Some Recent Advances in Nonlinear Inverse Scattering in 2D: Theory and Numerics

We survey our recently published results concerning scattering for where h is a quite general nonlinear analogue of the index of refraction. We will investigate direct scattering problem as well as several inverse scattering problems for this equation. We start by establishing sufficient conditions for the unique solvability of the direct scattering problem. Asymptotic behaviour of the scattering solutions is shown to give rise to scattering amplitude. Inverse problems are formulated as follows: extract information about the nonlinearity h from the knowledge of the scattering amplitude. At this point, one must specify more carefully the scattering data sets. We concentrate our attention to full data, backscattering data, fixed angle data and fixed energy data. The latter three data sets are collectively called limited data. For each data set, we define the inverse Born approximation and state theorems, which provide the recovery of main singularities of the function h 0 ( x ) = h ( x , 1). The key idea here is to show that the difference between the Born approximation and h 0 ( x ) is less singular than h 0 (x). For practical applications, one must be able to compute the inverse Born approxima‐ tions numerically. To this end, we proceed as follows. We formulate the computation as a solution of an underdetermined linear system. Due to the ill-posed nature of the system, regularization methods are used. Examples are given to illustrate the effectiveness of the method.


Introduction
We deal with the generalized nonlinear Schrödinger equation: where E denotes the electromagnetic field in two-dimensional case, ∆ is the two-dimensional Laplacian and h describes in a general form the nonlinear contribution to the index of refraction. Considering harmonic time dependence E(x, t) = e − iωt u(x) with frequency ω > 0, we obtain the steady-state nonlinear equation with fixed energy: (1) where k 2 = ω and fixed, and u denotes the complex amplitude of the field. Concerning the nonlinearity h(x, s), we pose the basic assumptions.
where c ρ and c ρ ′ are constants and (2) Here, L σ p (ℝ 2 ) denotes the weighted Lebesgue space with the norm.
The main practical example (it can be considered as the motivation of this research) of such type equations (1) is the equation of the form: with real number k, complex valued function q 1 (x) ∈ L 2 and real-valued function q 2 (x) ∈ L 2 , and parameter r ≥ 0. A particular nonlinearity in (3) of cubic type (r = 0) can be met in the context of a Kerr-like nonlinear dielectric film, while the case when r > 0 corresponds to the saturation model (see [1][2][3][4]).
We consider the inverse scattering problems for (1). For these purposes, we are interested of the scattering solutions to (1), i.e. solutions of the form (4) where , the unit sphere in ℝ 2 , u 0 (x, k, θ) = e ik(x,θ) is the incident wave and u sc (x, k, θ) is the scattered wave. The scattered wave must satisfy the Sommerfeld radiation condition at infinity: for fixed k > 0 and uniformly in . In that case, these solutions are the unique solutions of the Lippmann-Schwinger equation. (5) where G k + is the outgoing fundamental solution of the operator − ∆ − k 2 , i.e. the kernel of the integral operator (−∆ − k 2 − i0) − 1 . It is equal to where H 0 (1) denotes the Hankel function of order zero and first kind.
The following main results concerning the direct scattering problem were proved in [5].
Under the basic assumptions and (2) for h, it is proved that for any ρ > 1 there is k 0 > 0 such that (ii) the backscattering data: (iii) the fixed angle data: and (iv) the fixed energy data: We use the following notations for the direct and inverse Fourier transforms: where (ξ, x) denotes the inner product in ℝ 2 , i.e. (ξ, x) = ξ 1 x 1 + ξ 2 x 2 . By C > 0, we denote a generic constant that may change from one step to another. By H t (ℝ 2 ) = W 2 t (ℝ 2 ), t ∈ ℝ we denote the standard L 2 based Sobolev space. A weighted Sobolev space W p,σ 1 (ℝ 2 ) is defined here by The following notation for the characteristic function is used:

Inverse scattering problems
The direct scattering theory described above can also be reversed. The inverse scattering theory treats the function h as unknown and attempts to recover it from the knowledge of the scattering amplitude A for different data. Usually, the model in (1) is probed with one or more incident plane waves u 0 and the resulting scattered waves are measured at a distance. This gives rise to several different scattering data sets which can be used to recover the unknowns.
The inverse backscattering problem for (1) was treated in [6]. Also for (1), the recovery of unknown function h is possible from the full scattering data. In addition to the two-dimensional studies mentioned above, certain particular nonlinear cases of (1) have been investigated in other dimensions too. In one-space dimension, we refer to [7] and the references therein. In higher dimensions n ≥ 3 we are only aware of [8,9]. Similar problems with formally more general equation but with bounded h are considered in [10] and [11].
Our point of view is that the nonlinearity may contain local singularities in the space coordinate x, and therefore we work in the frame of weighted Lebesgue spaces. These local singularities can be recovered from the scattering amplitude using the method of Born approximation. As a unifying result, we obtain mathematically more general results that have far wider applicability in physical experiments.

