Recovering the Dirichlet-to-Neumann map in inverse scattering problems using integral equation methods

Consider the reconstruction of Dirichlet-to-Neumann map(D-to-N map) from the far-field patterns of the scattered waves in inverse scattering problems, which is the first step in detecting the obstacle boundary by the probe method using far-field measurements corresponding to all incident plane waves. In principle, this problem can be reduced to solving an integral equation of the second kind with the kernels involving the derivatives of the scattered waves for point sources. Based on the mixed reciprocity principle, we propose two simple and feasible numerical schemes for reconstructing D-to-N map. Compared with the well-known obstacle boundary recovering schemes using the simulation of D-to-N map directly, the proposed schemes give the possible ways to realizing the probe methods using practical far-field data, with the advantage of no numerical differentiation for scattered wave in their implementations. We present some numerical examples for the D-to-N map, showing the validity and stability of our schemes.     

Linearized inverse problem for the Dirichlet-to-Neumann map on differential forms

For a compact n-dimensional Riemannian manifold (M,g) with boundary i:∂M⊂M, the Dirichlet-to-Neumann (DN) map Λg:Ωk(∂M)→Ωn−k−1(∂M) is defined on exterior differential forms by Λgφ=i^∗(?dω), where ω solves the boundary value problem Δω=0, i^∗ω=φ, i^∗δω=0. For a symmetric second rank tensor field h on M, let be the Gateaux derivative of the DN map in the direction h. We study the question: for a given (M,g), how large is the subspace of tensor fields h satisfying ? Potential tensor fields belong to the subspace since the DN map is invariant under isomeries fixing the boundary. For a manifold of an even dimension n, the DN map on (n/2−1)-forms is conformally invariant, therefore spherical tensor fields belong to the subspace in the case of k=n/2−1. The manifold is said to be Ωk-rigid if there is no other h satisfying . We prove that the Ωk-rigidity is equivalent to the density of the range of some bilinear form on the space of exact harmonic fields.     

Stability estimates in inverse scattering

An algorithm is given for calculating the solution to the 3D inverse scattering problem with noisy discrete fixed energy data. The error estimates for the calculated solution are derived. The methods developed are of a general nature and can be used in many applications: in nondestructive evaluation and remote sensing, in geophysical exploration, medical diagnostics, and technology.     

Stability estimate for the hyperbolic inverse boundary value problem by local Dirichlet-to-Neumann map

In this paper we consider the stability of the inverse problem of determining a function q(x) in a wave equation in a bounded smooth domain in Rn from boundary observations. This information is enclosed in the hyperbolic (dynamic) Dirichlet-to-Neumann map associated to the solutions to the wave equation. We prove in the case of n?2 that q(x) is uniquely determined by the range restricted to a subboundary of the Dirichlet-to-Neumann map whose stability is a type of double logarithm.     

On the reconstruction of media inhomogeneity by inverse wave scattering model

Consider the reconstruction of the complex refraction index of an object, which is immersed in a known homogeneous background, from the knowledge of scattered waves of the point sources outside of the object. We firstly establish the uniqueness for this inverse problem, which provides the theoretical basis for the reconstruction scheme. Then based on the contrast source inversion(CSI) method, we propose an algorithm determining the refraction index and the artificial wave sources alternately by a dynamic iterative scheme. The algorithm defines the iterates by solving a series of minimization problems with uniformly convex penalty terms, which are allowed to be non-smooth to include L1 and total variation like functionals, ensuring the reconstruction quality when the unknown refraction index has the special features such as sparsity and discontinuity. By choosing the regularizing parameter automatically, the algorithm is terminated in terms of discrepancy principle. The convergence property of the iterative sequence is rigorously proven. Numerical implementations demonstrate the validity of the proposed algorithm.     

Stability estimate for an inverse problem for the magnetic Schrödinger equation from the Dirichlet-to-Neumann map

We consider the problem of stability estimate of the inverse problem of determining the magnetic field entering the magnetic Schrödinger equation in a bounded smooth domain of Rn with input Dirichlet data, from measured Neumann boundary observations. This information is enclosed in the dynamical Dirichlet-to-Neumann map associated to the solutions of the magnetic Schrödinger equation. We prove in dimension n?2 that the knowledge of the Dirichlet-to-Neumann map for the magnetic Schrödinger equation measured on the boundary determines uniquely the magnetic field and we prove a Hölder-type stability in determining the magnetic field induced by the magnetic potential.     

Direct and inverse multichannel scattering problems

We consider small oscillations of a system of pairwise interacting particles in an external field near a stable equilibrium. The system is assumed to consist of finitely many channels, i.e., semi-infinite linear chains of particles, attached to a scatterer, which is a finite system of interacting particles. Direct and inverse scattering problems are considered. In particular, an algorithm finding characteristics of the channels on the basis of scattering data is given.     

