Inverse nodal and inverse spectral problems for discontinuous boundary value problems

Inverse nodal and inverse spectral problems are studied for second-order differential operators on a finite interval with discontinuity conditions inside the interval. Uniqueness theorems are proved, and a constructive procedure for the solution is provided.     

A decomposition scheme for acoustic obstacle scattering in a multilayered medium

We consider the acoustic wave scattering by an impenetrable obstacle embedded in a multilayered background medium, which is modelled by a linear system constituted by the Helmholtz equations with different wave numbers and the transmission conditions across the interfaces. The aim of this article is to construct an efficient computing scheme for the scattered waves for this complex scattering process, with a rigorous mathematical analysis. First, we construct a set of functions by a series of coupled transmission problems, which are proven to be well-defined. Then, the solution to our complex scattering in each layer is decomposed as the summation in terms of these functions, which are essentially the contributions from two interfaces enclosing this layer. These contributions physically correspond to the scattered fields for simple scattering problems, which do not involve the multiple scattering and are coupled via the boundary conditions. Finally, we propose an iteration scheme to compute the wave field in each layer decoupling the multiple scattering effects, with the advantage that only the solvers for the well-known transmission problems and an obstacle scattering problem in a homogeneous background medium are applied. The convergence property of this iteration scheme is proven.     

The use of the linear sampling method for obtaining super-resolution effects in Born approximation

A two-step reconstruction scheme is introduced to solve fixed frequency inverse scattering problems in Born approximation conditions. The aim of the approach is to achieve super-resolution effects by constraining the inversion method to exploit some a priori knowledge on the scatterer. Therefore, the first step is to apply the linear sampling method to the far-field data in order to obtain an estimate of the support of the inhomogeneity. The second step is to apply the projected Landweber method to the linearized scattering equation in order to obtain super-resolution effects via out-of-band extrapolation. The effectiveness of the approach, which has a rather wide applicability power, is tested in the case of a two-dimensional problem for some scatterers of simple geometry.     

Inverse spectral problems for differential pencils on the half-line with turning points

The inverse spectral problem of recovering pencils of second-order differential operators on the half-line with turning points is studied. We establish properties of the spectral characteristics, give a formulation of the inverse problem, prove a uniqueness theorem and provide a constructive procedure for the solution of the inverse problem.     

Inverse spectral problems for Sturm-Liouville operators with singular potentials. Part III: Reconstruction by three spectra

We solve the inverse spectral problem of recovering the singular potential from W⁻¹₂(0,1) of a Sturm-Liouville operator by its spectra on the three intervals [0,1], [0,a], and [a,1] for some a∈(0,1). Necessary and sufficient conditions on the spectral data are derived, and uniqueness of the solution is analyzed.     

The inverse eigenvalue problem for a weighted helmholtz equation

Given the m lowest eigenvalues, we seek to recover an approximation to the density function ρ in the weighted Helmholtz equation -Δ=λρu on a rectangle with Dirchlet boundary conditions. The density ρ is assumed to be symmetric with respect to the midlines of the rectangle. Projection of the boundary value problem and the unknown density function onto appropriate vector spaces leads to a matrix inverse problem. Solutions of the matrix inverse problem exist provided that the reciprocals of the prescribed eigenvalues are close to the reciprocals of the simple eigenvalues of the base problem with ρ = 1. The matrix inverse problem is solved by a fixed—point iterative method and a density function ρ^* is constructed which has the same m lowest eigenvalues as the unknown ρ. The algorithm can be modified when multiple base eigenvalues arise, although the success of the modification depends on the symmetry properties of the base eigenfunctions.     

