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We analyze the factorization method (introduced by Kirsch in 1998 to solve inverse scattering problems at fixed frequency from the far field operator) for a general class of boundary conditions that generalizes impedance boundary conditions. For instance, when the surface impedance operator is of pseudo-differential type, our main result stipulates that the factorization method works if the order of this operator is different from one and the operator is Fredholm of index zero with non-negative imaginary part. We also provide some validating numerical examples for boundary operators of second order with discussion on the choice of the test function. [ABSTRACT FROM AUTHOR]

Akduman, Haddar and Kress [Akduman I, Kress R. Electrostatic imaging via conformal mapping. Inverse Prob. 2002;18:1659–1672; Haddar H, Kress R. Conformal mappings and inverse boundary value problems. Inverse Prob. 2005;21:935–953; Kress R. Inverse Dirichlet problem and conformal mapping. Math.Comput. Simul. 2004;66:255–265] have employed a conformal mapping technique for the inverse problem to reconstruct a perfectly conducting inclusion in a homogeneous background medium from Cauchy data for electrostatic imaging, that is, for solving an inverse boundary value problem for the Laplace equation. We propose an extension of this approach to inverse obstacle scattering for time-harmonic waves, that is, to the solution of an inverse boundary value problem for the Helmholtz equation. The main idea is to use the conformal mapping algorithm in an iterative procedure to obtain Cauchy data for a Laplace problem from the given Cauchy data for the Helmholtz problem. We present the foundations of the method together with a convergence result and exhibit the feasibility of the method via numerical examples. [ABSTRACT FROM PUBLISHER]

We consider inverse obstacle scattering problems for the wave equation with Robin or Neumann boundary conditions. The problem of reconstructing the geometry of such obstacles from measurements of scattered waves in the time domain is tackled using a time domain linear sampling method. This imaging technique yields a picture of the scatterer by solving a linear operator equation involving the measured data for many right-hand sides given by singular solutions to the wave equation. We analyse this algorithm for causal and smooth impulse shapes, we discuss the effect of different choices of the singular solutions used in the algorithm, and finally we propose a fast FFT-based implementation. [ABSTRACT FROM AUTHOR]

We use the Factorization method to retrieve the shape of cracks with impedance boundary conditions from far fields associated with incident plane waves at a fixed frequency. This work is an extension of the study initiated by Kirsch and Ritter [Inverse Problems, 16, pp. 89-105, 2000] where the case of sound soft cracks is considered. We address here the scalar problem and provide theoretical validation of the method when the impedance boundary conditions hold on both sides of the crack. We then deduce an inversion algorithm and present some validating numerical results in the case of simply and multiply connected cracks. [ABSTRACT FROM AUTHOR]

We use the linear sampling method (LSM) to identify a crack with impedance boundary conditions from far-field measurements at a fixed frequency. This article extends the work of Cakoni–Colton [F. Cakoni and D. Colton,The linear sampling method for cracks, Inverse Probl. 19 (2003), pp. 279–295] where LSM has been used to reconstruct a crack with impedance boundary conditions on one side of the crack and a Dirichlet boundary condition on the other one. In addition, we present two methods to also reconstruct the impedance parameters whence the geometry is known. The first one is based on the interpretation of the indicator function produced by the LSM, while the second one is a natural approach based on the integral representation of the far-field in terms of densities on the crack geometry. The performance of the different reconstruction methods is illustrated through numerical examples in a 2D setting of the scattering problem. [ABSTRACT FROM AUTHOR]

We consider the interior transmission problem corresponding to the inverse scattering by an inhomogeneous (possibly anisotropic) media in which an impenetrable obstacle with Dirichlet boundary conditions is embedded. Our main focus is to understand the associated eigenvalue problem, more specifically to prove that the transmission eigenvalues form a discrete set and show that they exist. The presence of Dirichlet obstacle brings new difficulties to already complicated situation dealing with a non-selfadjoint eigenvalue problem. In this paper, we employ a variety of variational techniques under various assumptions on the index of refraction as well as the size of the Dirichlet obstacle. [ABSTRACT FROM AUTHOR]