Let us set
In the subsections that follow we consider the inverse problems of recovering information about h 0 from the knowledge of full data D, backscattering data D B , fixed angle data D A and fixed energy data D E .

Full scattering data
The inverse problem with full data D was investigated in [5]. Here we summarize the main results without proofs.
Theorem 1 (Saito's formula). Under the basic assumptions and (7) for the function h we have, where the limit is valid in the sense of distributions for 4/3 < p ≤ 2 and pointwise (even uniformly) for 2 < p ≤ ∞.
Corollary 1 (Uniqueness). Let σ be as in (7). Consider the scattering problems for two sets of potentials h and . If the scattering amplitudes coincide for some sequence k j → ∞ and for all θ′, , then holds in the sense of distributions for 4/3 < p ≤ ∞.
Corollary 2 (Representation formula). Let σ be as in (7). Then the representation holds in the sense of distributions for 4/3 < p ≤ ∞.

Remark 1.
In addition to providing the above results, Saito's formula might be applied numerically too. It can be written as a convolution equation: where the function f can be computed from the full scattering data in principle. A numerical inversion of this equation would yield a full recovery of h 0 but this is an open problem as far as we know. What is more, unlike the representation formula above it holds pointwise in the important case of bounded (p = ∞) nonlinearities.
We assume that the function x ↦ h(x, s) is real-valued and recall that For reasons of purely technical nature we define the scattering solutions u(x, k, θ) for negative k as These are the unique solutions of the integral equation: provided that h is real-valued. This allows us to extend A to negative k ≤ − k 0 by We also put A(k, θ′, θ) = 0 for |k| ≤ k 0 . Splitting we have that or where F denotes the Fourier transform (8). Using the basic assumptions for the function h and (6), we can easily obtain that We have used here the fact that the basic assumptions for the function h guarantee that the functions α and β both belong to Hence, for k large, we have approximately that (10) These considerations and real valuedness of h suggest and justify the following definition: We define the inverse Born approximation q B via the equality (11) which is understood in the sense of tempered distributions. In order to recover main singularities of h 0 from q B , we must study their difference and show that it is locally less singular than h 0 . To this end, we have the following main result from [5].

Remark 2.
Theorem 2 means that, for 4/3 < p < ∞, the main singularities of h 0 can be recovered from the inverse scattering Born approximation q B with full data D. In the case of p = ∞, we have no singularities but may have finite jumps. Under such circumstances, we record the following special case.
Corollary 3. If a piecewise smooth compactly supported function h 0 contains a jump over a smooth curve, then the curve and the height function of the jump are uniquely determined by the full scattering data. Especially, for the function h 0 being the characteristic function of a smooth bounded domain, this domain is uniquely determined by the full scattering data.
Concluding, in this part of the work, the uniqueness of the direct problem for the nonlinearities h satisfying the appropriate properties was proved. These properties allow local singularities and do not require compact support, but rather some sufficient decay at infinity. Under similar properties, we were also able to establish the asymptotic behaviour of scattering solutions, which gives us the scattering data so we can investigate the inverse scattering problems. Note that both results were proved without assuming smallness of the norm of the nonlinearity as is necessary in dimensions three and higher.
What can we regard as the main result of this section is the Saito's formula since it implies a uniqueness result and a representation formula for our unknown function h 0 . In addition, we managed to extract more information about the nonlinearity by applying the method of Born approximation. More precisely, the main singularities (or jumps over smooth curve) of h 0 can be recovered from the Born approximation q B which corresponds to the full scattering data D.

