Inverse Backscattering Problem Research Articles

Inverse scattering for a random potential

In this paper, we consider an inverse problem for the [Formula: see text]-dimensional random Schrödinger equation [Formula: see text]. We study the scattering of plane waves in the presence of a potential [Formula: see text] which is assumed to be a Gaussian random function such that its covariance is described by a pseudodifferential operator. Our main result is as follows: given the backscattered far field, obtained from a single realization of the random potential [Formula: see text], we uniquely determine the principal symbol of the covariance operator of [Formula: see text]. Especially, for [Formula: see text] this result is obtained for the full nonlinear inverse backscattering problem. Finally, we present a physical scaling regime where the method is of practical importance.

The inverse backscattering for Schrödinger operators for potentials with noncompact support

We consider the inverse backscattering problem for the Schrödinger operator H = −Δ + V on , n ≥ 3, as well as the higher‐order Schrödinger operator ( − Δ)m + V, m = 2,3,…. We show that in some suitable Banach spaces, the map from the potential to the backscattering amplitude is a local diffeomorphism. This kind of problem (for m = 1) was studied by Eskin and Ralston [Comm. Math. Phys., 124(2), 169‐215 (1989)], where they assumed that . In this paper, we replace the assumption on V with certain decay assumption at infinity.

Well-posedness of the Goursat problem and stability for point source inverse backscattering

We show logarithmic stability for the point source inverse backscattering problem under the assumption of angularly controlled potentials. Radial symmetry implies Hölder stability. Importantly, we also show that the point source equation is well-posed and also that the associated characteristic initial value problem, or Goursat problem, is well-posed. These latter results are difficult to find in the literature in the form required by the stability proof.

Inverse backscattering problem for perturbations of biharmonic operator

We consider the inverse backscattering problem for a biharmonic operator with two lower order perturbations in two and three dimensions. The inverse Born approximation is used to recover jumps and singularities of an unknown combination of potentials. Numerical examples are given to illustrate the practical usefulness of the method.

Uniqueness for the inverse backscattering problem for angularly controlled potentials

We consider the problem of recovering a smooth, compactly supported potential on from its backscattering data. We show that if two such potentials have the same backscattering data and the difference of the two potentials has controlled angular derivatives, then the two potentials are identical. In particular, if two potentials differ by a finite linear combination of spherical harmonics with radial coefficients and have the same backscattering data then the two potentials are identical.

The back-scattering problem in three dimensions

In this thesis we study the (inverse) back-scattering problem for the Schr"odinger operator in $R^3$. We introduce the back-scattering transform $B(v)$ of a real-valued potential $vin C_0^infty(R^3)$, and prove that the back-scattering data associated to $v$ determine $B(v)$. Under the assumption that the Schr"odinger operator $H_v=-Delta +v$ has no eigenvectors in $L^2(R^3)$ it is shown that $B(v)$ may be expressed in terms of the wave group $K_v(t)=sin(tsqrt{H_v})/sqrt{H_v}$. We prove also that the mapping $vmapsto B(v)$ is a homeomorphism in a neighbourhood of the origin in the Banach space $X_0^r$, which is the completion of $C_0^infty(R^3;R)$ w.r.t. the norm $fmapstosum_{|a|=1}|d^af|_{L^1}$.

Inverse backscattering problem for the generalized nonlinear Schrödinger operator in two dimensions

The inverse Born approximation is studied for the generalized nonlinear two-dimensional Schrödinger equation where the real-valued unknown potentials αl belong to for some 2 ⩽ p ⩽ ∞ and their Fourier transforms belong to the space for some 1 < s < 2. We prove that the leading-order singularities of the sum of the unknown potentials can be recovered from the inverse Born approximation. We also prove that the approximation is equal to the true function up to sum of two functions: one of them from the Sobolev space Ht while the other being a continuous function.

Linearizing non-linear inverse problems and an application to inverse backscattering

We propose an abstract approach to prove local uniqueness and conditional Hölder stability to non-linear inverse problems by linearization. The main condition is that, in addition to the injectivity of the linearization A, we need a stability estimate for A as well. That condition is satisfied in particular, if A ∗ A is an elliptic pseudo-differential operator. We apply this scheme to show uniqueness and Hölder stability for the inverse backscattering problem for the acoustic equation near a constant sound speed.

Inverse backscattering for the Schrödinger equation in 2D

We study the inverse backscattering problem for the Schrödinger equation in two dimensions. We prove that, for a non-smooth potential in 2D, the main singularities up to 1/2 of the derivative of the potential are contained in the Born approximation (diffraction tomography approximation) constructed from the backscattering data. We measure singularities in the scale of Hilbertian Sobolev spaces.

