathematically be formulated as Cauchy problems for elliptic partial differential equations; examples include a classical thermostatics problem which consists of recovering the temperature in a given domain when it's distribution and heat flux are known over the accessible region of the boundary [23,5,4,25], electrostatics problem encountered in electric impedance tomography [13,12], corrosion detection [18,17,1], inverse scattering problems [11,22,3]. Cauchy problems for elliptic equations as with many other inverse problems are known to be ill-posed.…”

Section: Introductionmentioning

confidence: 99%

Non-linear inverse geothermal problems

Wokiyi¹

2017

View full text Add to dashboard Cite

The inverse geothermal problem consist of estimating the temperature distribution below the earth's surface using temperature and heat-flux measurements on the earth's surface. The problem is important since temperature governs a variety of the geological processes including formation of magmas, minerals, fosil fuels and also deformation of rocks. Mathematical this problem is formulated as a Cauchy problem for an non-linear elliptic equation and since the thermal properties of the rocks depend strongly on the temperature, the problem is non-linear. This problem is ill-posed in the sense that it does not satisfy atleast one of Hadamard's definition of well-posedness.We formulated the problem as an ill-posed non-linear operator equation which is defined in terms of solving a well-posed boundary problem. We demonstrate existence of a unique solution to this well-posed problem and give stability estimates in appropriate function spaces. We show that the operator equation is well-defined in appropriate function spaces.Since the problem is ill-posed, regularization is needed to stabilize computations. We demostrate that Tikhonov regularization can be implemented efficiently for solving the operator equation. The algorithm is based on having a code for solving a wellposed problem related to the operator equation. In this study we demostrate that the algorithm works efficiently for 2D calculations but can also be modified to work for 3D calculations.i

show abstract

A Qualitative Approach to Inverse Scattering Theory

Cited by 237 publications

References 118 publications

Time domain scattering and inverse scattering problems in a locally perturbed half-plane

Time domain scattering and inverse scattering problems in a locally perturbed half-plane

Revisiting generalized FEM: a Petrov–Galerkin enrichment based FEM interpolation for Helmholtz problem

Non-linear inverse geothermal problems

Contact Info

Product

Resources

About