On approximation in total variation penalization for image reconstruction and inverse problems

In this paper, we examine some theoretical issues associated with the use of total variation based image reconstruction. Our investigations are motivated by problems of inverse interferome-try, in which laser light phase shifts are used to reconstruct medium density profiles in flow field sensing. The reconstruction problem is posed as a residual minimization with total variation reg-ularization applied to handle the inherent ill-posedness. We consider numerical approximations of these penalized minimal residual problems, and analyze some approximation strategies and their properties. The standard definition of total variation leads to inconsistent approximations, with piecewise constant basis functions, so we consider alternative definitions, which preserve the needed compactness and produce convergent approximations.     

Carleman estimates and inverse problems for Dirac operators

We consider limiting Carleman weights for Dirac operators and prove corresponding Carleman estimates. In particular, we show that limiting Carleman weights for the Laplacian also serve as limiting weights for Dirac operators. As an application we consider the inverse problem of recovering a Lipschitz continuous magnetic field and electric potential from boundary measurements for the Pauli Dirac operator. M. Salo is supported by the Academy of Finland. L. Tzou is supported by the Doctoral Post-Graduate Scholarship from the Natural Science and Engineering Research Council of Canada. This article was written while L. Tzou was visiting the University of Helsinki and TKK, whose hospitality is gratefully acknowledged. The authors would like to thank András Vasy and Lauri Ylinen for useful comments.     

Some inverse scattering problems on star-shaped graphs

Having in mind applications to the fault-detection/diagnosis of lossless electrical networks, here we consider some inverse scattering problems for Schrödinger operators over star-shaped graphs. We restrict ourselves to the case of minimal experimental setup consisting in measuring, at most, two reflection coefficients when an infinite homogeneous (potential-less) branch is added to the central node. First, by studying the asymptotic behavior of only one reflection coefficient in the high-frequency limit, we prove the identifiability of the geometry of this star-shaped graph: the number of edges and their lengths. Next, we study the potential identification problem by inverse scattering, noting that the potentials represent the inhomogeneities due to the soft faults in the network wirings (potentials with bounded H¹-norms). The main result states that, under some assumptions on the geometry of the graph, the measurement of two reflection coefficients, associated to two different sets of boundary conditions at the external vertices of the tree, determines uniquely the potentials; it can be seen as a generalization of the theorem of the two boundary spectra on an interval.     

First eigenvalue estimates of Dirichlet-to-Neumann operators on graphs

Following Escobar (J Funct Anal 150(2):544–556, 1997) and Jammes (Ann l’Inst Fourier 65(3):1381–1385, 2015), we introduce two types of isoperimetric constants and give lower bound estimates for the first nontrivial eigenvalues of Dirichlet-to-Neumann operators on finite graphs with boundary respectively.     

Marching schemes for inverse acoustic scattering problems

Summary For the numerical solution of inverse Helmholtz problems the boundary value problem for a Helmholtz equation with spatially variable wave number has to be solved repeatedly. For large wave numbers this is a challenge. In the paper we reformulate the inverse problem as an initial value problem, and describe a marching scheme for the numerical computation that needs only n² log n operations on an n × n grid. We derive stability and error estimates for the marching scheme. We show that the marching solution is close to the low-pass filtered true solution. We present numerical examples that demonstrate the efficacy of the marching scheme.     

On spherical-wave scattering by a spherical scatterer and related near-field inverse problems

A spherical acoustic wave is scattered by a bounded obstacle.A generalization of the ‘optical theorem’ (whichrelates the scattering cross-section to the far-field patternin the forward direction for an incident plane wave) is proved.For a spherical scatterer, low-frequency results are obtainedby approximating the known exact solution (separation of variables).In particular, a closed-form approximation for the scatteredwavefield at the source of the incident spherical wave is obtained.This leads to the explicit solution of some simple near-fieldinverse problems, where both the source and coincident receiverare located at several points in the vicinity of a small sphere.     

On the inverse stochastic reconstruction problem

We consider the reconstruction problem in the class of stochastic differential equations of Itô type on the basis of given motion properties that depend only on part of the variables. We determine the set of controls providing necessary and sufficient conditions for the existence of a given integral manifold.     

Analysis of subspace migrations in limited-view inverse scattering problems

An analysis of subspace migration that occurs in the limited-view inverse scattering problem is considered herein. Based on the structure of singular vectors associated with the nonzero singular values of the multi-static response matrix, we establish a relationship between the subspace migration imaging function and Bessel functions of integer order of the first kind. The revealed structure and numerical examples show why subspace migration is applicable for imaging of small scatterers in limited-view inverse scattering problems.