Inverse Spectral Problems in Rectangular Domains

We consider the Schrödinger operator ? Δ + q in domains of the form R = {x ∈ ? ⁿ : 0 ≤ x _i  ≤ a _i , i = 1,…, n} with either Dirichlet or Neumann boundary conditions on the faces of R, and study the constraints on q imposed by fixing the spectrum of ? Δ + q with these boundary conditions. We work in the space of potentials, q, which become real-analytic on ? ⁿ when they are extended evenly across the coordinate planes and then periodically. Our results have the corollary that there are no continuous isospectral deformations for these operators within that class of potentials. This work is based on new formulas for the trace of the wave group in this setting. In addition to the inverse spectral results these formulas lead to asymptotic expansions for the traces of the wave and heat kernels on rectangular domains.     

An inverse pollution problem in porous media

In this paper, we deal with an inverse source problem of integro-differential parabolic equations, which comes from nonlinear pollution problems in porous media. We study the existence and uniqueness of solutions for the direct problem as well as the existence of quasisolutions of the inverse source problem in an appropriate class of admissible source functions.     

The inverse electromagnetic scattering problem for a partially coated dielectric

We use the linear sampling method to determine the shape and surface conductivity of a partially coated dielectric infinite cylinder from knowledge of the far field pattern of the scattered TM polarized electromagnetic wave at fixed frequency. A mathematical justification of the method is provided based on the use of a complete family of solutions. Numerical examples are given showing the efficiency of our method.     

On imaging obstacles inside inhomogeneous media

We show that an obstacle inside a known inhomogeneous medium can be determined from measurements of the scattering amplitude at one frequency, without a priori knowledge of the boundary condition. We also show that an obstacle inside a known inhomogeneous anisotropic conducting medium can be determined from electrostatic current and voltage measurements on the boundary of a domain containing the obstacle. Moreover, two obstacles with boundary measurements which are merely comparable as operators must be identical. The first part of the paper gives an extension of the factorization method which may be of independent interest and also yields a new reconstruction procedure.     

On inverse spectral problem for non-selfadjoint Sturm-Liouville operator on a finite interval

An inverse spectral problem is studied for a non-selfadjoint Sturm-Liouville operator on a finite interval with an arbitrary behavior of the spectrum. The spectral data introduced generalize the classical discrete spectral data corresponding to the specification of the spectral function in the selfadjoint case. The connection with other types of spectral characteristics is investigated and a uniqueness theorem is proved. A constructive procedure for solving the inverse problem is given.     

New characterizations of solutions in inverse scattering theory

This paper introduces a new approach to characterize the shape ω of a scattering medium (either an acoustically soft obstacle or an inhomogeneous medium) by the far field data. In contrast to the Linear Sampling Method normality of the far field operator is not needed. Therefore, also scattering by limited far field data and absorbing media can be treated. While in the Linear Sampling Method the points in the interior of ω are characterized by the solution of an integral equation of the first kind, for our new method a constrained optimization problem has to be solved. Although this new approach is more time consuming some numerical experiments at the end of this paper show the practicability of the method.     

The recovery of even polynomial potentials

We consider the problem of reconstructing an even polynomial potential from one set of spectral data of a Sturm-Liouville problem. We show that we can recover an even polynomial of degree 2m from m+1 given Taylor coefficients of the characteristic function whose zeros are the eigenvalues of one spectrum. The idea here is to represent the solution as a power series and identify the unknown coefficients from the characteristic function. We then compute these coefficients by solving a nonlinear algebraic system, and provide numerical examples at the end. Because of its algebraic nature, the method applies also to non self-adjoint problems.     

Locating an obstacle in a 3D finite depth ocean using the convex scattering support

We consider an inverse scattering problem in a 3D homogeneous shallow ocean. Specifically, we describe a simple and efficient inverse method which can compute an approximation of the vertical projection of an immersed obstacle. This reconstruction is obtained from the far-field patterns generated by illuminating the obstacle with a single incident wave at a given fixed frequency. The technique is based on an implementation of the theory of the convex scattering support [S. Kusiak, J. Sylvester, The scattering support, Commun. Pure Appl. Math. (2003) 1525–1548]. A few examples are presented to show the feasibility of the method.