A monochromatic, i.e., fixed-frequency, back-scattering kernel measured at all angles does not uniquely determine the index of refraction in an inhomogeneous medium, nor can it guarantee any upper bound on the support of the inhomogeneity. We show that it is possible to associate with any such kernel its convex back-scattering support, a convex set which must be a subset of the convex hull of the support of any inhomogeneity with that back-scattering kernel. For the Born approximation, we further demonstrate that there is an inhomogeneity supported in any neighborhood of the convex back-scattering support which has exactly that back-scattering kernel. Last, we discuss a practical implementation of these results and include a numerical example. [ABSTRACT FROM AUTHOR]

We consider the inverse scattering problem of determining the shape of an obstacle in R from knowledge of the time harmonic incident electromagnetic wave and the far-field pattern of the scattered wave with frequency in the resonance region. The approach used is the linear sampling method which does not require a priori knowledge of either the boundary condition or the connectivity of the scattering obstacle. Numerical examples are given for the case of both a simply connected perfect conductor and multiply connected obstacles satisfying an impedance boundary condition on the boundary of one component and satisfying a perfect conducting boundary condition on the boundary of another component. [ABSTRACT FROM AUTHOR]

Abstract: We show that the Herglotz wave function with kernel the Tikhonov regularized solution of the far field equation becomes unbounded as the regularization parameter tends to zero iff the wavenumber k belongs to a discrete set of values. When the scatterer is such that the total field vanishes on the boundary, these values correspond to the square root of Dirichlet eigenvalues for −Δ. When the scatterer is a nonabsorbing inhomogeneous medium these values correspond to so-called transmission eigenvalues. [Copyright &y& Elsevier]

Diffusion magnetic resonance imaging (dMRI) is an imaging modality that probes the diffusion characteristics of a sample via the application of magnetic field gradient pulses. The dMRI signal from a heterogeneous sample includes the contribution of the water proton magnetization from all spatial positions in a voxel. If the voxel can be spatially divided into different Gaussian diffusion compartments with inter-compartment exchange governed by linear kinetics, then the dMRI signal can be approximated using the macroscopic Karger model, which is a system of coupled ordinary differential equations (ODEs), under the assumption that the duration of the diffusion-encoding gradient pulses is short compared to the diffusion time (the narrow pulse assumption). Recently, a new macroscopic model of the dMRI signal, without the narrow pulse restriction, was derived from the Bloch–Torrey partial differential equation (PDE) using periodic homogenization techniques. When restricted to narrow pulses, this new homogenized model has the same form as the Karger model. We conduct a numerical study of the new homogenized model for voxels that are made up of periodic copies of a representative volume that contains spherical and cylindrical cells of various sizes and orientations and show that the signal predicted by the new model approaches the reference signal obtained by solving the full Bloch–Torrey PDE in O ( ε 2 ) , where ε is the ratio between the size of the representative volume and a measure of the diffusion length. When the narrow gradient pulse assumption is not satisfied, the new homogenized model offers a much better approximation of the full PDE signal than the Karger model. Finally, preliminary results of applying the new model to a voxel that is not made up of periodic copies of a representative volume are shown and discussed. [ABSTRACT FROM AUTHOR]

In this paper, we develop a conceptually unified approach for characterizing and determining scattering poles and interior eigenvalues for a given scattering problem. Our approach explores a duality stemming from interchanging the roles of incident and scattered fields in our analysis. Both sets are related to the kernel of the relative scattering operator mapping incident fields to scattered fields, corresponding to the exterior scattering problem for the interior eigenvalues and the interior scattering problem for scattering poles. Our discussion includes the scattering problem for a Dirichlet obstacle where duality is between scattering poles and Dirichlet eigenvalues, and the inhomogeneous scattering problem where the duality is between scattering poles and transmission eigenvalues. Our new characterization of the scattering poles suggests a numerical method for their computation in terms of scattering data for the corresponding interior scattering problem., Competing Interests: We declare we have no competing interests., (© 2020 The Author(s).)

We use homogenization theory to establish a new macroscopic model for the complex transverse water proton magnetization in a voxel due to diffusion-encoding magnetic field gradient pulses in the case of biological tissue with impermeable membranes. In this model, new higher-order diffusion tensors emerge and offer more information about the structure of the biological tissues. We explicitly solve the macroscopic model to obtain an ordinary differential equation for the diffusion MRI signal that has similar structure as diffusional kurtosis imaging models. We finally present some validating numerical results on synthetic examples showing the accuracy of the model with respect to signals obtained by solving the Bloch-Torrey equation., (Copyright © 2018 Elsevier Inc. All rights reserved.)