Backscattering and fixed angle data
In this section, we consider backscattering data D B and fixed angle scattering data D A following [12]. Using (10) we introduce the inverse backscattering and inverse fixed angle scattering Born approximations q B b and q B θ 0 as follows: where θ 0 is fixed.
Furthermore, we assume in addition that the nonlinearity h possesses the Taylor expansion: where uniformly in s ∈ (0, s 0 ), s 0 > 0 and with η 1 , η 2 ∈ L σ p ( R 2 ) , where σ as in (7). From this we obtain the expansion: and η ∈ L σ p (ℝ 2 ) with the same σ as above.
Again, the main result for the recovery of main singularities is formulated as the following theorem.
Theorem 3. Let σ be as in (7)  Let us sketch the main ideas in the proof of Theorem 3. Using the definition, we may expand the difference in several terms as In straightforward manner one sees that q ∞ θ 0 ∈ C ∞ (ℝ 2 ) and q 2 The first nonlinear term q 1 θ 0 cannot be analyzed directly from its definition. Instead, we first proved the representation where F 4 -1 denotes the four-dimensional inverse Fourier transform. This formula might have independent interest too, but primarily it allows us to prove the following regularity: the term q 1 θ 0 belongs to the space If we combine all these steps, we obtain Theorem 3 for fixed angle scattering.
The inverse backscattering Born approximation is treated similarly. Namely, we write The latter four terms have exactly the same regularity results as their counterparts in fixed angle scattering. For the first nonlinear term, the representation is now The additional assumption in Theorem 3 implies that h 0 ∈ L 1 s ′ (ℝ 2 ).This explains why we restrict s′ < p. By Sobolev embedding, we obtain with some positive ∈ < min{1/p, 1 − 2/p}. Hence, h 0 is locally more singular than either of these differences. That's why both Born approximations recover the main singularities of h 0 . On the other hand, we may perform the comparison also in the scale of Sobolev spaces. Indeed, if h 0 ∈ H comp r (ℝ 2 ) with some 0 < r < 1, then for any t < 2r if 0 < r ≤ 1/3 and for any t < (1 + r)/2 if 1/3 < r < 1. In both cases this t can be made bigger than r. It means that we can reconstruct all singularities from Sobolev space H r , 0 < r < 1 from data D B and D A by the method of Born approximation. Concluding, in this section we proved that all singularities and jumps (in the absence of a uniqueness theorem) of the nonlinearity h can be recovered from the inverse scattering Born approximation corresponding to fixed angle scattering and backscattering data D A and D B , respectively. No assumptions about the smallness of the norm of nonlinearity h were used as it were in previous publications even in the linear case.

Fixed energy data
The two-dimensional fixed energy problemx1 was a long-standing open problem. In the case of linear Schrödinger operator, the first uniqueness and reconstruction algorithm was proved by Nachman [13] via ∂ -methods for potentials of conductivity type. Sun and Uhlmann [14] proved uniqueness for potentials satisfying nearness conditions to each other. The question of global uniqueness for the linear Schrödinger equation with fixed energy was settled only in 2008 by Bukhgeim [15] for compactly supported potentials from L p , p > 2. This result has recently been improved and extended to related inverse problems (see for example [16] and [17]). Note that Grinevich and Novikov [18] proved that in two dimensions there are nonzero real potentials of the Schwartz class with zero scattering amplitude at fixed positive energy.
Thus, the compactness of the supports of the potentials is very natural condition in our considerations.
The results of this section are proved in [19] and they slightly generalize the linear case to a special type of nonlinearity. It turned out that (as we can see in this section) inverse fixed energy scattering problem is much more difficult than the others.
In fixed energy scattering problem, instead of the scattering solutions (4) we need the so-called complex geometrical optics solutions. Complex geometrical optics (CGO) solutions or exponentially growing solutions of the form: (14) with R decaying at infinity for |z| large for the homogeneous Schrödinger equation (15) are obtained as follows. Substituting (14) into (15) yields It means that using the Faddeev Green's function as the fundamental solution of the differential operator we see that the function R solves the integral equation (16) It remains to establish unique solvability of this equation. To this end we again use iterations in the sense of next theorem.
Turning now to the inverse fixed energy scattering problem, we define the scattering transform by and T h (ξ) = 0 for |ξ| < 2 C 0 . Here z = . What is more, we have the uniform limit T h (ξ) = lim j→∞ T h , j (ξ), where We point out that the scattering transform is somehow an auxiliary object (see D E in Introduction). But it is connected to the scattering amplitude as follows. It is well known that the scattering amplitude at fixed energy uniquely determines the Dirichlet-to-Neumann map Λ h -k 0 2 which in turn uniquely determines the scattering transform (see the details, for example, in [19] and [20]). Recall that Λ h f = ∂ ν u, where u satisfies the Dirichlet problem: Here, Ω is a domain with boundary ∂Ω and outward unit normal vector ν.
Next, we define the inverse fixed energy Born approximation by (17) In contrast to the preceding inverse problems, we now set the unknown function to be (18) In linear case h 0 is the actual potential appearing in the Schrödinger equation, but otherwise the connection to physical scatterers is not known to us.
We assume that the nonlinearity h admits the Taylor expansion where |∂ s h(x, e 0 (1 + s))| s = 0 | ≤ β 1 (x) and O(β 1 (x)s 2 ) with β 1 (x) ∈ L 2 (Ω) and with small s in the neighbourhood of zero and where O is uniform in x ∈ Ω and such s.