Inverse backscattering in even dimensions

This paper investigates the inverse backscattering problem for a potential in even dimensions. Our results are obtained through the study of a modified backscattering map. We use the wave equation approach here and extend Melrose and Uhlmann's method [6] to even dimensions.

Inverse backscattering problem for Maxwell's equations

In this paper we consider the inverse backscattering problem for Maxwell's equations in a nonmagnetic inhomogeneous medium, i.e. the magnetic permeability is a fixed constant. We show that the electric permittivity e is uniquely determined by the trace of the backscattering kernel S(s, - θ, 0) for all s∈R, 0∈S 2 provided that it is a priori close to a constant.

Inverse Backscattering Problem for the Acoustic Equation in Even Dimensions

We show that the sound speedc(x) of the acoustic wave equation in any even dimension can be uniquely determined by the backscattering data provided that it is close to a constant. In the three-dimensional case, P. Stefanov and G. Uhlmann (SIAM J. Math. Anal.28,1997, 1191–1204) have proved a similar result. Their method takes advantage of the inversion formula for the Radon transform in odd dimensions being a local operator. This is not true in even dimensions. Moreover, the odd-dimensional Lax and Phillips modified Radon transform fails to work in even dimensions. In this paper, we overcome these difficulties and prove an even-dimensional version of Stefanov and Uhlmann's result.

Stability estimate for an inverse acoustic backscattering problem

We prove a stability estimate for an inverse acoustic backscattering problem. This inverse problem consists of the determination of the sound speed, which is in the principal part of the equation, by the backscattering data. Previously, the stability estimates for inverse problems dealing with the unknown coefficient in the principal part of the equation were of logarithmic or Hölder type. Here we are able to obtain a Lipschitz-type estimate for this inverse backscattering problem.

Inverse Backscattering for the Acoustic Equation

In this paper, we consider the inverse backscattering problem for the acoustic equation. The problem is to determine the sound speed of a medium by measuring the response to sound waves in the backscattering direction, i.e., by measuring the echoes. We prove that one can uniquely identify the sound speed from this information if it is a priori close to the constant sound speed.

A uniqueness result for the inverse back-scattering problem

Let q1(t,x), q2(t,x) belong to C(Rt;W1, infinity (Rx3)), qi(t,x)=0 for mod x mod > rho with some rho >0 and let KiHash be the corresponding generalized scattering kernels, i=1,2. The author proves that if q1>or=q2 and if K1Hash (s',- omega 0; s, omega 0)=K2Hash (s', - omega 0;s, omega 0) for some omega 0 in S2, then q1=q2. As a corollary, the author gets the following result. Let Vi(x) in Linfinity (R3), Vi has compact support and suppose that - Delta +Vi has no bound states, i=1, 2. Let ai be the scattering amplitude related to Vi, i=1, 2. Suppose that V1>or=V2 and for some omega 0 the author has a1(k,- omega 0, omega 0)=a2(k,- omega 0, omega 0) for all k. Then V1=V2.

A comparison between the Born and Rytov approximations for the inverse backscattering problem

We compare the Born and Rytov approximations in solving the inverse acoustic backscattering problem, i.e., determining medium properties from reflections. For the one‐dimensional problem, we show that the Rytov approximation is generally better than the Born approximation in predicting sound speed changes, while both methods have the same error in determining the positions of reflectors. This is shown analytically for simple models and numerically for more general models. The performance of the Rytov approximation is degraded when low‐velocity regions are present in the medium being probed. The accuracy of the inversion depends on the manner in which the sound speed perturbation is linearized. The location of the receiver affects the accuracy of the inversion, and, in the case of the Rytov approximation, best results are obtained when the receiver is at the interface between the known and unknown regions. Furthermore, the Rytov method is less sensitive to the choice of reference sound speed used in the inversion than is the Born approximation.

The inverse backscattering problem in three dimensions

This article is a study of the mapping from a potentialq(x) onR 3 to the backscattering amplitude associated with the Hamiltonian −Δ+q(x). The backscattering amplitude is the restriction of the scattering amplitudea(θ, ω, k), (θ, ω, k)εS 2×S 2×ℝ+, toa(θ,−θ, k). We show that in suitable (complex) Banach spaces the map fromq(x) toa(x/|x|, −x/|x|, |x|) is usually a local diffeomorphism. Hence in contrast to the overdetermined problem of recoveringq from the full scattering amplitude the inverse backscattering problem is well posed.