Suppose in addition that the nonlinearity h satisfies the asymptotic expansions
where . Then we have the following main result concerning the recovery of singularities of h 0 (x) defined by (18).

Theorem 5.
Under the foregoing conditions for the potential function h, it is true that for any t < 1 modulo C ∞ (ℝ 2 ) -functions.

Remark 3.
The embedding theorem for Sobolev spaces says that the difference q B f -h 0 belongs to L q (R 2 ) for any q < ∞ modulo C ∞ (R 2 ) functions. It means that all singularities from L loc p ( R 2 ) , p < ∞ of unknown function h 0 can be obtained exactly by the Born approximation which corresponds to the inverse scattering problem with fixed positive energy.
We note that under fixed energy data we have some additional assumptions on h. This limits the applicability of the main result to, for example, saturation type nonlinearities. In particular, cubic nonlinearity is excluded from these considerations. Moreover, the unknown function h 0 has no direct connection to original scatterers in nonlinear cases.

Numerical examples
Here we discuss the numerical computation of the Born approximation first proposed in [21] for backscattering and fixed angle data. We assume that the scatterer h 0 (x) is supported in the rectangle R 0 = -1, Here Ω, Ω 1 , Ω 2 are the shifted ellipses rotated in angle Θ theta (counter-clockwise) detailed in Table 1 and shown in Figure 1.  Consider the Born approximation for full data (11) in the form (19) To discretize the unknown function f we divide the rectangle R 0 into N = n × n disjoint subrectangles r j of equal size, i.e.
Then, we represent f on R 0 in piecewise constant form: where f j 's are the unknown values on r j 's. Substitution into (19) yields Evaluating this at some points k = k s , θ ′ = θ t ′ and θ = θ p leads us to If we denote M = N 1 N 2 N 3 , we may form one linearized index l = l(s, t, p), l = 1, …, M and write the latter equation as (20) where E ( j, l) = ∫ r j e ik s (θ t ′ -θ p , y) dy are easily evaluated and g l = A(k s , θ t ′ , θ p ) needs to be computed.
The computation of g l is carried out using numerical integration, see [21] for details. In matrix form (20) is clearly Ef = g.
For backscattering data and fixed angle data the system (20) is modified accordingly. We note that the system Ef = g does not depend on the scatterer but only on data type and measurement setup.
The fixed energy case differs from the first three data sets. In fixed energy inversion we approximate the scattering transform as is evaluated by numerical integration, see [19,21] for details. Then the inverse Born approximation (17) is computed similarly to (19).
We use N = 2500 and N 1 = 12 for each data set. For full data we use N 2 = N 3 = 6. For backscattering and fixed angle data, we use N 2 = 24. For fixed energy scattering we use m = 40 and s = 6.
In all four cases we obtain the linear system Ef = g whose coefficient matrix E is of size M × N. The data g is corrupted with zero mean Gaussian noise with standard deviation σ = 0.01 max |g|.
The size M as well as the ranks r(E) and (approximate) condition numbers log 10 κ(E) measuring the ill-posedness of the linear system EF = g are shown in Table 2 Table 2. Matrix sizes, ranks and condition numbers As the linear system is rank-deficient and ill-posed, we use regularization method to solve it.
More precisely, we apply the total variation method (TV) which is defined as where the matrix L implements the differences between neighbouring elements in horizontal and vertical directions (for details, see [21]). As in [21] we formulate this minimization problem as a quadratic problem in standard form for more efficient solution. As the regularization parameter we use δ = 2 ⋅ 10 − 3 for D E and δ = 10 − 3 otherwise.
All computations are carried out in a UNIX system with 512 GB of memory and 40 logical CPU cores, each running at 2.8 GHz. The software platform is MATLAB R2015b. We have used 12 workers in parallel in computing the right-hand side g. We note that a desktop PC with 8 logical cores running at 3.4 GHz and 16 GB of memory is also capable of our computations, but with 7 workers it is considerably slower in computing g.
The computational times to both form and solve the linear system are shown in Table 3. We point out that the right-hand side g contains synthetic data and actual physical measurements may take longer or shorter time. The contour plots of scatterers h 0 (x) and their TV reconstructions via Born approximation from full data, backscattering data and fixed angle data (with θ 0 = (1, 0)) are shown in Figures 2-5 for all examples. For fixed energy scattering we only consider the linear Example 1, since otherwise we do not have direct comparison to a scatterer. The TV reconstruction is shown in Figure 6. In each figure solid white line indicates the true geometry of the scatterer.
We see that the location of the scatterer is located quite nicely in all cases. The shape of the scatterer is best seen from full data and backscattering data. By computing the Born approximation from full angle data, we close an open problem from [5].